We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} with Betti number b1\documentclass[12pt]{minimal}
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\begin{document}$$b_1$$\end{document}, the order of vanishing of the Ruelle zeta function at zero equals 4-b1\documentclass[12pt]{minimal}
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\begin{document}$$4-b_1$$\end{document}, while in the hyperbolic case it is equal to 4-2b1\documentclass[12pt]{minimal}
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\begin{document}$$4-2b_1$$\end{document}. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document} with harmonic 1-forms on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}.
Massachusetts Institute of Technology (MIT)publisher-imprint-nameSpringervolume-issue-count3issue-article-count7issue-toc-levels0issue-pricelist-year2022issue-copyright-holderSpringer-Verlag GmbH Germany, part of Springer Natureissue-copyright-year2022article-contains-esmNoarticle-numbering-styleContentOnlyarticle-registration-date-year2022article-registration-date-month2article-registration-date-day9article-toc-levels0toc-levels0volume-typeRegularjournal-productNonStandardArchiveJournalnumbering-styleContentOnlyarticle-grants-typeOpenChoicemetadata-grantOpenAccessabstract-grantOpenAccessbodypdf-grantOpenAccessbodyhtml-grantOpenAccessbibliography-grantOpenAccessesm-grantOpenAccessonline-firstfalsepdf-file-referenceBodyRef/PDF/222_2022_Article_1108.pdfpdf-typeTypesettarget-typeOnlinePDFissue-online-date-year2022issue-online-date-month6issue-online-date-day10issue-print-date-year2022issue-print-date-month6issue-print-date-day10issue-typeRegulararticle-typeOriginalPaperjournal-subject-primaryMathematicsjournal-subject-secondaryMathematics, generaljournal-subject-collectionMathematics and Statisticsopen-accesstrue
B. Delarue: formerly known as Benjamin Küster.
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Let (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document} be a compact connected oriented 3-dimensional Riemannian manifold of negative sectional curvature. The Ruelle zeta functionζR(λ)=∏γ(1-eiλTγ),Imλ≫1\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \zeta _{\mathrm R}(\lambda )=\prod _{\gamma }\big (1-e^{i\lambda T_{\gamma }}\big ),\quad {{\,\mathrm{Im}\,}}\lambda \gg 1 \end{aligned}$$\end{document}is a converging product for Imλ\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Im}\,}}\lambda $$\end{document} large enough and continues meromorphically to λ∈C\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \in {\mathbb {C}}$$\end{document} as proved by Giulietti–Liverani–Pollicott [34] and Dyatlov–Zworski [20]. Here the product is taken over all primitive closed geodesics γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} on (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document} and Tγ\documentclass[12pt]{minimal}
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\begin{document}$$T_{\gamma }$$\end{document} is the length of γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document}.
In this paper we study the order of vanishing of ζR\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _{\mathrm R}$$\end{document} at λ=0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda =0$$\end{document}, defined as the unique integer mR(0)\documentclass[12pt]{minimal}
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\begin{document}$$m_{\mathrm R}(0)$$\end{document} such that λ-mR(0)ζR(λ)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda ^{-m_{\mathrm R}(0)}\zeta _{\mathrm R}(\lambda )$$\end{document} is holomorphic and nonzero at 0. Our main result is
Theorem 1
Let (Σ,gH)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g_H)$$\end{document} be a compact connected oriented hyperbolic 3-manifold and b1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$b_1(\Sigma )$$\end{document} be the first Betti number of Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}. Then:
For (Σ,gH)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g_H)$$\end{document} we have mR(0)=4-2b1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$m_{\mathrm R}(0)=4-2b_1(\Sigma )$$\end{document}.
There exists an open and dense set O⊂C∞(Σ;R)\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {O}}\subset C^\infty (\Sigma ;{\mathbb {R}})$$\end{document} such that for any b∈O\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {b}}\in {\mathscr {O}}$$\end{document}, there exists ε>0\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon >0$$\end{document} such that for any τ∈(-ε,ε)\{0}\documentclass[12pt]{minimal}
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\begin{document}$$\tau \in (-\varepsilon ,\varepsilon ){\setminus } \{0\}$$\end{document} and gτ:=e-2τbgH\documentclass[12pt]{minimal}
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\begin{document}$$g_\tau :=e^{-2\tau {\mathbf {b}}}g_H$$\end{document}, the manifold (Σ,gτ)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g_\tau )$$\end{document} has mR(0)=4-b1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$m_{\mathrm R}(0)=4-b_1(\Sigma )$$\end{document}.
Part 1 of Theorem 1 was proved by Fried [25, Theorem 3] using the Selberg trace formula. The novelty is part 2, which says that for generic conformal perturbations of the hyperbolic metric the order of vanishing ofζR\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _{\mathrm R}$$\end{document}equals4-b1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$4-b_1(\Sigma )$$\end{document}. In particular, when b1(Σ)>0\documentclass[12pt]{minimal}
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\begin{document}$$b_1(\Sigma )>0$$\end{document} (fulfilled in many cases, in particular for mapping tori over pseudo-Anosov maps [24, Theorem 13.4]), mR(0)\documentclass[12pt]{minimal}
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\begin{document}$$m_{\mathrm R}(0)$$\end{document} is not topologically invariant. Theorem 1 is the first result on instability of the order of vanishing of ζR\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _{\mathrm R}$$\end{document} at 0 for Riemannian metrics. It is in contrast to the 2-dimensional case, where Dyatlov–Zworski [21] showed that mR(0)=b1(Σ)-2\documentclass[12pt]{minimal}
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\begin{document}$$m_{\mathrm R}(0)=b_1(\Sigma )-2$$\end{document} for any compact connected oriented negatively curved surface (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document}, and is complementary to a recent breakthrough on the (acyclic) Fried conjecture by Dang–Guillarmou–Rivière–Shen [16], see §1.3 below.
A result similar to Theorem 1 holds for contact perturbations of SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document}, see Theorem 4 in §4.
Outline of the proof
We now outline the proof of Theorem 1. We use the microlocal approach to Pollicott–Ruelle resonances and dynamical zeta functions, which we review here – see §2 for details and §1.3 for a historical overview. Let M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$M=S\Sigma $$\end{document} be the sphere bundle of (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document} and X∈C∞(M;TM)\documentclass[12pt]{minimal}
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\begin{document}$$X\in C^\infty (M;TM)$$\end{document} be the generator of the geodesic flow. The geodesic flow is a contact flow, i.e. there exists a 1-form α∈C∞(M;T∗M)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \in C^\infty (M;T^*M)$$\end{document} such that ιXα=1\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X\alpha =1$$\end{document}, ιXdα=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _Xd\alpha =0$$\end{document}, and α∧dα∧dα\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \wedge d\alpha \wedge d\alpha $$\end{document} is a nonvanishing volume form. When g has negative curvature, the geodesic flow is Anosov, i.e. the tangent spaces TρM\documentclass[12pt]{minimal}
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\begin{document}$$T_\rho M$$\end{document} decompose into a direct sum of the flow, unstable, and stable subspaces. Denote by Eu∗\documentclass[12pt]{minimal}
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\begin{document}$$E_u^*$$\end{document}, Es∗\documentclass[12pt]{minimal}
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\begin{document}$$E_s^*$$\end{document} the dual unstable/stable subbundles of the cotangent bundle T∗M\documentclass[12pt]{minimal}
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\begin{document}$$T^* M$$\end{document}, that is, Eu∗\documentclass[12pt]{minimal}
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\begin{document}$$E_u^*$$\end{document}, Es∗\documentclass[12pt]{minimal}
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\begin{document}$$E_s^*$$\end{document} are the annihilators of unstable/stable plus flow directions; these define closed conic subsets of T∗M\documentclass[12pt]{minimal}
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\begin{document}$$T^*M$$\end{document}.
Define the spaces of resonantk-forms at 0Res0k:={u∈D′(M;Ωk)∣ιXu=0,LXu=0,WF(u)⊂Eu∗}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{Res}\,}}^k_0:=\{u\in {\mathcal {D}}'(M;\Omega ^k)\mid \iota _X u=0,\ {\mathcal {L}}_X u=0,\ {{\,\mathrm{WF}\,}}(u)\subset E_u^*\}. \end{aligned}$$\end{document}Here Ωk\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^k$$\end{document} is the (complexified) bundle of k-forms, LX=dιX+ιXd\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X=d\iota _X+\iota _Xd$$\end{document} is the Lie derivative with respect to X, and for any distribution u∈D′(M;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {D}}'(M;\Omega ^k)$$\end{document} we denote by WF(u)⊂T∗M\0\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{WF}\,}}(u)\subset T^*M{\setminus } 0$$\end{document} the wavefront set of u, see for instance [38, Chapter 8]. The wavefront set condition makes Res0k\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^k_0$$\end{document} into a finite dimensional space, which is a consequence of the interpretation of Res0k\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}_0^k$$\end{document} as the eigenspace at 0 of the operator Pk,0:=-iLX\documentclass[12pt]{minimal}
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\begin{document}$$P_{k,0}:=-i{\mathcal {L}}_X$$\end{document} acting on certain anisotropic Sobolev spaces tailored to the flow (see [29, Theorem 1.7] and [21, Lemma 2.2]). We similarly define the spaces of generalized resonantk-forms at 0Res0k,ℓ:={u∈D′(M;Ωk)∣ιXu=0,LXℓu=0,WF(u)⊂Eu∗},Res0k,∞:=⋃ℓ≥1Res0k,ℓ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&{{\,\mathrm{Res}\,}}^{k,\ell }_0:=\{u\in {\mathcal {D}}'(M;\Omega ^k)\mid \iota _X u=0,\ {\mathcal {L}}_X^\ell u=0,\ {{\,\mathrm{WF}\,}}(u)\subset E_u^*\},\nonumber \\&{{\,\mathrm{Res}\,}}^{k,\infty }_0:=\bigcup _{\ell \ge 1} {{\,\mathrm{Res}\,}}^{k,\ell }_0. \end{aligned}$$\end{document}The semisimplicity condition for k-forms states that Res0k,∞=Res0k\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\infty }_0={{\,\mathrm{Res}\,}}^k_0$$\end{document}, which means that the operator Pk,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{k,0}$$\end{document} has no nontrivial Jordan blocks at 0. We also have the dual spaces of generalized coresonantk-forms at 0, replacing Eu∗\documentclass[12pt]{minimal}
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\begin{document}$$E_u^*$$\end{document} with Es∗\documentclass[12pt]{minimal}
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\begin{document}$$E_s^*$$\end{document} in the wavefront set condition:Res0∗k,ℓ:={u∗∈D′(M;Ωk)∣ιXu∗=0,LXℓu∗=0,WF(u)⊂Es∗}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{Res}\,}}^{k,\ell }_{0*}:=\{u_*\in {\mathcal {D}}'(M;\Omega ^k)\mid \iota _X u_*=0,\ {\mathcal {L}}_X^\ell u_*=0,\ {{\,\mathrm{WF}\,}}(u)\subset E_s^*\}. \end{aligned}$$\end{document}Since Eu∗∩Es∗={0}\documentclass[12pt]{minimal}
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\begin{document}$$E_u^*\cap E_s^*=\{0\}$$\end{document}, wavefront set calculus makes it possible to define u∧u∗\documentclass[12pt]{minimal}
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\begin{document}$$u\wedge u_*$$\end{document} as a distributional differential form as long as WF(u)⊂Eu∗\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{WF}\,}}(u)\subset E_u^*$$\end{document}, WF(u∗)⊂Es∗\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{WF}\,}}(u_*)\subset E_s^*$$\end{document}.
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\begin{document}$$\begin{aligned} m_{\mathrm R}(0)=\sum _{k=0}^4 (-1)^k\dim {{\,\mathrm{Res}\,}}^{k,\infty }_0. \end{aligned}$$\end{document}Thus the problem reduces to understanding the spaces Res0k,∞\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\infty }_0$$\end{document} for k=0,1,2,3,4\documentclass[12pt]{minimal}
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\begin{document}$$k=0,1,2,3,4$$\end{document}. The proof of Theorem 1 computes their dimensions, listed in the table below, from which the formulas for mR(0)\documentclass[12pt]{minimal}
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\begin{document}$$m_{\mathrm R}(0)$$\end{document} follow immediately. See Theorem 2 in §3 for the hyperbolic case and Theorem 3 in §4, as well as §4.4, for the case of generic perturbations.
Note that the semisimplicity condition holds for k=0,1,3,4\documentclass[12pt]{minimal}
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\begin{document}$$k=0,1,3,4$$\end{document} in both the hyperbolic case and for generic perturbations. However, semisimplicity fails for k=2\documentclass[12pt]{minimal}
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\begin{document}$$k=2$$\end{document} in the hyperbolic case (assuming b1(Σ)>0\documentclass[12pt]{minimal}
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\begin{document}$$b_1(\Sigma )>0$$\end{document}), and it is restored for generic perturbations. Also, since b2(M)=b1(Σ)+1\documentclass[12pt]{minimal}
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\begin{document}$$b_2(M) = b_1(\Sigma ) + 1$$\end{document} (see (2.28)), we may interpret the dimension of Res02\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}_0^2$$\end{document} in the perturbed case as the ‘topological part’ coming from the bijection with the de Rham cohomology group H2(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$H^2(M;C)$$\end{document} and the extra invariant form dα\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha $$\end{document}.
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\begin{document}$$k=0,4$$\end{document} of the above table are well-known: the semisimplicity condition holds and Res00\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^0_0$$\end{document}, Res04\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^4_0$$\end{document} are spanned by 1, dα∧dα\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge d\alpha $$\end{document}, see Lemma 2.4. One can also see that the map u↦dα∧u\documentclass[12pt]{minimal}
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\begin{document}$$u\mapsto d\alpha \wedge u$$\end{document} gives an isomorphism from Res01,ℓ\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{1,\ell }_0$$\end{document} to Res03,ℓ\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{3,\ell }_0$$\end{document}. Thus it remains to understand the spaces Res0k,∞\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\infty }_0$$\end{document} for k=1,2\documentclass[12pt]{minimal}
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\begin{document}$$k=1,2$$\end{document} and this is where the situation gets more complicated.
The spaces Res0k∩kerd\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^k_0\cap \ker d$$\end{document} of resonant states that are closed forms play a distinguished role in our argument. Similarly to [21] we introduce linear maps πk\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k$$\end{document} from Res0k∩kerd\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^k_0\cap \ker d$$\end{document} to the de Rham cohomology groups Hk(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$H^k(M;{\mathbb {C}})$$\end{document}, see (2.61). We show that the map π1\documentclass[12pt]{minimal}
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\begin{document}$$\pi _1$$\end{document} is an isomorphism, see Lemma 2.8. This gives the dimension of the space of closed forms in Res01\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_0$$\end{document}: since b1(M)=b1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$b_1(M)=b_1(\Sigma )$$\end{document},dim(Res01∩kerd)=b1(Σ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \dim ({{\,\mathrm{Res}\,}}^1_0\cap \ker d)=b_1(\Sigma ). \end{aligned}$$\end{document}In the hyperbolic case, the other b1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$b_1(\Sigma )$$\end{document}-dimensional space of non-closed forms in Res01\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_0$$\end{document} is obtained by rotating the closed forms by π/2\documentclass[12pt]{minimal}
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\begin{document}$$\pi /2$$\end{document} in the dual unstable space, see §3.3. This rotation commutes with the geodesic flow because the geodesic flow is conformal on the stable/unstable spaces, see (3.7). This additional symmetry, which is only present in the hyperbolic case, is related to the presence of a closed 2-form ψ∈C∞(M;Ω2)\documentclass[12pt]{minimal}
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\begin{document}$$\psi \in C^\infty (M;\Omega ^2)$$\end{document} which is invariant under the geodesic flow and is not a multiple of dα\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha $$\end{document}, see §3.2.3. The space Res02\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^2_0$$\end{document} is spanned by dα\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha $$\end{document}, ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document}, and the differentials du where u are the non-closed forms in Res01\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_0$$\end{document}, see §3.4. We also show in §3.4 that each du∈d(Res01)\documentclass[12pt]{minimal}
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\begin{document}$$du\in d({{\,\mathrm{Res}\,}}^1_0)$$\end{document} lies in the range of LX\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X$$\end{document}, producing b1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$b_1(\Sigma )$$\end{document} Jordan blocks for the operator P2,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{2,0}$$\end{document}.
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\begin{document}$$g_\tau =e^{-2\tau {\mathbf {b}}}g_H$$\end{document}, we use first variation techniques and make the following nondegeneracy assumption (see §4.4): for the spaces Res01,Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_0,{{\,\mathrm{Res}\,}}^1_{0*}$$\end{document} and the contact form α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} defined using the hyperbolic metric gH\documentclass[12pt]{minimal}
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\begin{document}$$g_H$$\end{document}, and denoting by πΣ:M=SΣ→Σ\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Sigma :M=S\Sigma \rightarrow \Sigma $$\end{document} the projection map, we assume that(du,du∗)↦∫M(πΣ∗b)α∧du∧du∗defines a nondegenerate pairingond(Res01)×d(Res0∗1).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (du,du_*)\mapsto & {} \int _M (\pi _\Sigma ^*{\mathbf {b}})\alpha \wedge du\wedge du_*\quad \text {defines a nondegenerate pairing}\nonumber \\&\quad \text {on}\quad d({{\,\mathrm{Res}\,}}^1_0)\times d({{\,\mathrm{Res}\,}}^1_{0*}). \end{aligned}$$\end{document}Under the assumption (1.3), we show that the non-closed 1-forms in Res01\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_0$$\end{document} move away once τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} becomes nonzero (i.e. they turn into generalized resonant states for nonzero Pollicott–Ruelle resonances), see §4.1. Thus for 0<|τ|<ε\documentclass[12pt]{minimal}
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\begin{document}$$0<|\tau |<\varepsilon $$\end{document} all the resonant 1-forms are closed and we get dimRes01=b1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$\dim {{\,\mathrm{Res}\,}}^1_0=b_1(\Sigma )$$\end{document}. Further analysis shows that semisimplicity is restored for k=2\documentclass[12pt]{minimal}
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\begin{document}$$k=2$$\end{document} and dimRes02=b1(Σ)+2\documentclass[12pt]{minimal}
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\begin{document}$$\dim {{\,\mathrm{Res}\,}}^2_0=b_1(\Sigma )+2$$\end{document}.
It remains to show that the nondegeneracy assumption (1.3) holds for a generic choice of the conformal factor b∈C∞(Σ;R)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {b}}\in C^\infty (\Sigma ;{\mathbb {R}})$$\end{document}. The difficulty here is that b\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {b}}$$\end{document} can only depend on the point in Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} and not on elements of SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document} which is where α∧du∧du∗\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \wedge du\wedge du_*$$\end{document} lives. We reduce (1.3) to the following statement on nontriviality of pushforwards (see Proposition 4.10): for each real-valued resonant 1-form for the hyperbolic metric u∈Res01\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^1_0$$\end{document} we havedu≠0⟹πΣ∗(α∧du∧J∗(du))≠0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} du\ne 0\quad \Longrightarrow \quad \pi _{\Sigma *}^{}(\alpha \wedge du\wedge {\mathcal {J}}^*(du))\ne 0. \end{aligned}$$\end{document}Here J:(x,v)↦(x,-v)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {J}}:(x,v)\mapsto (x,-v)$$\end{document} is the antipodal map on M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$M=S\Sigma $$\end{document} and πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}$$\end{document} is the pushforward of differential k-forms on M to (k-2)\documentclass[12pt]{minimal}
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\begin{document}$$(k-2)$$\end{document}-forms on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} obtained by integrating along the fibers, see (2.19).
The statement (1.4) concerns resonant 1-forms for the hyperbolic metric g=gH\documentclass[12pt]{minimal}
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\begin{document}$$g=g_H$$\end{document}, which are relatively well-understood. However, it is complicated by the fact that πΣ∗(α∧du∧J∗(du))\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}(\alpha \wedge du\wedge {\mathcal {J}}^*(du))$$\end{document} is merely a distribution, so we cannot hope to show it is nonzero by evaluating its value at some point. Instead we pair it with functions in C∞(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$C^\infty (\Sigma )$$\end{document} which have to be chosen carefully so that we can compute the pairing. More precisely, we prove the following identity (Theorem 5 in §5):Q4F=-16Δg|σ|g2whereπΣ∗(α∧du∧J∗(du))=Fdvolg.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Q_4F=-\textstyle {1\over 6}\Delta _g|\sigma |_g^2\quad \text {where}\quad \pi _{\Sigma *}^{}(\alpha \wedge du\wedge {\mathcal {J}}^*(du))=F\,d{{\,\mathrm{vol}\,}}_g. \end{aligned}$$\end{document}Here dvolg\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document}, Δg\documentclass[12pt]{minimal}
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\begin{document}$$Q_4:{\mathcal {D}}'(\Sigma )\rightarrow C^\infty (\Sigma )$$\end{document} is a naturally defined smoothing operator, andσ:=πΣ∗(dα∧u)∈C∞(Σ;T∗Σ)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sigma :=\pi _{\Sigma *}^{}(d\alpha \wedge u)\in C^\infty (\Sigma ;T^*\Sigma ) \end{aligned}$$\end{document}is proved to be a nonzero harmonic 1-form on (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document}. The identity (1.5) implies the nontriviality statement (1.4): if F=0\documentclass[12pt]{minimal}
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\begin{document}$$F=0$$\end{document} then |σ|g2\documentclass[12pt]{minimal}
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\begin{document}$$|\sigma |_g^2$$\end{document} is constant, but hyperbolic 3-manifolds do not admit harmonic 1-forms of nonzero constant length as shown in Appendix A. This finishes the proof of Theorem 1.
If one is interested instead in conformal perturbations of the contact form α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}, then one needs to show that α∧du∧du∗\documentclass[12pt]{minimal}
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\begin{document}$$du_*\ne 0$$\end{document}. The latter follows from the full support property for Pollicott–Ruelle resonant states proved by Weich [54]. See Theorem 4 in §4 for details.
We finally note that it would have been possible to introduce a flat unitary twist in our discussion. Namely, we can consider a Hermitian vector bundle over Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} endowed with a unitary flat connection A. Resonant spaces can be defined using the operator dA\documentclass[12pt]{minimal}
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\begin{document}$$d_{A}$$\end{document} and the holonomy of A provides a way to twist the Ruelle zeta function as well, we refer to [12] for details. We do not pursue this extension here in order to simplify the presentation.
A conjecture
Theorem 1 can be interpreted as follows: the hyperbolic metric has non-closed resonant states due to the extra symmetries, and by destroying these symmetries we make all resonant states closed. We thus make the following conjecture about generic contact Anosov flows:
Conjecture 1
Let M be a compact 2n+1\documentclass[12pt]{minimal}
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\begin{document}$$2n+1$$\end{document} dimensional manifold and α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} a contact 1-form on M such that the corresponding flow is Anosov with orientable stable/unstable bundles. Define the spaces Res0k\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^k_0$$\end{document}, 0≤k≤2n\documentclass[12pt]{minimal}
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\begin{document}$$0\le k\le 2n$$\end{document}, by (1.2) and let πk:Res0k∩kerd→Hk(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k:{{\,\mathrm{Res}\,}}^k_0\cap \ker d\rightarrow H^k(M;{\mathbb {C}})$$\end{document} be defined by (2.61). Then for a generic choice of α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} we have:
the semisimplicity condition holds in all degrees k=0,⋯,2n\documentclass[12pt]{minimal}
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\begin{document}$$k=0,\dots ,2n$$\end{document};
Denoting by bk(M)\documentclass[12pt]{minimal}
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\begin{document}$$b_k(M)$$\end{document} the k-th Betti number of M, we then havedimRes0k=∑j=0⌊k/2⌋bk-2j(M),0≤k≤n;dimRes02n-k=dimRes0k\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \dim {{\,\mathrm{Res}\,}}^k_0=\sum _{j=0}^{\lfloor k/2\rfloor } b_{k-2j}(M),\quad 0\le k\le n;\quad \dim {{\,\mathrm{Res}\,}}^{2n-k}_0=\dim {{\,\mathrm{Res}\,}}^k_0\nonumber \\ \end{aligned}$$\end{document}and the order of vanishing of the Ruelle zeta function at 0 is given by (see [20, (2.5)])mR(0)=∑k=02n(-1)k+ndimRes0k=∑k=0n(-1)k+n(n+1-k)bk(M).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_{\mathrm R}(0)=\sum _{k=0}^{2n}(-1)^{k+n}\dim {{\,\mathrm{Res}\,}}^k_0=\sum _{k=0}^n(-1)^{k+n}(n+1-k)b_k(M). \end{aligned}$$\end{document}
The proof of part 2 of Theorem 1 (see Theorem 3 in §4, as well as §4.4) shows that Conjecture 1 holds for n=2\documentclass[12pt]{minimal}
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\begin{document}$$n=2$$\end{document} and geodesic flows of generic nearly hyperbolic metrics (while the conjecture is stated for generic metrics that do not have to be nearly hyperbolic). Moreover, [21] shows that Conjecture 1 holds for n=1\documentclass[12pt]{minimal}
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\begin{document}$$n=1$$\end{document} and any contact Anosov flow.
Note that the conditions (1) and (2) of Conjecture 1 imply (3). Indeed, by the work of Dang–Rivière [18, Theorem 2.1] the cohomology of the complex (Resk,∞,d)\documentclass[12pt]{minimal}
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\begin{document}$$({{\,\mathrm{Res}\,}}^{k,\infty },d)$$\end{document}, with Resk,∞\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\infty }$$\end{document} defined in (2.38) below with λ0:=0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0:=0$$\end{document}, is isomorphic to the de Rham cohomology of M (with the isomorphism mapping each closed form in Resk,∞\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\infty }$$\end{document} to its cohomology class). By (2.43) and the semisimplicity condition (1), we have Resk,∞=Res0k⊕(α∧Res0k-1)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\infty }={{\,\mathrm{Res}\,}}^k_0\oplus (\alpha \wedge {{\,\mathrm{Res}\,}}^{k-1}_0)$$\end{document}. By condition (2), we have d(u+α∧v)=dα∧v\documentclass[12pt]{minimal}
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\begin{document}$$d(u+\alpha \wedge v)=d\alpha \wedge v$$\end{document} for all u∈Res0k\documentclass[12pt]{minimal}
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\begin{document}$$v\in {{\,\mathrm{Res}\,}}^{k-1}_0$$\end{document}. If k≤n\documentclass[12pt]{minimal}
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\begin{document}$$k\le n$$\end{document}, then dα∧:Res0k-1→Res0k+1\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge :{{\,\mathrm{Res}\,}}^{k-1}_0\rightarrow {{\,\mathrm{Res}\,}}^{k+1}_0$$\end{document} is injective, so Resk,∞∩kerd=Res0k\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\infty }\cap \ker d={{\,\mathrm{Res}\,}}^k_0$$\end{document} and d(Resk-1,∞)=dα∧Res0k-2\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}^{k-1,\infty })=d\alpha \wedge {{\,\mathrm{Res}\,}}^{k-2}_0$$\end{document}. This gives condition (3).
Note also that for n=2\documentclass[12pt]{minimal}
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\begin{document}$$n=2$$\end{document} the set of contact forms satisfying Conjecture 1 is open in C∞(M;T∗M)\documentclass[12pt]{minimal}
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\begin{document}$$C^\infty (M;T^{*}M)$$\end{document}. Indeed, by the perturbation theory discussed in §4.1, more specifically (4.18), if we take a sufficiently small perturbation of a contact form satisfying Conjecture 1, then dimRes01,∞≤b1(M)\documentclass[12pt]{minimal}
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\begin{document}$$\dim {{\,\mathrm{Res}\,}}^{1,\infty }_0\le b_1(M)$$\end{document} and dimRes02,∞≤b2(M)+1\documentclass[12pt]{minimal}
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\begin{document}$$\dim {{\,\mathrm{Res}\,}}^{2,\infty }_0\le b_2(M)+1$$\end{document}. By Lemma 2.8 we see that semisimplicity holds for k=1\documentclass[12pt]{minimal}
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\begin{document}$$k=1$$\end{document} and d(Res01)=0\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}^1_0)=0$$\end{document}. Then Lemma 2.11 together with Lemma 2.4 give all the conclusions of Conjecture 1. A similar argument might work in the case of higher n. Thus the main task in proving the conjecture is to show that (1) and (2) hold on a dense set of contact forms.
One can make a similar conjecture for geodesic flows of generic negatively curved compact orientable n+1\documentclass[12pt]{minimal}
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\begin{document}$$n+1$$\end{document}-dimensional Riemannian manifolds (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document}, with M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$M=S\Sigma $$\end{document}. In particular, if n=2m\documentclass[12pt]{minimal}
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\begin{document}$$n=2m$$\end{document} is even, then Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} is odd dimensional and thus has Euler characteristic 0. By the Gysin exact sequence we have bk(M)=bk(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$b_k(M)=b_k(\Sigma )$$\end{document} for 0≤k<n\documentclass[12pt]{minimal}
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\begin{document}$$b_n(M)=b_n(\Sigma )+b_0(\Sigma )$$\end{document}. Moreover, by Poincaré duality we have bk(Σ)=bn+1-k(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$b_k(\Sigma )=b_{n+1-k}(\Sigma )$$\end{document}. Thus (1.7) becomesmR(0)=b0(Σ)+∑k=0m(-1)k(2m+1-2k)bk(Σ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_{\mathrm R}(0)=b_0(\Sigma )+\sum _{k=0}^m (-1)^k(2m+1-2k)b_k(\Sigma ). \end{aligned}$$\end{document}This is in contrast to the hyperbolic case, where by [25, Theorem 3]mR(0)=∑k=0m(-1)k(2m+2-2k)bk(Σ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_{\mathrm R}(0)=\sum _{k=0}^m (-1)^k(2m+2-2k)b_k(\Sigma ). \end{aligned}$$\end{document}Note that we only expect Conjecture 1 to hold for generic flows/metrics rather than, say, all non-hyperbolic metrics: for n=2\documentclass[12pt]{minimal}
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\begin{document}$$n=2$$\end{document} the proof of Theorem 1 uses first variation which by the Implicit Function Theorem suggests that there is a ‘singular submanifold’ of metrics passing through the hyperbolic metric on which Conjecture 1 fails.
Previous work
The treatment of Pollicott–Ruelle resonances of an Anosov flow as eigenvalues of the generator of the flow on anisotropic Banach and Hilbert spaces has been developed by many authors, including Baladi [3], Baladi–Tsujii [9], Blank–Keller–Liverani [5], Butterley–Liverani [6], Gouëzel–Liverani [33], and Liverani [46, 47] (some of the above papers considered the related setting of Anosov maps). In this paper we use the microlocal approach to dynamical resonances, introduced by Faure–Sjöstrand [29] and developed further by Dyatlov–Zworski [20]; see also Faure–Roy–Sjöstrand [28], Dyatlov–Guillarmou [15], as well as Dang–Rivière [17] and Meddane [48] for the treatment of Morse–Smale and Axiom A flows.
The study of the relation of the vanishing order mR(0)\documentclass[12pt]{minimal}
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\begin{document}$$m_{\mathrm R}(0)$$\end{document} to the topology of the underlying manifold M has a long history, going back to the works of Fried [25, 26] for geodesic flows on hyperbolic manifolds. The paper [25] also related the leading coefficient of ζR\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _{\mathrm R}$$\end{document} at 0 to Reidemeister torsion, which is a topological invariant of M. It considered the more general setting of a twisted zeta function corresponding to a unitary representation. One advantage of such twists is that one can choose the representation so that the twisted de Rham complex is acyclic, i.e. has no cohomology, and then one expects ζR\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _{\mathrm R}$$\end{document} to be holomorphic and nonvanishing at 0.
In [27, p. 66] Fried conjectured a formula relating the Reidemeister torsion with the value ζR(0)\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _{\mathrm R}(0)$$\end{document} for geodesic flows on all compact locally homogeneous manifolds with acyclic representations. Fried’s conjecture was proved by Shen [53] for compact locally symmetric reductive manifolds, following earlier contributions by Bismut [4] and Moscovici–Stanton [49]. The abovementioned works [4, 25, 26, 49, 53] used representation theory and Selberg trace formulas, which do not extend beyond the class of locally symmetric manifolds.
In recent years much progress has been made on understanding the relation between the behavior of ζR\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _{\mathrm R}$$\end{document} at 0, as well as the dimensions of Res0k,ℓ\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\ell }_0$$\end{document}, with topological invariants for general (not locally symmetric) negatively curved Riemannian manifolds and Anosov flows:
Dyatlov–Zworski [21] computed mR(0)\documentclass[12pt]{minimal}
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\begin{document}$$m_{\mathrm R}(0)$$\end{document} for any contact Anosov flow in dimension 3 with orientable stable/unstable bundles, including geodesic flows on compact oriented negatively curved surfaces;
Dang–Rivière [18, Theorem 2.1] showed that the chain complex (Res∙,∞,d)\documentclass[12pt]{minimal}
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\begin{document}$$({{\,\mathrm{Res}\,}}^{\bullet , \infty }, d)$$\end{document}, where Resk,∞=Resk,∞(0)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\infty }={{\,\mathrm{Res}\,}}^{k,\infty }(0)$$\end{document} is defined in (2.39) below, is homotopy equivalent to the usual de Rham complex and hence their cohomologies agree. One can see that Conjecture 1 is compatible with this result, using (2.43) and the fact that (dα∧)k:Ω0n-k→Ω0n+k\documentclass[12pt]{minimal}
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\begin{document}$$(d\alpha \wedge )^k: \Omega _0^{n-k} \rightarrow \Omega _0^{n + k}$$\end{document} is a bundle isomorphism for 0≤k≤n\documentclass[12pt]{minimal}
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\begin{document}$$0\le k\le n$$\end{document};
Hadfield [35] showed a result similar to [21] for geodesic flows on negatively curved surfaces with boundary;
Dang–Guillarmou–Rivière–Shen [16] computed dimRes0k,∞\documentclass[12pt]{minimal}
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\begin{document}$$\dim {{\,\mathrm{Res}\,}}^{k,\infty }_0$$\end{document} for hyperbolic 3-manifolds and proved Fried’s formula relating ζR(0)\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _{\mathrm R}(0)$$\end{document} to Reidemeister torsion for nearly hyperbolic 3-manifolds in the acyclic case; see also Chaubet–Dang [11];
Küster–Weich [44] obtained several results on geodesic flows on compact hyperbolic manifolds and their perturbations, in particular showing that dimRes01=b1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$\dim {{\,\mathrm{Res}\,}}^1_0=b_1(\Sigma )$$\end{document} when dimΣ≠3\documentclass[12pt]{minimal}
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\begin{document}$$\dim \Sigma \ne 3$$\end{document};
Cekić–Paternain [12] studied volume preserving Anosov flows in dimension 3, giving the first example of a situation where mR(0)\documentclass[12pt]{minimal}
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\begin{document}$$m_{\mathrm R}(0)$$\end{document} jumps under perturbations of the flow and thus is not topologically invariant;
Borns-Weil–Shen [10] proved a result similar to [21] for nonorientable stable/unstable bundles.
Our Theorem 1 gives a jump in mR(0)\documentclass[12pt]{minimal}
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\begin{document}$$m_{\mathrm R}(0)$$\end{document} for geodesic flows on 3-manifolds and indicates that the situation for the hyperbolic case is different from that in the case of generic metrics. We stress that it is more difficult to obtain results for generic metric perturbations (such as Theorem 1) than for generic perturbations of contact forms (such as Theorem 4 in §4) due to the more restricted nature of metric perturbations.
One of our main technical results (Theorem 5) bears (limited) similarities to known pairing formulas for Patterson–Sullivan distributions such as those established by Anantharaman–Zelditch [2], Hansen–Hilgert–Schröder [37], Dyatlov–Faure–Guillarmou [14], and Guillarmou–Hilgert–Weich [32]. We briefly discuss this in the Remark after Theorem 5.
Structure of the paper
§2 discusses contact Anosov flows on 5-manifolds and sets up the scene for the rest of the paper. In particular, it introduces Pollicott–Ruelle resonances, (co-)resonant states, dynamical zeta functions, de Rham cohomology, and geodesic flows. It also proves various general lemmas about the maps πk\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k$$\end{document} and semisimplicity.
§3 gives a complete description of generalized resonant states at 0 for hyperbolic 3-manifolds, proving part 1 of Theorem 1. The approach in this section is geometric, as opposed to the algebraic route taken in [25] and [16].
§4 discusses contact perturbations of geodesic flows on hyperbolic 3-manifolds. It proves Theorem 3 which is a general perturbation statement using the nondegeneracy condition (1.3), as well as Theorem 4 on generic contact perturbations. It also gives the proof of part 2 of Theorem 1, relying on the key identity (1.5).
§5 contains the proof of the identity (1.5) (stated in Theorem 5), using a change of variables, a regularization procedure, and the results of §3.
Finally, Appendix A gives a proof of the fact that hyperbolic 3-manifolds have no nonzero harmonic 1-forms of constant length.
Contact 5-dimensional flows
In this section we study general contact Anosov flows on 5-dimensional manifolds. Some of the statements below apply to non-contact Anosov flows and to other dimensions, however we use the setting of 5-dimensional contact flows for uniformity of presentation.
Contact Anosov flows
Assume that M is a compact connected 5-dimensional C∞\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \in C^\infty (M;T^*M)$$\end{document} is a contact 1-form on M, namelydvolα:=α∧dα∧dα≠0everywhere.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d{{\,\mathrm{vol}\,}}_\alpha :=\alpha \wedge d\alpha \wedge d\alpha \ne 0\quad \text {everywhere}. \end{aligned}$$\end{document}We fix the orientation on M by requiring that dvolα\documentclass[12pt]{minimal}
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\begin{document}$$d{{\,\mathrm{vol}\,}}_\alpha $$\end{document} be positively oriented. Let X∈C∞(M;TM)\documentclass[12pt]{minimal}
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\begin{document}$$X\in C^\infty (M;TM)$$\end{document} be the associated Reeb field, that is the unique vector field satisfyingιXα=1,ιXdα=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \iota _X \alpha =1,\quad \iota _X d\alpha =0. \end{aligned}$$\end{document}Note that this immediately implies (where LX\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X$$\end{document} denotes the Lie derivative)LXα=dιXα+ιXdα=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {L}}_X\alpha =d\iota _X\alpha +\iota _Xd\alpha =0. \end{aligned}$$\end{document}We assume that the flow generated by X,φt:=etX:M→M,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varphi _t:=e^{tX}:M\rightarrow M, \end{aligned}$$\end{document}is an Anosov flow, namely there exists a continuous flow/unstable/stable decomposition of the tangent spaces to M,TρM=E0(ρ)⊕Eu(ρ)⊕Es(ρ),ρ∈M,E0(ρ):=RX(ρ)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} T_\rho M=E_0(\rho )\oplus E_u(\rho )\oplus E_s(\rho ),\quad \rho \in M,\quad E_0(\rho ):={\mathbb {R}} X(\rho ) \end{aligned}$$\end{document}and there exist constants C,θ>0\documentclass[12pt]{minimal}
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\begin{document}$$C,\theta >0$$\end{document} and a smooth norm |∙|\documentclass[12pt]{minimal}
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\begin{document}$$|\bullet |$$\end{document} on the fibers of TM such that for all ρ∈M\documentclass[12pt]{minimal}
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\begin{document}$$\rho \in M$$\end{document}, ξ∈TρM\documentclass[12pt]{minimal}
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\begin{document}$$\xi \in T_\rho M$$\end{document}, and t|dφt(ρ)ξ|≤Ce-θ|t|·|ξ|ift≤0,ξ∈Eu(ρ)ort≥0,ξ∈Es(ρ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} |d\varphi _t(\rho )\xi |\le Ce^{-\theta |t|}\cdot |\xi |\quad \text {if}\quad {\left\{ \begin{array}{ll} t\le 0,&{} \xi \in E_u(\rho )\quad \text {or}\\ t\ge 0,&{} \xi \in E_s(\rho ).\end{array}\right. } \end{aligned}$$\end{document}The flow/unstable/stable decomposition gives rise to the dual decomposition of the cotangent spaces to M,Tρ∗M=E0∗(ρ)⊕Eu∗(ρ)⊕Es∗(ρ),E0∗:=(Eu⊕Es)⊥,Eu∗:=(E0⊕Eu)⊥,Es∗:=(E0⊕Es)⊥.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} T_\rho ^* M=E_0^*(\rho )\oplus E_u^*(\rho )\oplus E_s^*(\rho ),&\quad E_0^*:=(E_u\oplus E_s)^\perp ,\\ E_u^*:=(E_0\oplus E_u)^\perp ,&\quad E_s^*:=(E_0\oplus E_s)^\perp . \end{aligned} \end{aligned}$$\end{document}Since LXα=0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X\alpha =0$$\end{document}, we see from (2.3) that α|Eu⊕Es=0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha |_{E_u\oplus E_s}=0$$\end{document} and thusE0∗=Rα.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} E_0^*={\mathbb {R}}\alpha . \end{aligned}$$\end{document}Since α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} is a contact form and dα\documentclass[12pt]{minimal}
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\begin{document}$$E_u\times E_u$$\end{document} and on Es×Es\documentclass[12pt]{minimal}
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\begin{document}$$E_s\times E_s$$\end{document} (as follows from (2.3) and the fact that LXdα=0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_Xd\alpha =0$$\end{document}), we have dimEu=dimEs=2\documentclass[12pt]{minimal}
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\begin{document}$$\dim E_u=\dim E_s=2$$\end{document}.
Bundles of differential forms
We define the vector bundles over MΩk:=∧k(T∗M),Ω0k:={ω∈Ωk∣ιXω=0}≃∧k(Eu∗⊕Es∗).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Omega ^k:=\wedge ^k (T^*M),\quad \Omega ^k_0:=\{\omega \in \Omega ^k\mid \iota _X\omega =0\}\simeq \wedge ^k(E_u^*\oplus E_s^*). \end{aligned}$$\end{document}Note that smooth sections of Ωk\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^k$$\end{document} are differential k-forms on M.
We use the de Rham cohomology groupsHk(M;C):={u∈C∞(M;Ωk)∣du=0}{dv∣v∈C∞(M;Ωk-1)}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} H^k(M;{\mathbb {C}}):={\{u\in C^\infty (M;\Omega ^k)\mid du=0\}\over \{dv\mid v\in C^\infty (M;\Omega ^{k-1})\}}. \end{aligned}$$\end{document}Unless otherwise stated, we will always take Ωk\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^k$$\end{document} to be complexified. We define the Betti numbersbk(M):=dimHk(M;C).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} b_k(M):=\dim H^k(M;{\mathbb {C}}). \end{aligned}$$\end{document}Since M is connected and by Poincaré duality we haveb0(M)=1,bk(M)=b5-k(M).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} b_0(M)=1,\quad b_k(M)=b_{5-k}(M). \end{aligned}$$\end{document}The bundles Ωk\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^k$$\end{document} and Ω0k\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^k_0$$\end{document} are related as follows:Ωk≃Ω0k⊕Ω0k-1\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Omega ^k\simeq \Omega ^k_0\oplus \Omega ^{k-1}_0 \end{aligned}$$\end{document}with the canonical isomorphism and its inverse given byu↦(u-α∧ιXu,ιXu),(v,w)↦v+α∧w.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u\mapsto (u-\alpha \wedge \iota _X u,\iota _X u),\quad (v,w)\mapsto v+\alpha \wedge w. \end{aligned}$$\end{document}Denote by dα∧\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge $$\end{document} the map u↦dα∧u\documentclass[12pt]{minimal}
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\begin{document}$$u\mapsto d\alpha \wedge u$$\end{document} and by dα∧2\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge ^2$$\end{document} the map u↦dα∧dα∧u\documentclass[12pt]{minimal}
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\begin{document}$$u\mapsto d\alpha \wedge d\alpha \wedge u$$\end{document}, then we have linear isomorphisms (as both maps are injective and image and domain have the same dimension)dα∧:Ω01→Ω03,dα∧2:Ω00→Ω04.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d\alpha \wedge :\Omega ^1_0\rightarrow \Omega ^3_0,\quad d\alpha \wedge ^2:\Omega ^0_0\rightarrow \Omega ^4_0. \end{aligned}$$\end{document}We also have a nondegenerate bilinear pairing between sections of Ω0k\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^k_0$$\end{document} and Ω04-k\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^{4-k}_0$$\end{document} given byu∈C∞(M;Ω0k),u∗∈C∞(M;Ω04-k)↦⟨⟨u,u∗⟩⟩:=∫Mα∧u∧u∗\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u\in C^\infty (M;\Omega ^k_0),\ u_*\in C^\infty (M;\Omega ^{4-k}_0)\ \mapsto \ \langle \!\langle u,u_*\rangle \!\rangle :=\int _M \alpha \wedge u\wedge u_*\nonumber \\ \end{aligned}$$\end{document}which in particular identifies the dual space to L2(M;Ω0k)\documentclass[12pt]{minimal}
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\begin{document}$$L^2(M;\Omega ^k_0)$$\end{document} with L2(M;Ω04-k)\documentclass[12pt]{minimal}
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\begin{document}$$L^2(M;\Omega ^{4-k}_0)$$\end{document}. If A:C∞(M;Ω0k)→D′(M;Ω0k)\documentclass[12pt]{minimal}
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\begin{document}$$A^T:C^\infty (M;\Omega ^{4-k}_0)\rightarrow {\mathcal {D}}'(M;\Omega ^{4-k}_0)$$\end{document} satisfying⟨⟨Au,u∗⟩⟩=⟨⟨u,ATu∗⟩⟩for allu∈C∞(M;Ω0k),u∗∈C∞(M;Ω04-k).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \!\langle Au,u_*\rangle \!\rangle =\langle \!\langle u,A^Tu_*\rangle \!\rangle \quad \text {for all}\quad u\in C^\infty (M;\Omega ^k_0),\ u_*\in C^\infty (M;\Omega ^{4-k}_0).\nonumber \\ \end{aligned}$$\end{document}
Geodesic flows
A large class of examples of contact Anosov flows is given by geodesic flows on negatively curved manifolds, which is the setting of the main results of this paper. More precisely, assume that (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} on M as follows: for all ξ∈T(x,v)M\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \alpha (x,v),\xi \rangle =\langle v,d\pi _\Sigma (x,v)\xi \rangle _{g}. \end{aligned}$$\end{document}Then α\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _t$$\end{document} is the geodesic flow, and dvolα\documentclass[12pt]{minimal}
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\begin{document}$$d{{\,\mathrm{vol}\,}}_\alpha $$\end{document} is the standard Liouville volume form up to a constant, see for instance [52, §1.3.3]. If the metric g has negative sectional curvature, then the flow φt\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _t$$\end{document} is Anosov, see for instance [42, Theorem 3.9.1].
We have the time reversal involutionJ:M→M,J(x,v)=(x,-v)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {J}}:M\rightarrow M,\quad {\mathcal {J}}(x,v)=(x,-v) \end{aligned}$$\end{document}which is an orientation reversing diffeomorphism satisfyingJ∗α=-α,J∗X=-X,φt∘J=J∘φ-t\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {J}}^*\alpha =-\alpha ,\quad {\mathcal {J}}^*X=-X,\quad \varphi _t\circ {\mathcal {J}}={\mathcal {J}}\circ \varphi _{-t} \end{aligned}$$\end{document}and the differential of J\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {J}}$$\end{document} maps E0,Eu,Es\documentclass[12pt]{minimal}
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\begin{document}$$E_0,E_u,E_s$$\end{document} into E0,Es,Eu\documentclass[12pt]{minimal}
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\begin{document}$$E_0,E_s,E_u$$\end{document}.
Horizontal and vertical spaces
Recall from (2.2) that an Anosov flow induces a splitting of the tangent bundle TM into the flow, unstable, and stable subbundles. For geodesic flows there is another splitting, into horizontal and vertical subbundles, which we briefly review here. See [52, §1.3.1] for more details.
Let (x,v)∈M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$(x,v)\in M=S\Sigma $$\end{document}. The vertical space at (x, v) is the tangent space to the fiber SxΣ\documentclass[12pt]{minimal}
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\begin{document}$$S_x\Sigma $$\end{document}:V(x,v):=kerdπΣ(x,v)⊂T(x,v)M.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathbf {V}}(x,v):=\ker d\pi _\Sigma (x,v)\ \subset \ T_{(x,v)}M. \end{aligned}$$\end{document}To define a complementary horizontal subspace of T(x,v)M\documentclass[12pt]{minimal}
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\begin{document}$$T_{(x,v)}M$$\end{document}, we use the metric. The connection map of the metric is the unique bundle homomorphism K:TM→TΣ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {K}}:TM\rightarrow T\Sigma $$\end{document} covering the map πΣ\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Sigma $$\end{document} such that for any curve on M written asρ(t)=(x(t),v(t)),x(t)∈Σ,v(t)∈Sx(t)Σ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \rho (t)=(x(t),v(t)),\quad x(t)\in \Sigma ,\quad v(t)\in S_{x(t)}\Sigma \end{aligned}$$\end{document}we haveK(ρ(t))ρ˙(t)=Dtv(t)∈Tx(t)Σ,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {K}}(\rho (t)){\dot{\rho }}(t)={\mathbf {D}}_t v(t)\ \in \ T_{x(t)}\Sigma , \end{aligned}$$\end{document}where Dtv(t)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {D}}_tv(t)$$\end{document} denotes the Levi–Civita covariant derivative of the vector field v(t) along the curve x(t) (see e.g. [13, Proposition 2.2] for a precise definition). Note that since dt⟨v(t),v(t)⟩g=0\documentclass[12pt]{minimal}
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\begin{document}$$d_t\langle v(t),v(t)\rangle _g=0$$\end{document}, the range of K(x,v)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {K}}(x,v)$$\end{document} is g-orthogonal to v.
We now define the horizontal space asH(x,v):=kerK(x,v)⊂T(x,v)M.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathbf {H}}(x,v):=\ker {\mathcal {K}}(x,v)\ \subset \ T_{(x,v)}M. \end{aligned}$$\end{document}We have the splittingT(x,v)M=H(x,v)⊕V(x,v),dimH(x,v)=3,dimV(x,v)=2\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} T_{(x,v)}M={\mathbf {H}}(x,v)\oplus {\mathbf {V}}(x,v),\quad \dim {\mathbf {H}}(x,v)=3,\quad \dim {\mathbf {V}}(x,v)=2 \end{aligned}$$\end{document}and the isomorphisms (here {v}⊥\documentclass[12pt]{minimal}
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\begin{document}$$\{v\}^\perp $$\end{document} is the g-orthogonal complement of v in TxΣ\documentclass[12pt]{minimal}
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\begin{document}$$T_x\Sigma $$\end{document})dπΣ(x,v):H(x,v)→TxΣ,K(x,v):V(x,v)→{v}⊥\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d\pi _\Sigma (x,v):{\mathbf {H}}(x,v)\rightarrow T_x\Sigma ,\quad {\mathcal {K}}(x,v):{\mathbf {V}}(x,v)\rightarrow \{v\}^\perp \end{aligned}$$\end{document}which together give the following isomorphism T(x,v)M→TxΣ⊕{v}⊥\documentclass[12pt]{minimal}
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\begin{document}$$T_{(x,v)}M\rightarrow T_x\Sigma \oplus \{v\}^\perp $$\end{document}:ξ↦(ξH,ξV),ξH=dπΣ(x,v)ξ,ξV=K(x,v)ξ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \xi \mapsto (\xi _H,\xi _V),\quad \xi _H=d\pi _\Sigma (x,v)\xi ,\quad \xi _V={\mathcal {K}}(x,v)\xi . \end{aligned}$$\end{document}We use the map (2.15) to identify T(x,v)M\documentclass[12pt]{minimal}
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\begin{document}$$T_{(x,v)}M$$\end{document} with TxΣ⊕{v}⊥\documentclass[12pt]{minimal}
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\begin{document}$$T_x\Sigma \oplus \{v\}^\perp $$\end{document}.
Under the identification (2.15), the contact form α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} and its differential satisfy (see [52, Proposition 1.24])α(x,v)(ξ)=⟨ξH,v⟩g,dα(x,v)(ξ,η)=⟨ξV,ηH⟩g-⟨ξH,ηV⟩g.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \alpha (x,v)(\xi )&=\langle \xi _H,v\rangle _g,\\ d\alpha (x,v)(\xi ,\eta )&=\langle \xi _V,\eta _H\rangle _g-\langle \xi _H,\eta _V\rangle _g. \end{aligned} \end{aligned}$$\end{document}Using the splitting (2.15), we define the Sasaki metric⟨∙,∙⟩S\documentclass[12pt]{minimal}
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\begin{document}$$\langle \bullet ,\bullet \rangle _S$$\end{document} on M as follows:⟨ξ,η⟩S:=⟨ξH,ηH⟩g+⟨ξV,ηV⟩g.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \xi ,\eta \rangle _S:=\langle \xi _H,\eta _H\rangle _g+\langle \xi _V,\eta _V\rangle _g. \end{aligned}$$\end{document}We finally remark that the generator X of the geodesic flow has the following form under the isomorphism (2.15):X(x,v)H=v,X(x,v)V=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} X(x,v)_H=v,\quad X(x,v)_V=0. \end{aligned}$$\end{document}
De Rham cohomology of the sphere bundle
We now describe the de Rham cohomology of M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$M=S\Sigma $$\end{document} in terms of the cohomology of Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}. To relate the two, we use the pullback operatorsπΣ∗:C∞(Σ;Ωk)→C∞(M;Ωk),0≤k≤3\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _\Sigma ^*:C^\infty (\Sigma ;\Omega ^k)\rightarrow C^\infty (M;\Omega ^k),\quad 0\le k\le 3 \end{aligned}$$\end{document}and the pushforward operators defined by integrating along the fibers of SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document}πΣ∗:C∞(M;Ωk)→C∞(Σ;Ωk-2),2≤k≤5.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _{\Sigma *}^{}:C^\infty (M;\Omega ^k)\rightarrow C^\infty (\Sigma ;\Omega ^{k-2}),\quad 2\le k\le 5. \end{aligned}$$\end{document}Here the orientation on each fiber SxΣ\documentclass[12pt]{minimal}
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\begin{document}$$S_x\Sigma $$\end{document} is induced by the orientation on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}: if v,v1,v2\documentclass[12pt]{minimal}
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\begin{document}$$x\in \Sigma $$\end{document}πΣ∗ω(x)(X1,⋯,Xk-2)=∫SxΣιX~k-2⋯ιX~1ω.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _{\Sigma *}^{}\omega (x)(X_1,\dots ,X_{k-2})=\int _{S_x\Sigma } \iota _{{\widetilde{X}}_{k-2}}\dots \iota _{{\widetilde{X}}_1}\omega . \end{aligned}$$\end{document}Another characterization of πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$(0,v_2)$$\end{document} written using the horizontal/vertical decomposition (2.15), where v,v1,v2\documentclass[12pt]{minimal}
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\begin{document}$$v,v_1,v_2$$\end{document} is a positively oriented g-orthonormal basis on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}.
The pushforward map has the following properties (see for instance [8, Propositions 6.14.1 and 6.15] for the related case of vector bundles):dπΣ∗=πΣ∗d,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d\pi _{\Sigma *}^{}&=\pi _{\Sigma *}^{}d, \end{aligned}$$\end{document}πΣ∗(ω1∧(πΣ∗ω2))=(πΣ∗ω1)∧ω2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _{\Sigma *}^{}\big (\omega _1\wedge (\pi _\Sigma ^*\omega _2)\big )&=(\pi _{\Sigma *}^{}\omega _1)\wedge \omega _2. \end{aligned}$$\end{document}Note that the maps πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}$$\end{document}, πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Sigma ^*$$\end{document} can also be defined on distributional forms. For πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}$$\end{document} this follows from the fact that pushforward is always well-defined on distributions as long as the fibers are compact and for the pullback πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Sigma ^*$$\end{document} this follows from the fact that πΣ\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Sigma $$\end{document} is a submersion [38, Theorem 6.1.2].
Since the map J\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {J}}$$\end{document} defined in (2.12) is an orientation reversing diffeomorphism of the fibers of SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document}, we also haveπΣ∗(J∗ω)=-πΣ∗ω.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _{\Sigma *}^{}({\mathcal {J}}^*\omega )=-\pi _{\Sigma *}^{}\omega . \end{aligned}$$\end{document}Since pullbacks commute with the differential d, and by (2.22), the operations πΣ∗,πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Sigma ^*,\pi _{\Sigma *}^{}$$\end{document} induce maps on de Rham cohomology, which we denote by the same letters:πΣ∗:Hk(Σ;C)→Hk(M;C),πΣ∗:Hk(M;C)→Hk-2(Σ;C).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _\Sigma ^*:H^k(\Sigma ;{\mathbb {C}})\rightarrow H^k(M;{\mathbb {C}}),\quad \pi _{\Sigma *}^{}:H^k(M;{\mathbb {C}})\rightarrow H^{k-2}(\Sigma ;{\mathbb {C}}). \end{aligned}$$\end{document}From the Gysin exact sequence (see for instance [8, Proposition 14.33], where the Euler class is zero since Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} is three-dimensional; alternatively one can use Künneth formulas and the fact that every compact orientable 3-manifold is parallelizable) we have isomorphismsπΣ∗:H1(Σ;C)→H1(M;C),πΣ∗:H4(M;C)→H2(Σ;C)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _\Sigma ^*:H^1(\Sigma ;{\mathbb {C}})\rightarrow H^1(M;{\mathbb {C}}),\quad \pi _{\Sigma *}^{}:H^4(M;{\mathbb {C}})\rightarrow H^2(\Sigma ;{\mathbb {C}}) \end{aligned}$$\end{document}and the exact sequences0→H2(Σ;C)→πΣ∗H2(M;C)→πΣ∗H0(Σ;C)→0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&0\rightarrow H^2(\Sigma ;{\mathbb {C}})\xrightarrow {\pi _\Sigma ^*} H^2(M;{\mathbb {C}})\xrightarrow {\pi _{\Sigma *}^{}} H^0(\Sigma ;{\mathbb {C}})\rightarrow 0, \end{aligned}$$\end{document}0→H3(Σ;C)→πΣ∗H3(M;C)→πΣ∗H1(Σ;C)→0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&0\rightarrow H^3(\Sigma ;{\mathbb {C}})\xrightarrow {\pi _\Sigma ^*} H^3(M;{\mathbb {C}})\xrightarrow {\pi _{\Sigma *}^{}} H^1(\Sigma ;{\mathbb {C}})\rightarrow 0. \end{aligned}$$\end{document}In particular, we get formulas for the Betti numbers of the sphere bundle M:b0(M)=b5(M)=1,b1(M)=b4(M)=b1(Σ),b2(M)=b3(M)=b1(Σ)+1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} b_0(M)= & {} b_5(M)=1,\quad b_1(M)=b_4(M)=b_1(\Sigma ),\nonumber \\&b_2(M)=b_3(M)=b_1(\Sigma )+1. \end{aligned}$$\end{document}
Pollicott–Ruelle resonances
We now review the theory of Pollicott–Ruelle resonances in the present setting. Define the first order differential operatorsPk:=-iLX:C∞(M;Ωk)→C∞(M;Ωk),Pk,0:=-iLX:C∞(M;Ω0k)→C∞(M;Ω0k).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} P_k&\,:=-i{\mathcal {L}}_X:C^\infty (M;\Omega ^k)\rightarrow C^\infty (M;\Omega ^k),\\ P_{k,0}&\,:=-i{\mathcal {L}}_X:C^\infty (M;\Omega ^k_0)\rightarrow C^\infty (M;\Omega ^k_0). \end{aligned} \end{aligned}$$\end{document}Note that Pk,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{k,0}$$\end{document} is the restriction of Pk\documentclass[12pt]{minimal}
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\begin{document}$$P_k$$\end{document} to C∞(M;Ω0k)\documentclass[12pt]{minimal}
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\begin{document}$$C^\infty (M;\Omega ^k_0)$$\end{document} which is the space of all u∈C∞(M;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$u\in C^\infty (M;\Omega ^k)$$\end{document} which satisfy ιXu=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _Xu=0$$\end{document}.
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\begin{document}$$\lambda \in {\mathbb {C}}$$\end{document} with Imλ\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Im}\,}}\lambda $$\end{document} large enough, the integralRk(λ):=i∫0∞eiλte-itPkdt:L2(M;Ωk)→L2(M;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R_k(\lambda ):=i\int _0^\infty e^{i\lambda t}e^{-itP_k}\,dt:L^2(M;\Omega ^k)\rightarrow L^2(M;\Omega ^k) \end{aligned}$$\end{document}converges and defines a bounded operator on L2\documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document} which is holomorphic in λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}. Here the evolution group e-itPk\documentclass[12pt]{minimal}
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\begin{document}$$e^{-itP_k}$$\end{document} is given by e-itPku=φ-t∗u\documentclass[12pt]{minimal}
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\begin{document}$$e^{-itP_k}u=\varphi _{-t}^*u$$\end{document}. It is straightforward to check that Rk(λ)\documentclass[12pt]{minimal}
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\begin{document}$$R_k(\lambda )$$\end{document} is the L2\documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document}-resolvent of Pk\documentclass[12pt]{minimal}
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\begin{document}$$P_k$$\end{document}:Rk(λ)=(Pk-λ)-1:L2(M;Ωk)→L2(M;Ωk),Imλ≫1,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R_k(\lambda )=(P_k-\lambda )^{-1}:L^2(M;\Omega ^k)\rightarrow L^2(M;\Omega ^k),\quad {{\,\mathrm{Im}\,}}\lambda \gg 1, \end{aligned}$$\end{document}where we treat Pk\documentclass[12pt]{minimal}
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\begin{document}$$P_k$$\end{document} as an unbounded operator on L2\documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document} with domain {u∈L2(M;Ωk)∣Pku∈L2(M;Ωk)}\documentclass[12pt]{minimal}
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\begin{document}$$\{u\in L^2(M;\Omega ^k)\mid P_ku\in L^2(M;\Omega ^k)\}$$\end{document} and Pku\documentclass[12pt]{minimal}
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\begin{document}$$P_ku$$\end{document} is defined in the sense of distributions.
Meromorphic continuation
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\begin{document}$$\varphi _t$$\end{document} is an Anosov flow, the resolvent Rk(λ)\documentclass[12pt]{minimal}
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\begin{document}$$R_k(\lambda )$$\end{document} admits a meromorphic continuationRk(λ):C∞(M;Ωk)→D′(M;Ωk),λ∈C,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R_k(\lambda ):C^\infty (M;\Omega ^k)\rightarrow {\mathcal {D}}'(M;\Omega ^k),\quad \lambda \in {\mathbb {C}}, \end{aligned}$$\end{document}see for instance [20, §3.2] and [29, Theorems 1.4, 1.5]. The proof of this continuation shows that Rk(λ)\documentclass[12pt]{minimal}
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\begin{document}$$R_k(\lambda )$$\end{document} acts on certain anisotropic Sobolev spaces adapted to the stable/unstable decompositions, see e.g. [20, §3.1]; this makes it possible to compose the operator Rk(λ)\documentclass[12pt]{minimal}
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\begin{document}$$R_k(\lambda )$$\end{document} with itself. Instead of introducing these spaces here, we use the spaces of distributionsDΓ′(M;Ωk):={u∈D′(M;Ωk)∣WF(u)⊂Γ},\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {D}}'_\Gamma (M;\Omega ^k):=\{u\in {\mathcal {D}}'(M;\Omega ^k)\mid {{\,\mathrm{WF}\,}}(u)\subset \Gamma \}, \end{aligned}$$\end{document}where Γ⊂T∗M\0\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{WF}\,}}(u)$$\end{document} denotes the wavefront set of a distribution u. These spaces come with a natural sequential topology, see [38, Definition 8.2.2].
We have the wavefront set property of Rk(λ)\documentclass[12pt]{minimal}
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\begin{document}$$R_k(\lambda )$$\end{document} proved in [20, (3.7)]:WF′(Rk(λ))⊂W:=Δ(T∗M)∪Υ+∪(Eu∗×Es∗),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{WF}\,}}'(R_k(\lambda ))\ \subset \ {\mathscr {W}}:=\Delta (T^*M)\cup \Upsilon _+\cup (E_u^*\times E_s^*), \end{aligned}$$\end{document}where Δ(T∗M)⊂T∗M×T∗M\documentclass[12pt]{minimal}
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\begin{document}$$\Delta (T^*M)\subset T^*M\times T^*M$$\end{document} is the diagonal and Υ+={(φt(x),dφt(x)-Tξ,x,ξ)∣t≥0,ξ(X(x))=0}\documentclass[12pt]{minimal}
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\begin{document}$$\Upsilon _+=\{(\varphi _t(x),d\varphi _t(x)^{-T} \xi ,x,\xi )\mid t\ge 0, \xi (X(x))=0\}$$\end{document}; for an operator B:C∞(M)→D′(M)\documentclass[12pt]{minimal}
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\begin{document}$$B:C^\infty (M) \rightarrow {\mathcal {D}}'(M)$$\end{document} with Schwartz kernel KB∈D′(M×M)\documentclass[12pt]{minimal}
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\begin{document}$$K_B \in {\mathcal {D}}'(M \times M)$$\end{document}, we denote WF′(B)={(x,ξ,y,-η)∣(x,ξ,y,η)∈WF(KB)}⊂T∗(M×M)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{WF}\,}}'(B) = \{(x, \xi , y, -\eta ) \mid (x, \xi , y, \eta ) \in {{\,\mathrm{WF}\,}}(K_B)\} \subset T^*(M \times M)$$\end{document}. The Schwartz kernel of Rk(λ)\documentclass[12pt]{minimal}
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\begin{document}$$R_k(\lambda )$$\end{document} is meromorphic in λ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'_{{\mathscr {W}}'}$$\end{document} where W′:={(x,ξ,y,-η)∣(x,ξ,y,η)∈W}\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {W}}':=\{(x,\xi ,y,-\eta )\mid (x,\xi ,y,\eta )\in {\mathscr {W}}\}$$\end{document}. By the wavefront set calculus [38, Theorem 8.2.13] and since Eu∗∩Es∗=0\documentclass[12pt]{minimal}
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\begin{document}$$R_k(\lambda )$$\end{document} defines a meromorphic family of continuous operatorsRk(λ):DEu∗′(M;Ωk)→DEu∗′(M;Ωk),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R_k(\lambda ):{\mathcal {D}}'_{E_u^*}(M;\Omega ^k)\rightarrow {\mathcal {D}}'_{E_u^*}(M;\Omega ^k), \end{aligned}$$\end{document}where we view Eu∗⊂T∗M\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'_{E_u^*}$$\end{document} by (2.31).
Note that differential operators (in particular, d,ιX,LX\documentclass[12pt]{minimal}
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\begin{document}$$d,\iota _X,{\mathcal {L}}_X$$\end{document}) define continuous maps on the regularity classes DEu∗′\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'_{E_u^*}$$\end{document}. We haveRk(λ)(Pk-λ)u=(Pk-λ)Rk(λ)u=ufor allu∈DEu∗′(M;Ωk).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R_k(\lambda )(P_k-\lambda )u=(P_k-\lambda )R_k(\lambda )u=u\quad \text {for all } u\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^k).\nonumber \\ \end{aligned}$$\end{document}For Imλ≫1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Im}\,}}\lambda \gg 1$$\end{document} and u∈C∞(M;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$u\in C^\infty (M;\Omega ^k)$$\end{document} this follows from (2.30); the general case follows from here by analytic continuation and since C∞\documentclass[12pt]{minimal}
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\begin{document}$$C^\infty $$\end{document} is dense in DEu∗′\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'_{E_u^*}$$\end{document}.
We also have the commutation relationsdRk(λ)u=Rk+1(λ)du,ιXRk(λ)u=Rk-1(λ)ιXufor allu∈DEu∗′(M;Ωk).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} dR_k(\lambda )u=R_{k+1}(\lambda )du,\quad \iota _X R_k(\lambda )u=R_{k-1}(\lambda )\iota _Xu\quad \text {for all}\quad u\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^k).\nonumber \\ \end{aligned}$$\end{document} As with (2.34) it suffices to consider the case Imλ≫1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Im}\,}}\lambda \gg 1$$\end{document} and u∈C∞(M;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$u\in C^\infty (M;\Omega ^k)$$\end{document}, in which (2.35) follows from (2.29) and the fact that d and ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} commute with φ-t∗\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _{-t}^*$$\end{document}.
The poles of the family of operators Rk(λ)\documentclass[12pt]{minimal}
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\begin{document}$$R_k(\lambda )$$\end{document} are called Pollicott–Ruelle resonances on k-forms. At each pole λ0∈C\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0\in {\mathbb {C}}$$\end{document} we have an expansion (see for instance [20, (3.6)])Rk(λ)=RkH(λ;λ0)-∑j=1Jk(λ0)(Pk-λ0)j-1Πk(λ0)(λ-λ0)j,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R_k(\lambda )=R^H_k(\lambda ;\lambda _0)-\sum _{j=1}^{J_k(\lambda _0)}{(P_k-\lambda _0)^{j-1}\Pi _k(\lambda _0)\over (\lambda -\lambda _0)^j}, \end{aligned}$$\end{document}where RkH(λ;λ0):DEu∗′(M;Ωk)→DEu∗′(M;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$R^H_k(\lambda ;\lambda _0):{\mathcal {D}}'_{E_u^*}(M;\Omega ^k)\rightarrow {\mathcal {D}}'_{E_u^*}(M;\Omega ^k)$$\end{document} is a family of operators holomorphic in a neighborhood of λ0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0$$\end{document}, Jk(λ0)≥1\documentclass[12pt]{minimal}
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\begin{document}$$J_k(\lambda _0)\ge 1$$\end{document} is an integer, and Πk(λ0):DEu∗′(M;Ωk)→DEu∗′(M;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$\Pi _k(\lambda _0):{\mathcal {D}}'_{E_u^*}(M;\Omega ^k)\rightarrow {\mathcal {D}}'_{E_u^*}(M;\Omega ^k)$$\end{document} is a finite rank operator commuting with Pk\documentclass[12pt]{minimal}
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\begin{document}$$P_k$$\end{document} and such that (Pk-λ0)Jk(λ0)Πk(λ0)=0\documentclass[12pt]{minimal}
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\begin{document}$$(P_k-\lambda _0)^{J_k(\lambda _0)}\Pi _k(\lambda _0)=0$$\end{document}.
Taking the expansions of (2.35) at λ0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0$$\end{document} we see thatdΠk(λ0)=Πk+1(λ0)d,ιXΠk(λ0)=Πk-1(λ0)ιX.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d\Pi _k(\lambda _0)=\Pi _{k+1}(\lambda _0)d,\quad \iota _X \Pi _k(\lambda _0)=\Pi _{k-1}(\lambda _0)\iota _X. \end{aligned}$$\end{document}
Resonant states
The range of the operator Πk(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$\Pi _k(\lambda _0)$$\end{document} is equal to the space of generalised resonant states (see for instance [20, Proposition 3.3])Resk,∞(λ0):=⋃ℓ≥1Resk,ℓ(λ0),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{Res}\,}}^{k,\infty }(\lambda _0):=\bigcup _{\ell \ge 1}{{\,\mathrm{Res}\,}}^{k,\ell }(\lambda _0), \end{aligned}$$\end{document}where we defineResk,ℓ(λ0):={u∈DEu∗′(M;Ωk)∣(Pk-λ0)ℓu=0}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{Res}\,}}^{k,\ell }(\lambda _0):=\{u\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^k)\mid (P_k-\lambda _0)^\ell u=0\}. \end{aligned}$$\end{document}We define the algebraic multiplicity of λ0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0$$\end{document} as a resonance on k-forms bymk(λ0):=rankΠk(λ0)=dimResk,∞(λ0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_k(\lambda _0):={{\,\mathrm{rank}\,}}\Pi _k(\lambda _0)=\dim {{\,\mathrm{Res}\,}}^{k,\infty }(\lambda _0). \end{aligned}$$\end{document}The geometric multiplicity is the dimension of the space of resonant statesResk(λ0):=Resk,1(λ0)={u∈DEu∗′(M;Ωk)∣(Pk-λ0)u=0}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{Res}\,}}^k(\lambda _0):={{\,\mathrm{Res}\,}}^{k,1}(\lambda _0)=\{u\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^k)\mid (P_k-\lambda _0)u=0\}. \end{aligned}$$\end{document}We say a resonance λ0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0$$\end{document} of Pk\documentclass[12pt]{minimal}
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\begin{document}$$P_k$$\end{document} is semisimple if the algebraic and geometric multiplicities coincide, that is Resk,∞(λ0)=Resk(λ0)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\infty }(\lambda _0)={{\,\mathrm{Res}\,}}^k(\lambda _0)$$\end{document}. This is equivalent to saying that Jk(λ0)=1\documentclass[12pt]{minimal}
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\begin{document}$$J_k(\lambda _0)=1$$\end{document} in (2.36). Another equivalent definition of semisimplicity isu∈DEu∗′(M;Ωk),(Pk-λ0)2u=0⟹(Pk-λ0)u=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^k),\ (P_k-\lambda _0)^2u=0\quad \Longrightarrow \quad (P_k-\lambda _0)u=0. \end{aligned}$$\end{document}We note that the operators Πk(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$\Pi _k(\lambda _0)$$\end{document} are idempotent. In fact, applying the Laurent expansion (2.36) at λ0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0$$\end{document} to u∈Resk,ℓ(λ1)\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^{k,\ell }(\lambda _1)$$\end{document} and using the identity Rk(λ)u=-∑j=0ℓ-1(λ-λ1)-j-1(Pk-λ1)ju\documentclass[12pt]{minimal}
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\begin{document}$$R_k(\lambda )u=-\sum _{j=0}^{\ell -1}(\lambda -\lambda _1)^{-j-1}(P_k-\lambda _1)^ju$$\end{document} we see thatΠk(λ0)Πk(λ1)=Πk(λ0)ifλ1=λ0,0ifλ1≠λ0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Pi _k(\lambda _0)\Pi _k(\lambda _1)={\left\{ \begin{array}{ll} \Pi _k(\lambda _0)&{}\text {if }\lambda _1=\lambda _0,\\ 0&{}\text {if }\lambda _1\ne \lambda _0. \end{array}\right. } \end{aligned}$$\end{document}
Operators on the bundles Ω0k\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^k_0$$\end{document}
The above constructions apply equally as well to the operators Pk,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{k,0}$$\end{document} (except that the operator d does not preserve sections of Ω0k\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^k_0$$\end{document}, so the first commutation relation in (2.37) does not hold, and the second one is trivial); we denote the resulting objects byRk,0(λ),Jk,0(λ0),Rk,0H(λ;λ0),Πk,0(λ0),Res0k,ℓ(λ0),mk,0(λ0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R_{k,0}(\lambda ),\ J_{k,0}(\lambda _0),\ R^H_{k,0}(\lambda ;\lambda _0),\ \Pi _{k,0}(\lambda _0),\ {{\,\mathrm{Res}\,}}^{k,\ell }_0(\lambda _0),\ m_{k,0}(\lambda _0). \end{aligned}$$\end{document}Under the isomorphism (2.7) the operator Pk\documentclass[12pt]{minimal}
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\begin{document}$$P_k$$\end{document} is conjugated to Pk,0⊕Pk-1,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{k,0}\oplus P_{k-1,0}$$\end{document}. Therefore (2.7) gives an isomorphismResk,ℓ(λ0)≃Res0k,ℓ(λ0)⊕Res0k-1,ℓ(λ0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{Res}\,}}^{k,\ell }(\lambda _0)\simeq {{\,\mathrm{Res}\,}}^{k,\ell }_0(\lambda _0)\oplus {{\,\mathrm{Res}\,}}^{k-1,\ell }_0(\lambda _0). \end{aligned}$$\end{document}Moreover, we get for all u∈DEu∗′(M;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^k)$$\end{document}Πk(λ0)u=Πk,0(λ0)(u-α∧ιXu)+α∧Πk-1,0(λ0)ιXu.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Pi _k(\lambda _0)u=\Pi _{k,0}(\lambda _0)(u-\alpha \wedge \iota _X u) +\alpha \wedge \Pi _{k-1,0}(\lambda _0)\iota _X u. \end{aligned}$$\end{document}Since LXdα=0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_Xd\alpha =0$$\end{document}, the operations (2.8) give rise to linear isomorphismsdα∧:Res01,ℓ(λ0)→Res03,ℓ(λ0),dα∧2:Res00,ℓ(λ0)→Res04,ℓ(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d\alpha \wedge :{{\,\mathrm{Res}\,}}^{1,\ell }_0(\lambda _0)\rightarrow {{\,\mathrm{Res}\,}}^{3,\ell }_0(\lambda _0),\quad d\alpha \wedge ^2:{{\,\mathrm{Res}\,}}^{0,\ell }_0(\lambda _0)\rightarrow {{\,\mathrm{Res}\,}}^{4,\ell }_0(\lambda _0)\nonumber \\ \end{aligned}$$\end{document}which in particular give the equalitiesm1,0(λ0)=m3,0(λ0),m0,0(λ0)=m4,0(λ0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_{1,0}(\lambda _0)=m_{3,0}(\lambda _0),\quad m_{0,0}(\lambda _0)=m_{4,0}(\lambda _0). \end{aligned}$$\end{document}
Transposes and coresonant states
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\begin{document}$${\mathcal {L}}_X\alpha =0$$\end{document} and ∫MLXω=0\documentclass[12pt]{minimal}
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\begin{document}$$\int _M {\mathcal {L}}_X\omega =0$$\end{document} for any 5-form ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}, we have(Pk,0)T=-P4-k,0,k=0,1,2,3,4,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (P_{k,0})^T=-P_{4-k,0}, \quad k = 0, 1, 2, 3, 4, \end{aligned}$$\end{document}where the transpose is defined using the pairing ⟨⟨∙,∙⟩⟩\documentclass[12pt]{minimal}
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\begin{document}$$\langle \!\langle \bullet ,\bullet \rangle \!\rangle $$\end{document}, see (2.10). Thus the transpose of the resolvent (Rk,0(λ))T\documentclass[12pt]{minimal}
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\begin{document}$$(R_{k,0}(\lambda ))^T$$\end{document} is the meromorphic continuation of the resolvent corresponding to the vector field -X\documentclass[12pt]{minimal}
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\begin{document}$$-X$$\end{document}; the latter generates an Anosov flow with the unstable and stable spaces switching roles compared to the ones for X. Similarly to (2.33) we have(Rk,0(λ))T:DEs∗′(M;Ω04-k)→DEs∗′(M;Ω04-k),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (R_{k,0}(\lambda ))^T:{\mathcal {D}}'_{E_s^*}(M;\Omega ^{4-k}_0)\rightarrow {\mathcal {D}}'_{E_s^*}(M;\Omega ^{4-k}_0), \end{aligned}$$\end{document}where DEs∗′\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'_{E_s^*}$$\end{document} is the space of distributional sections with wavefront set contained in Es∗\documentclass[12pt]{minimal}
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\begin{document}$$E_s^*$$\end{document}. Same applies to the transposes of the operators Rk,0H(λ;λ0)\documentclass[12pt]{minimal}
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\begin{document}$$R^H_{k,0}(\lambda ;\lambda _0)$$\end{document} and Πk,0(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$\Pi _{k,0}(\lambda _0)$$\end{document} appearing in (2.36). The range of (Πk,0(λ0))T\documentclass[12pt]{minimal}
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\begin{document}$$(\Pi _{k,0}(\lambda _0))^T$$\end{document} is the space of generalised coresonant statesRes0∗4-k,∞(λ0)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{4-k,\infty }_{0*}(\lambda _0)$$\end{document} whereRes0∗k,∞(λ0):=⋃ℓ≥1Res0∗k,ℓ(λ0),Res0∗k,ℓ(λ0):={u∗∈DEs∗′(M;Ω0k)∣(Pk,0+λ0)ℓu∗=0}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} {{\,\mathrm{Res}\,}}^{k,\infty }_{0*}(\lambda _0)&\,:=\bigcup _{\ell \ge 1}{{\,\mathrm{Res}\,}}^{k,\ell }_{0*}(\lambda _0),\\ {{\,\mathrm{Res}\,}}^{k,\ell }_{0*}(\lambda _0)&\,:=\{u_*\in {\mathcal {D}}'_{E_s^*}(M;\Omega ^k_0)\mid (P_{k,0}+\lambda _0)^\ell u_*=0\}. \end{aligned} \end{aligned}$$\end{document}The space of coresonant states is defined asRes0∗k(λ0):=Res0∗k,1(λ0)={u∗∈DEs∗′(M;Ω0k)∣(Pk,0+λ0)u∗=0}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{Res}\,}}^k_{0*}(\lambda _0):={{\,\mathrm{Res}\,}}^{k,1}_{0*}(\lambda _0)=\{u_*\in {\mathcal {D}}'_{E_s^*}(M;\Omega ^k_0)\mid (P_{k,0}+\lambda _0)u_*=0\}. \end{aligned}$$\end{document}Similarly to (2.45) we have the isomorphismsdα∧:Res0∗1,ℓ(λ0)→Res0∗3,ℓ(λ0),dα∧2:Res0∗0,ℓ(λ0)→Res0∗4,ℓ(λ0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d\alpha \wedge :{{\,\mathrm{Res}\,}}^{1,\ell }_{0*}(\lambda _0)\rightarrow {{\,\mathrm{Res}\,}}^{3,\ell }_{0*}(\lambda _0),\quad d\alpha \wedge ^2:{{\,\mathrm{Res}\,}}^{0,\ell }_{0*}(\lambda _0)\rightarrow {{\,\mathrm{Res}\,}}^{4,\ell }_{0*}(\lambda _0).\nonumber \\ \end{aligned}$$\end{document}In the special case when φt\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _t$$\end{document} is a geodesic flow with the time reversal map J\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {J}}$$\end{document} defined in (2.12), the pullback operator J∗\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {J}}^*$$\end{document} gives an isomorphism between DEu∗′(M;Ω0k)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'_{E_u^*}(M;\Omega ^k_0)$$\end{document} and DEs∗′(M;Ω0k)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'_{E_s^*}(M;\Omega ^k_0)$$\end{document}. Moreover, J∗Pk,0=-Pk,0J∗\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {J}}^*P_{k,0}=-P_{k,0}{\mathcal {J}}^*$$\end{document}. This gives rise to isomorphisms between the spaces of generalised resonant and coresonant statesJ∗:Res0k,ℓ(λ0)→Res0∗k,ℓ(λ0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {J}}^*:{{\,\mathrm{Res}\,}}^{k,\ell }_0(\lambda _0)\rightarrow {{\,\mathrm{Res}\,}}^{k,\ell }_{0*}(\lambda _0). \end{aligned}$$\end{document}
Coresonant states and pairing
Since Eu∗\documentclass[12pt]{minimal}
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\begin{document}$$E_u^*$$\end{document} and Es∗\documentclass[12pt]{minimal}
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\begin{document}$$E_s^*$$\end{document} intersect only at the zero section, we can define the product u∧u∗∈D′(M;Ω04)\documentclass[12pt]{minimal}
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\begin{document}$$u\wedge u_*\in {\mathcal {D}}'(M;\Omega ^4_0)$$\end{document} and thus the pairing ⟨⟨u,u∗⟩⟩\documentclass[12pt]{minimal}
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\begin{document}$$\langle \!\langle u,u_*\rangle \!\rangle $$\end{document} for any u∈DEu∗′(M;Ω0k)\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^k_0)$$\end{document}, u∗∈DEs∗′(M;Ω04-k)\documentclass[12pt]{minimal}
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\begin{document}$$u_*\in {\mathcal {D}}'_{E_s^*}(M;\Omega ^{4-k}_0)$$\end{document}, see [38, Theorem 8.2.10]. Note that this pairing is nondegenerate since both DEu∗′\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'_{E_u^*}$$\end{document} and DEs∗′\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'_{E_s^*}$$\end{document} contain C∞\documentclass[12pt]{minimal}
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\begin{document}$$C^\infty $$\end{document}, and the transpose formula (2.10) still holds since C∞\documentclass[12pt]{minimal}
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\begin{document}$$C^\infty $$\end{document} is dense in DEu∗′\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'_{E_u^*}$$\end{document} and in DEs∗′\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'_{E_s^*}$$\end{document}. In particular, we have a pairingu∈Res0k,∞(λ0),u∗∈Res0∗4-k,∞(λ0)↦⟨⟨u,u∗⟩⟩∈C.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u\in {{\,\mathrm{Res}\,}}^{k,\infty }_0(\lambda _0),\ u_*\in {{\,\mathrm{Res}\,}}^{4-k,\infty }_{0*}(\lambda _0) \quad \mapsto \quad \langle \!\langle u,u_*\rangle \!\rangle \in {\mathbb {C}}. \end{aligned}$$\end{document}This pairing is nondegenerate. Indeed, assume that u∈Res0k,∞(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^{k,\infty }_0(\lambda _0)$$\end{document} and ⟨⟨u,u∗⟩⟩=0\documentclass[12pt]{minimal}
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\begin{document}$$\langle \!\langle u,u_*\rangle \!\rangle =0$$\end{document} for all u∗∈Res0∗4-k,∞(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$u_*\in {{\,\mathrm{Res}\,}}^{4-k,\infty }_{0*}(\lambda _0)$$\end{document}. Since Res0∗4-k,∞(λ0)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{4-k,\infty }_{0*}(\lambda _0)$$\end{document} is the range of (Πk,0(λ0))T\documentclass[12pt]{minimal}
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\begin{document}$$(\Pi _{k,0}(\lambda _0))^T$$\end{document}, we have0=⟨⟨u,(Πk,0(λ0))Tφ⟩⟩=⟨⟨Πk,0(λ0)u,φ⟩⟩=⟨⟨u,φ⟩⟩for allφ∈C∞(M;Ω04-k),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} 0=\langle \!\langle u,(\Pi _{k,0}(\lambda _0))^T\varphi \rangle \!\rangle= & {} \langle \!\langle \Pi _{k,0}(\lambda _0)u,\varphi \rangle \!\rangle \nonumber \\= & {} \langle \!\langle u,\varphi \rangle \!\rangle \quad \text {for all } \varphi \in C^\infty (M;\Omega ^{4-k}_0), \end{aligned}$$\end{document}where the last equality follows from the fact that Πk,0(λ0)2=Πk,0(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$\Pi _{k,0}(\lambda _0)^2=\Pi _{k,0}(\lambda _0)$$\end{document} and u is in the range of Πk,0(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$\Pi _{k,0}(\lambda _0)$$\end{document}. It follows that u=0\documentclass[12pt]{minimal}
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\begin{document}$$u=0$$\end{document}. Similarly one can show that if ⟨⟨u,u∗⟩⟩=0\documentclass[12pt]{minimal}
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\begin{document}$$\langle \!\langle u,u_*\rangle \!\rangle =0$$\end{document} for some u∗∈Res0∗4-k,∞(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$u_*\in {{\,\mathrm{Res}\,}}^{4-k,\infty }_{0*}(\lambda _0)$$\end{document} and all u∈Res0k,∞(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^{k,\infty }_0(\lambda _0)$$\end{document}, then u∗=0\documentclass[12pt]{minimal}
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\begin{document}$$u_*=0$$\end{document}.
Consider the operators on finite dimensional spacesPk,0-λ0:Res0k,∞(λ0)→Res0k,∞(λ0),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} P_{k,0}-\lambda _0&:{{\,\mathrm{Res}\,}}^{k,\infty }_0(\lambda _0)\rightarrow {{\,\mathrm{Res}\,}}^{k,\infty }_0(\lambda _0), \end{aligned}$$\end{document}-P4-k,0-λ0:Res0∗4-k,∞(λ0)→Res0∗4-k,∞(λ0),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} -P_{4-k,0}-\lambda _0&:{{\,\mathrm{Res}\,}}^{4-k,\infty }_{0*}(\lambda _0)\rightarrow {{\,\mathrm{Res}\,}}^{4-k,\infty }_{0*}(\lambda _0), \end{aligned}$$\end{document}which are transposes of each other with respect to the pairing (2.51). The kernels of ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}-th powers of these operators are Res0k,ℓ(λ0)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\ell }_0(\lambda _0)$$\end{document} and Res0∗4-k,ℓ(λ0)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{4-k,\ell }_{0*}(\lambda _0)$$\end{document}, thus (using the isomorphisms (2.49))dimRes0k,ℓ(λ0)=dimRes0∗4-k,ℓ(λ0)=dimRes0∗k,ℓ(λ0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \dim {{\,\mathrm{Res}\,}}^{k,\ell }_0(\lambda _0)=\dim {{\,\mathrm{Res}\,}}^{4-k,\ell }_{0*}(\lambda _0)=\dim {{\,\mathrm{Res}\,}}^{k,\ell }_{0*}(\lambda _0). \end{aligned}$$\end{document}We now give a solvability result for the operators Pk,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{k,0}$$\end{document}. It follows from the Fredholm property of these operators on anisotropic Sobolev spaces but we present instead a proof using the Laurent expansion (2.36).
Lemma 2.1
Assume that w∈DEu∗′(M;Ω0k)\documentclass[12pt]{minimal}
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\begin{document}$$w\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^k_0)$$\end{document}. Then the equation(Pk,0-λ0)u=w,u∈DEu∗′(M;Ω0k)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (P_{k,0}-\lambda _0)u=w,\quad u\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^k_0) \end{aligned}$$\end{document}has a solution if and only if w satisfies the condition⟨⟨w,u∗⟩⟩=0for allu∗∈Res0∗4-k(λ0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \!\langle w,u_*\rangle \!\rangle =0\quad \text {for all}\quad u_*\in {{\,\mathrm{Res}\,}}^{4-k}_{0*}(\lambda _0). \end{aligned}$$\end{document}
Proof
First of all, if (2.55) has a solution u, then for each u∗∈Res0∗4-k(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$u_*\in {{\,\mathrm{Res}\,}}^{4-k}_{0*}(\lambda _0)$$\end{document} we have⟨⟨w,u∗⟩⟩=⟨⟨(Pk,0-λ0)u,u∗⟩⟩=-⟨⟨u,(P4-k,0+λ0)u∗⟩⟩=0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \!\langle w,u_*\rangle \!\rangle =\langle \!\langle (P_{k,0}-\lambda _0)u,u_*\rangle \!\rangle =-\langle \!\langle u,(P_{4-k,0}+\lambda _0)u_*\rangle \!\rangle =0, \end{aligned}$$\end{document}that is the condition (2.56) is satisfied.
Now, assume that w satisfies the condition (2.56); we show that (2.55) has a solution. We start with the special case when w∈Res0k,∞(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$w\in {{\,\mathrm{Res}\,}}^{k,\infty }_0(\lambda _0)$$\end{document}. We use the pairing (2.51) to identify the dual space to Res0k,∞(λ0)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\infty }_0(\lambda _0)$$\end{document} with Res0∗4-k,∞(λ0)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{4-k,\infty }_{0*}(\lambda _0)$$\end{document}. By (2.56), w is annihilated by the kernel of the operator (2.53). Therefore w is in the range of the operator (2.52), that is (2.55) has a solution u∈Res0k,∞(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^{k,\infty }_0(\lambda _0)$$\end{document}.
We now consider the case of general w satisfying (2.56). Taking the constant term in the Laurent expansion of the identity (2.34) at λ=λ0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda =\lambda _0$$\end{document}, we obtain(Pk,0-λ0)Rk,0H(λ0;λ0)w=w-Πk,0(λ0)w.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (P_{k,0}-\lambda _0)R_{k,0}^H(\lambda _0;\lambda _0)w=w-\Pi _{k,0}(\lambda _0)w. \end{aligned}$$\end{document}We have Πk,0(λ0)w∈Res0k,∞(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$\Pi _{k,0}(\lambda _0)w\in {{\,\mathrm{Res}\,}}^{k,\infty }_0(\lambda _0)$$\end{document} and it satisfies (2.56), thus (2.55) has a solution with this right-hand side. Writing w=Πk,0(λ0)w+(Id-Πk,0(λ0))w\documentclass[12pt]{minimal}
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\begin{document}$$w = \Pi _{k, 0}(\lambda _0)w + \big ({{\,\mathrm{Id}\,}}- \Pi _{k, 0}(\lambda _0)\big )w$$\end{document}, we may take as u the sum of this solution and Rk,0H(λ0;λ0)w\documentclass[12pt]{minimal}
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\begin{document}$$R_{k,0}^H(\lambda _0;\lambda _0)w$$\end{document}. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Lemma 2.1 implies the following criterion for semisimplicity:
Lemma 2.2
The semisimplicity condition (2.41) holds for the operator Pk,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{k,0}$$\end{document} if and only if the restriction of the pairing (2.51) to Res0k(λ0)×Res0∗4-k(λ0)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^k_0(\lambda _0)\times {{\,\mathrm{Res}\,}}^{4-k}_{0*}(\lambda _0)$$\end{document} is nondegenerate.
Proof
The condition (2.41) is equivalent to saying that the intersection of Res0k(λ0)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^k_0(\lambda _0)$$\end{document} with the range of the operator Pk,0-λ0:DEu∗′(M;Ω0k)→DEu∗′(M;Ω0k)\documentclass[12pt]{minimal}
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\begin{document}$$P_{k,0}-\lambda _0:{\mathcal {D}}'_{E_u^*}(M;\Omega ^k_0)\rightarrow {\mathcal {D}}'_{E_u^*}(M;\Omega ^k_0)$$\end{document} is trivial; that is, for each w∈Res0k(λ0)\{0}\documentclass[12pt]{minimal}
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\begin{document}$$w\in {{\,\mathrm{Res}\,}}^k_0(\lambda _0){\setminus } \{0\}$$\end{document} the equation (2.55) has no solution. By Lemma 2.1, this is equivalent to saying that w does not satisfy the condition (2.56), i.e. there exists v∈Res0∗4-k(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$v\in {{\,\mathrm{Res}\,}}^{4-k}_{0*}(\lambda _0)$$\end{document} such that ⟨⟨w,v⟩⟩≠0\documentclass[12pt]{minimal}
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\begin{document}$$\langle \!\langle w,v\rangle \!\rangle \ne 0$$\end{document}. This is equivalent to the nondegeneracy condition of the present lemma. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Zeta functions
We now discuss dynamical zeta functions. We assume that the unstable/stable bundles Eu,Es\documentclass[12pt]{minimal}
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\begin{document}$$E_u,E_s$$\end{document} are orientable (the non-orientable case is covered by [10]); this is true for the case of geodesic flows on orientable manifolds as follows from the fact that the vertical bundle trivially intersects the weak unstable bundle RX⊕Eu\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}X \oplus E_u$$\end{document} (see [34, Lemma B.1]).
We say γ:[0,Tγ]→M\documentclass[12pt]{minimal}
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\begin{document}$$\gamma :[0,T_\gamma ]\rightarrow M$$\end{document} is a closed trajectory of the flow φt\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (T_\gamma )=\gamma (0)$$\end{document}. We identify closed trajectories obtained by shifting t. The primitive period of a closed trajectory, denoted by Tγ♯\documentclass[12pt]{minimal}
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\begin{document}$$T_\gamma ^\sharp $$\end{document}, is the smallest positive t>0\documentclass[12pt]{minimal}
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\begin{document}$$t>0$$\end{document} such that γ(t)=γ(0)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (t)=\gamma (0)$$\end{document}. We say γ\documentclass[12pt]{minimal}
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\begin{document}$$T_\gamma =T_\gamma ^\sharp $$\end{document}.
Define the linearised Poincaré mapPγ:=dφ-Tγ(γ(0))|Eu⊕Es\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {P}}_\gamma :=d\varphi _{-T_\gamma }(\gamma (0))|_{E_u\oplus E_s}$$\end{document}. We have detPγ=1\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge d\alpha $$\end{document} to Eu⊕Es\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _t$$\end{document}-invariant nonvanishing 4-form. Since φt\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _t$$\end{document} is an Anosov flow, the map I-Pγ\documentclass[12pt]{minimal}
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\begin{document}$$I-{\mathcal {P}}_\gamma $$\end{document} is invertible (in fact Pγ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {P}}_\gamma $$\end{document} has no eigenvalues on the unit circle).
For 0≤k≤4\documentclass[12pt]{minimal}
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\begin{document}$$0\le k\le 4$$\end{document}, define the zeta functionζk(λ):=exp(-∑γTγ♯tr(∧kPγ)eiλTγTγdet(I-Pγ)),Imλ≫1,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \zeta _k(\lambda ):=\exp \bigg (-\sum _{\gamma }{T_\gamma ^\sharp {{\,\mathrm{tr}\,}}(\wedge ^k{\mathcal {P}}_\gamma )e^{i\lambda T_\gamma } \over T_\gamma \det (I-{\mathcal {P}}_\gamma )}\bigg ),\quad {{\,\mathrm{Im}\,}}\lambda \gg 1, \end{aligned}$$\end{document}where the sum is over all the closed trajectories γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document}. The series in (2.58) converges for sufficiently large Imλ\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Im}\,}}\lambda $$\end{document}, see e.g. [20, §2.2].
The zeta function ζk\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _k$$\end{document} continues holomorphically to λ∈C\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \in {\mathbb {C}}$$\end{document} and for each λ0∈C\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0$$\end{document} as a zero of ζk\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _k$$\end{document} is equal to mk,0(λ0)\documentclass[12pt]{minimal}
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\begin{document}$$m_{k,0}(\lambda _0)$$\end{document}, the algebraic multiplicity of λ0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0$$\end{document} as a resonance of the operator Pk,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{k,0}$$\end{document} defined similarly to (2.40) – see [20, §4] for the proof.
By Ruelle’s identity (see e.g. [20, (2.5)]) the Ruelle zeta function defined in (1.1) factorizes as follows:ζR(λ)=ζ0(λ)ζ2(λ)ζ4(λ)ζ1(λ)ζ3(λ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \zeta _{\mathrm R}(\lambda )={\zeta _0(\lambda )\zeta _2(\lambda )\zeta _4(\lambda )\over \zeta _1(\lambda )\zeta _3(\lambda )}. \end{aligned}$$\end{document}Using (2.46) we see that the order of vanishing of the function ζR\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0$$\end{document} is equal tomR(λ0)=∑k=04(-1)kmk,0(λ0)=2m0,0(λ0)-2m1,0(λ0)+m2,0(λ0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_{\mathrm R}(\lambda _0)= \sum _{k=0}^4 (-1)^k m_{k,0}(\lambda _0)=2m_{0,0}(\lambda _0)-2m_{1,0}(\lambda _0)+m_{2,0}(\lambda _0).\nonumber \\ \end{aligned}$$\end{document}
Resonance at 0
This paper focuses on the resonance at 0, which is why we henceforth put λ0:=0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0:=0$$\end{document} unless stated otherwise. For instance we writeRk,0H(λ):=Rk,0H(λ;0),Πk,0:=Πk,0(0),Res0k,ℓ:=Res0k,ℓ(0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R^H_{k,0}(\lambda ):=R^H_{k,0}(\lambda ;0),\quad \Pi _{k,0}:=\Pi _{k,0}(0),\quad {{\,\mathrm{Res}\,}}^{k,\ell }_0:={{\,\mathrm{Res}\,}}^{k,\ell }_0(0). \end{aligned}$$\end{document}Our main goal is to study the order of vanishing of the Ruelle zeta function at 0, which by (2.59) is equal tomR(0)=2m0,0(0)-2m1,0(0)+m2,0(0),mk,0(0)=dimRes0k,∞.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_{\mathrm R}(0)=2m_{0,0}(0)-2m_{1,0}(0)+m_{2,0}(0),\quad m_{k,0}(0)=\dim {{\,\mathrm{Res}\,}}^{k,\infty }_0. \end{aligned}$$\end{document}Since LX=dιX+ιXd\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X=d\iota _X+\iota _Xd$$\end{document}, the space of resonant states at 0 for the operator Pk,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{k,0}$$\end{document} isRes0k={u∈DEu∗′(M;Ωk)∣ιXu=0,ιXdu=0}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{Res}\,}}^k_0=\{u\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^k)\mid \iota _X u=0,\ \iota _X du=0\}. \end{aligned}$$\end{document}In particular, the exterior derivative defines an operator d:Res0k→Res0k+1\documentclass[12pt]{minimal}
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\begin{document}$$d:{{\,\mathrm{Res}\,}}^k_0\rightarrow {{\,\mathrm{Res}\,}}^{k+1}_0$$\end{document}. (Unfortunately this is no longer true for the spaces of generalised resonant states Res0k,ℓ\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\ell }_0$$\end{document} with ℓ≥2\documentclass[12pt]{minimal}
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\begin{document}$$\ell \ge 2$$\end{document}, since d does not necessarily map these to the kernel of ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document}.)
0-Forms and 4-forms
We first analyze the resonance at 0 for the operators P0,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{0,0}$$\end{document} and P4,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{4,0}$$\end{document}. The following regularity result is a special case of [21, Lemma 2.3] (see also [28, Lemma 4] for a similar statement in the case of Anosov maps):
Using Lemma 2.3 we show the following statement similar to [21, Lemma 3.2] (we note that it straightforwardly generalizes to other dimensions, which was known already to [46, Corollary 2.11]):
Lemma 2.4
The semisimplicity condition (2.41) holds at λ0=0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0=0$$\end{document} for the operators P0,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{0,0}$$\end{document}, P4,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{4,0}$$\end{document} andm0,0(0)=m4,0(0)=1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_{0,0}(0)=m_{4,0}(0)=1. \end{aligned}$$\end{document}Moreover, Res00=Res0∗0\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^0_0={{\,\mathrm{Res}\,}}^0_{0*}$$\end{document} is spanned by the constant function 1 and Res04=Res0∗4\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^4_0={{\,\mathrm{Res}\,}}^4_{0*}$$\end{document} is spanned by the form dα∧dα\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge d\alpha $$\end{document}.
Proof
We only give the proof for 0-forms (i.e. functions); the case of 4-forms follows from here using the isomorphisms (2.45), (2.49).
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^0_0$$\end{document}. Then Xu=0\documentclass[12pt]{minimal}
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\begin{document}$$Xu=0$$\end{document}, so Lemma 2.3 implies that u∈C∞(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _t$$\end{document}; the stable/unstable decomposition (2.4) gives that du∈E0∗\documentclass[12pt]{minimal}
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\begin{document}$$Xu=0$$\end{document}, this implies that du=0\documentclass[12pt]{minimal}
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\begin{document}$$du=0$$\end{document} and thus (since M is connected) u is constant. We have shown that Res00\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^0_0$$\end{document} is spanned by the function 1; applying the above argument to -X\documentclass[12pt]{minimal}
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\begin{document}$$-X$$\end{document} we see that Res0∗0\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^0_{0*}$$\end{document} is spanned by 1 as well.
To show the semisimplicity condition (2.41), assume that u∈DEu∗′(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$Xu\in {{\,\mathrm{Res}\,}}^0_0$$\end{document}, so Xu is constant. Together with the identity ∫M(Xu)dvolα=0\documentclass[12pt]{minimal}
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\begin{document}$$\int _M (Xu)\,d{{\,\mathrm{vol}\,}}_\alpha =0$$\end{document} this gives Xu=0\documentclass[12pt]{minimal}
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\begin{document}$$Xu=0$$\end{document} as needed. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Closed forms
We now study resonant states which are closed, that is elements of the spaceRes0k∩kerd={u∈DEu∗′(M;Ωk)∣ιXu=0,du=0}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{Res}\,}}^k_0\cap \ker d=\{u\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^k)\mid \iota _X u=0,\ du=0\}. \end{aligned}$$\end{document}We use a special case of [21, Lemma 2.1] which shows that de Rham cohomology in the spaces DEu∗′(M;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'_{E_u^*}(M;\Omega ^k)$$\end{document} is the same as the usual de Rham cohomology defined in (2.6):
Similarly to [21, §3.3] we introduce the linear mapπk:Res0k∩kerd→Hk(M;C),πk(u)=[v]Hkwhereu=v+dw,v∈C∞(M;Ωk),w∈DEu∗′(M;Ωk-1).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \pi _k:{{\,\mathrm{Res}\,}}^k_0\cap \ker d\rightarrow H^k(M;{\mathbb {C}}),\quad \pi _k(u)=[v]_{H^k} \\ \text {where}\quad u=v+dw,\quad v\in C^\infty (M;\Omega ^k),\quad w\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^{k-1}). \end{aligned} \end{aligned}$$\end{document}Here v, w exist by Lemma 2.5. To show that the map πk\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k$$\end{document} is well-defined, assume that u=v+dw=v′+dw′\documentclass[12pt]{minimal}
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\begin{document}$$u=v+dw=v'+dw'$$\end{document} where v,v′∈C∞(M;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$v,v'\in C^\infty (M;\Omega ^k)$$\end{document} and w,w′∈DEu∗′(M;Ωk-1)\documentclass[12pt]{minimal}
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\begin{document}$$w,w'\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^{k-1})$$\end{document}. Then d(w-w′)=v′-v∈C∞(M;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$d(w-w')=v'-v\in C^\infty (M;\Omega ^k)$$\end{document}, thus by Lemma 2.5 we may write w-w′=w1+dw2\documentclass[12pt]{minimal}
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\begin{document}$$w-w'=w_1+dw_2$$\end{document} where w1∈C∞(M;Ωk-1)\documentclass[12pt]{minimal}
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\begin{document}$$w_1\in C^\infty (M;\Omega ^{k-1})$$\end{document}, w2∈DEu∗′(M;Ωk-2)\documentclass[12pt]{minimal}
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\begin{document}$$w_2\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^{k-2})$$\end{document}. Then v′-v=dw1\documentclass[12pt]{minimal}
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\begin{document}$$v'-v=dw_1$$\end{document} where w1\documentclass[12pt]{minimal}
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\begin{document}$$w_1$$\end{document} is smooth, so [v]Hk=[v′]Hk\documentclass[12pt]{minimal}
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\begin{document}$$[v]_{H^k}=[v']_{H^k}$$\end{document}.
Similar arguments apply to the spaces Res0∗k∩kerd\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k}_{0*}\cap \ker d$$\end{document} of closed coresonant k-forms; we denote the corresponding maps byπk∗:Res0∗k∩kerd→Hk(M;C).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _{k*}:{{\,\mathrm{Res}\,}}^{k}_{0*}\cap \ker d\rightarrow H^k(M;{\mathbb {C}}). \end{aligned}$$\end{document}From Lemma 2.4 we see that π0\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0$$\end{document} is an isomorphism and hence by (2.45) that π4=0\documentclass[12pt]{minimal}
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\begin{document}$$\pi _4=0$$\end{document}.
We now establish several properties of the spaces Res0k∩kerd\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^k_0\cap \ker d$$\end{document} and the maps πk\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k$$\end{document}; some of these are extensions of the results of [21, §3.3].
The first containment is immediate. For the second one, assume that u∈Res0k∩kerd\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^k_0\cap \ker d$$\end{document} and πk(u)=0\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k(u)=0$$\end{document}. Then u=v+dw\documentclass[12pt]{minimal}
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\begin{document}$$u=v+dw$$\end{document} where v∈C∞(M;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$v\in C^\infty (M;\Omega ^k)$$\end{document} satisfies [v]Hk=0\documentclass[12pt]{minimal}
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\begin{document}$$[v]_{H^k}=0$$\end{document} and w∈DEu∗′(M;Ωk-1)\documentclass[12pt]{minimal}
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\begin{document}$$w\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^{k-1})$$\end{document}. We have v=dζ\documentclass[12pt]{minimal}
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\begin{document}$$v=d\zeta $$\end{document} for some ζ∈C∞(M;Ωk-1)\documentclass[12pt]{minimal}
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\begin{document}$$\zeta \in C^\infty (M;\Omega ^{k-1})$$\end{document} and by (2.37)u=Πku=Πkd(ζ+w)=dΠk-1(ζ+w).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u=\Pi _ku=\Pi _k d(\zeta +w)=d\Pi _{k-1}(\zeta +w). \end{aligned}$$\end{document}Therefore u∈d(Resk-1,∞)\documentclass[12pt]{minimal}
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\begin{document}$$u\in d({{\,\mathrm{Res}\,}}^{k-1,\infty })$$\end{document}. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
We note that the case k=0\documentclass[12pt]{minimal}
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\begin{document}$$k = 0$$\end{document} of the following lemma holds trivially.
Lemma 2.7
Assume that for some k all the coresonant states in Res0∗5-k\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{5-k}_{0*}$$\end{document} are exact forms. Then the map πk\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k$$\end{document} is onto.
Proof
Take arbitrary v∈C∞(M;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$v\in C^\infty (M;\Omega ^k)$$\end{document} such that dv=0\documentclass[12pt]{minimal}
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\begin{document}$$dv=0$$\end{document}. We will construct u∈Res0k∩kerd\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^k_0\cap \ker d$$\end{document} such that πk(u)=[v]Hk\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k(u)=[v]_{H^k}$$\end{document} by puttingu:=v+dwfor somew∈DEu∗′(M;Ω0k-1).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u:=v+dw\quad \text {for some}\quad w\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^{k-1}_0). \end{aligned}$$\end{document}Such u is automatically closed, so we only need to choose w so that ιXu=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _Xu=0$$\end{document}, that isιXdw=LXw=-ιXv\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \iota _Xdw={\mathcal {L}}_Xw=-\iota _X v \end{aligned}$$\end{document}where the first equality is immediate because ιXw=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X w=0$$\end{document}.
To solve (2.62), we use Lemma 2.1. It suffices to check that the condition (2.56) holds:⟨⟨ιXv,u∗⟩⟩=0for allu∗∈Res0∗5-k.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \!\langle \iota _X v,u_*\rangle \!\rangle =0\quad \text {for all}\quad u_*\in {{\,\mathrm{Res}\,}}^{5-k}_{0*}. \end{aligned}$$\end{document}We compute⟨⟨ιXv,u∗⟩⟩=∫Mα∧(ιXv)∧u∗=∫Mv∧u∗=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \!\langle \iota _X v,u_*\rangle \!\rangle =\int _M \alpha \wedge (\iota _X v)\wedge u_* =\int _M v\wedge u_*=0. \end{aligned}$$\end{document}Here in the second equality we used that ιXu∗=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X u_*=0$$\end{document} (thus ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} of the 5-forms on both sides are the same) and in the last equality we used that v is closed and, by the assumption of the lemma, u∗\documentclass[12pt]{minimal}
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\begin{document}$$u_*$$\end{document} is exact. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
We only consider the case of π1\documentclass[12pt]{minimal}
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\begin{document}$$\pi _1$$\end{document}, with π1∗\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{1*}$$\end{document} handled similarly. To show that π1\documentclass[12pt]{minimal}
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\begin{document}$$\pi _1$$\end{document} is one-to-one, we use Lemma 2.6 and the fact that Res0,∞=Res00\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{0,\infty }={{\,\mathrm{Res}\,}}^0_0$$\end{document} consists of constant functions by Lemma 2.4. To show that π1\documentclass[12pt]{minimal}
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\begin{document}$$\pi _1$$\end{document} is onto, it suffices to use Lemma 2.7: by Lemma 2.4, the space Res0∗4\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^4_{0*}$$\end{document} is spanned by dα∧dα=d(α∧dα)\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge d\alpha =d(\alpha \wedge d\alpha )$$\end{document}. □\documentclass[12pt]{minimal}
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Lemma 2.9
We have d(Res03)=d(Res0∗3)=0\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}^3_0)=d({{\,\mathrm{Res}\,}}^3_{0*})=0$$\end{document}.
Proof
We only consider the case of Res03\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^3_0$$\end{document}, with Res0∗3\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^3_{0*}$$\end{document} handled similarly. Assume that u∈Res03\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^3_0$$\end{document}. Then du∈Res04\documentclass[12pt]{minimal}
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\begin{document}$$du\in {{\,\mathrm{Res}\,}}^4_0$$\end{document}, so by Lemma 2.4 we have du=cdα∧dα\documentclass[12pt]{minimal}
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\begin{document}$$du=cd\alpha \wedge d\alpha $$\end{document} for some constant c. It remains to use thatc∫Mdvolα=∫Mα∧du=∫Mdα∧u=0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} c\int _M d{{\,\mathrm{vol}\,}}_\alpha =\int _M \alpha \wedge du=\int _M d\alpha \wedge u=0, \end{aligned}$$\end{document}where in the second equality we integrated by parts and in the third equality we used that ιX(dα∧u)=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X(d\alpha \wedge u)=0$$\end{document}, thus dα∧u=0\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge u=0$$\end{document}. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
We also have the following nondegeneracy result for the pairing between closed resonant and coresonant forms when k=1\documentclass[12pt]{minimal}
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\begin{document}$$k=1$$\end{document}:
Lemma 2.10
The pairing induced by ⟨⟨∙,∙⟩⟩\documentclass[12pt]{minimal}
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\begin{document}$$\langle \!\langle \bullet ,\bullet \rangle \!\rangle $$\end{document} on (Res01∩kerd)×(dα∧(Res0∗1∩kerd))\documentclass[12pt]{minimal}
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\begin{document}$$({{\,\mathrm{Res}\,}}^1_0\cap \ker d)\times (d\alpha \wedge ({{\,\mathrm{Res}\,}}^1_{0*}\cap \ker d))$$\end{document} is nondegenerate.
Proof
We show the following stronger statement: for each closed but not exact v∈C∞(M;Ω1)\documentclass[12pt]{minimal}
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\begin{document}$$v\in C^\infty (M;\Omega ^1)$$\end{document},Re⟨⟨π1-1([v]H1),dα∧π1∗-1([v¯]H1)⟩⟩<0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{Re}\,}}\langle \!\langle \pi _1^{-1}([v]_{H^1}),d\alpha \wedge \pi _{1*}^{-1}([{\overline{v}}]_{H^1})\rangle \!\rangle < 0. \end{aligned}$$\end{document}Here we used that the map π1\documentclass[12pt]{minimal}
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\begin{document}$$\pi _1$$\end{document} is an isomorphism, as shown in Lemma 2.8. We haveπ1-1([v]H1)=v+df,π1∗-1([v¯]H1)=v+dg¯,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _1^{-1}([v]_{H^1})=v+df,\quad \pi _{1*}^{-1}([{\overline{v}}]_{H^1})=\overline{v+dg}, \end{aligned}$$\end{document}where f∈DEu∗′(M;C),g∈DEs∗′(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$f\in {\mathcal {D}}'_{E_u^*}(M;{\mathbb {C}}),g\in {\mathcal {D}}'_{E_s^*}(M;{\mathbb {C}})$$\end{document} satisfyXf=Xg=-ιXv.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Xf=Xg=-\iota _X v. \end{aligned}$$\end{document}We computeRe⟨⟨π1-1([v]H1),dα∧π1∗-1([v¯]H1)⟩⟩=Re∫Mα∧dα∧(v+df)∧(v+dg¯)=Re∫Mα∧dα∧(df∧v¯+v∧dg¯+df∧dg¯)=Re∫Mdα∧dα∧(fv¯-g¯v-g¯df)=Re∫M(fιXv¯-g¯ιXv-(Xf)g¯)dvolα=-Re⟨Xf,f⟩L2(M;dvolα).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} {{\,\mathrm{Re}\,}}\langle \!\langle \pi _1^{-1}([v]_{H^1}),d\alpha&\wedge \pi _{1*}^{-1}([{\overline{v}}]_{H^1})\rangle \!\rangle \\&={{\,\mathrm{Re}\,}}\int _M \alpha \wedge d\alpha \wedge (v+df)\wedge (\overline{v+dg})\\&={{\,\mathrm{Re}\,}}\int _M \alpha \wedge d\alpha \wedge (df\wedge {\overline{v}}+v\wedge d{\overline{g}}+df\wedge d{\overline{g}})\\&={{\,\mathrm{Re}\,}}\int _M d\alpha \wedge d\alpha \wedge (f{\overline{v}}-{\overline{g}} v-{\overline{g}} df)\\&={{\,\mathrm{Re}\,}}\int _M \big (f\iota _X {\overline{v}}-{\overline{g}}\iota _X v-(Xf)\overline{g}\big )d{{\,\mathrm{vol}\,}}_\alpha \\&=-{{\,\mathrm{Re}\,}}\langle Xf,f\rangle _{L^2(M;d{{\,\mathrm{vol}\,}}_\alpha )}. \end{aligned} \end{aligned}$$\end{document}Here in the second line we used that Re(v∧v¯)=0\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Re}\,}}(v\wedge {\overline{v}})=0$$\end{document}. In the third line we integrated by parts and used that dv=0\documentclass[12pt]{minimal}
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\begin{document}$$dv=0$$\end{document}. In the fourth line we used that ιXdα=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X d\alpha =0$$\end{document} (the 5-forms under the integral are equal as can be seen by taking ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} of both sides). In the last line we used the identity (2.64).
Thus, if (2.63) fails, we have Re⟨Xf,f⟩L2(M;dvolα)≤0\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Re}\,}}\langle Xf,f\rangle _{L^2(M;d{{\,\mathrm{vol}\,}}_\alpha )}\le 0$$\end{document} which by Lemma 2.3 implies that f∈C∞(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$f\in C^\infty (M;{\mathbb {C}})$$\end{document} and thus u:=π1-1([v]H1)\documentclass[12pt]{minimal}
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\begin{document}$$u:=\pi _1^{-1}([v]_{H^1})$$\end{document} lies in Res01∩C∞(M;Ω1)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_0\cap C^\infty (M;\Omega ^1)$$\end{document}. Now the fact that u is invariant under the flow φt\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _t$$\end{document} and the stable/unstable decomposition (2.4) imply that u∈E0∗\documentclass[12pt]{minimal}
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\begin{document}$$u\in E_0^*$$\end{document} at each point, and the fact that ιXu=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X u=0$$\end{document} then gives u=0\documentclass[12pt]{minimal}
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\begin{document}$$u=0$$\end{document}. This shows that v is exact, giving a contradiction. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
We finally give the following result in the case when all forms in Res01\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_0$$\end{document} are closed:
Lemma 2.11
Assume that Res01\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_0$$\end{document} consists of closed forms, i.e. d(Res01)=0\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}^1_0)=0$$\end{document}. Then:
The semisimplicity condition (2.41) holds at λ0=0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0=0$$\end{document} for the operators P1,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{1,0}$$\end{document} and P3,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{3,0}$$\end{document}.
Lemma 2.11 does not provide full information on the resonance at 0 since it does not prove the semisimplicity condition for the operator P2,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{2,0}$$\end{document}, and only assumes that resonant forms Res01\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}_0^1$$\end{document} are closed (in fact we will see that d(Res01)≠0\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}_0^1) \ne 0$$\end{document} and P2,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{2, 0}$$\end{document} is not semisimple in the hyperbolic case when b1(M)>0\documentclass[12pt]{minimal}
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\begin{document}$$b_1(M)>0$$\end{document}, see § 3).
Proof
1. Since dim(Res01∩kerd)=dim(Res0∗1∩kerd)\documentclass[12pt]{minimal}
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\begin{document}$$\dim ({{\,\mathrm{Res}\,}}^1_0\cap \ker d)=\dim ({{\,\mathrm{Res}\,}}^1_{0*}\cap \ker d)$$\end{document} by Lemma 2.8, and dimRes01=dimRes0∗1\documentclass[12pt]{minimal}
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\begin{document}$$\dim {{\,\mathrm{Res}\,}}^1_0=\dim {{\,\mathrm{Res}\,}}^1_{0*}$$\end{document} by (2.54), we have d(Res0∗1)=0\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}^1_{0*})=0$$\end{document}. By (2.49) we have Res0∗3=dα∧Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^3_{0*}=d\alpha \wedge {{\,\mathrm{Res}\,}}^1_{0*}$$\end{document}. Now Lemma 2.10 shows that ⟨⟨∙,∙⟩⟩\documentclass[12pt]{minimal}
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\begin{document}$$\langle \!\langle \bullet ,\bullet \rangle \!\rangle $$\end{document} defines a nondegenerate pairing on Res01×Res0∗3\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_0\times {{\,\mathrm{Res}\,}}^3_{0*}$$\end{document}, which by Lemma 2.2 shows that the semisimplicity condition (2.41) holds at λ0=0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0=0$$\end{document} for the operator P1,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{1,0}$$\end{document}. By (2.45) semisimplicity holds for P3,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{3,0}$$\end{document} as well.
2. We first show that Res02\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^2_0$$\end{document} consists of closed forms. Assume that ζ∈Res02\documentclass[12pt]{minimal}
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\begin{document}$$\zeta \in {{\,\mathrm{Res}\,}}^2_0$$\end{document}, then dζ∈Res03\documentclass[12pt]{minimal}
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\begin{document}$$d\zeta \in {{\,\mathrm{Res}\,}}^3_0$$\end{document}. By (2.45), dζ=dα∧u\documentclass[12pt]{minimal}
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\begin{document}$$d\zeta =d\alpha \wedge u$$\end{document} for some u∈Res01\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^1_0$$\end{document}. Take arbitrary u∗∈Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$$u_*\in {{\,\mathrm{Res}\,}}^1_{0*}$$\end{document}. Then⟨⟨u,dα∧u∗⟩⟩=∫Mα∧dζ∧u∗=∫Mdα∧ζ∧u∗=0\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \!\langle u,d\alpha \wedge u_*\rangle \!\rangle =\int _M \alpha \wedge d\zeta \wedge u_* =\int _M d\alpha \wedge \zeta \wedge u_*=0 \end{aligned}$$\end{document}Here in the second equality we integrate by parts and use that du∗=0\documentclass[12pt]{minimal}
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\begin{document}$$du_*=0$$\end{document}; in the last equality we use that ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} applied to the 5-form under the integral is equal to 0. Now by Lemma 2.10 we have u=0\documentclass[12pt]{minimal}
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\begin{document}$$u=0$$\end{document}, which means that dζ=0\documentclass[12pt]{minimal}
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\begin{document}$$d\zeta =0$$\end{document} as needed.
Next, by Lemma 2.6 we have kerπ2⊂d(Res1,∞)\documentclass[12pt]{minimal}
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\begin{document}$$\ker \pi _2\subset d({{\,\mathrm{Res}\,}}^{1,\infty })$$\end{document}. By (2.43), Lemma 2.4, and the fact that Res01,∞=Res01\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{1,\infty }_0={{\,\mathrm{Res}\,}}^1_0$$\end{document} we have Res1,∞=Res01⊕Cα\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{1,\infty }={{\,\mathrm{Res}\,}}^1_0\oplus {\mathbb {C}}\alpha $$\end{document}. Since d(Res01)=0\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}^1_0)=0$$\end{document} and dα∈kerπ2\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \in \ker \pi _2$$\end{document}, we see that kerπ2\documentclass[12pt]{minimal}
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\begin{document}$$\ker \pi _2$$\end{document} is spanned by dα\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha $$\end{document}.
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\begin{document}$$\pi _2$$\end{document} is onto, it suffices to use Lemma 2.7: since all elements of Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_{0*}$$\end{document} are closed, all elements of Res0∗3=dα∧Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^3_{0*}=d\alpha \wedge {{\,\mathrm{Res}\,}}^1_{0*}$$\end{document} are exact.
3. This follows immediately from the above statements and Lemma 2.8. To show that π3=0\documentclass[12pt]{minimal}
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\begin{document}$$\pi _3=0$$\end{document} we note that Res03=dα∧Res01\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^3_0=d\alpha \wedge {{\,\mathrm{Res}\,}}^1_0$$\end{document} consists of exact forms. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Summary
We now briefly summarize the contents of this section. Lemma 2.2 will often be used to interpret the semisimplicity condition (2.41) via the more tractable nondegeneracy of the pairing (2.9). Next, Lemma 2.4 provides us with a definitive understanding of Res00,∞\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}_0^{0, \infty }$$\end{document} and Res04,∞\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}_0^{4, \infty }$$\end{document}, which by the isomorphisms (2.49) reduces the problem to studying Res01,∞\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}_0^{1, \infty }$$\end{document} and Res02,∞\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}_0^{2, \infty }$$\end{document}. As Theorem 1 shows, this is a complicated question, but Lemma 2.8 says that Res01∩kerd\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}_0^1 \cap \ker d$$\end{document} is ‘stably topological’, that is, it is always mapped isomorphically by π1\documentclass[12pt]{minimal}
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\begin{document}$$H^1(M)$$\end{document}. Moreover, if one can show d(Res01)=0\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}_0^1) = 0$$\end{document}, Lemma 2.11 shows that semisimplicity for 1-forms is valid, which will be used in the perturbed picture in § 4. Under the same assumption, we also know that Res02\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}_0^2$$\end{document} is spanned by the ‘topological part’ π2-1(H2(M))\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha $$\end{document}. Thus, to compute (2.59) it suffices to study conditions under which forms in Res01\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}_0^1$$\end{document} are closed, and semisimplicity conditions for P2,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{2, 0}$$\end{document}. This will be done in two steps: in § 3 we will first develop a detailed understanding when φt\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _t$$\end{document} is the geodesic flow of a hyperbolic 3-manifold, and later in § 4 we will study the perturbed picture.
Resonant states for hyperbolic 3-manifolds
In this section we study in detail the Pollicott–Ruelle resonant states at 0 for geodesic flows on hyperbolic 3-manifolds. The theorem below summarizes the main results. Here Res0k=Res0k,1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^k_0={{\,\mathrm{Res}\,}}^{k,1}_0$$\end{document} are the spaces of resonant k-forms, Res0k,ℓ\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\ell }_0$$\end{document} are the spaces of generalized resonant k-forms (see §2.4), and πk:Res0k∩kerd→Hk(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k:{{\,\mathrm{Res}\,}}^k_0\cap \ker d\rightarrow H^k(M;{\mathbb {C}})$$\end{document} are the maps defined in (2.61). The maps πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}$$\end{document} are defined in §2.2.2.
Theorem 2
Let M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _t$$\end{document} be the geodesic flow on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}. Then:
There exists a 2-form ψ∈C∞(M;Ω02)\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _t$$\end{document}.
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\begin{document}$$d{\mathcal {C}}_\psi $$\end{document}.
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\begin{document}$$[\pi _\Sigma ^*d{{\,\mathrm{vol}\,}}_g]_{H^3}$$\end{document}.
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\begin{document}$$d\alpha \wedge {\mathcal {C}}_\psi $$\end{document} onto the space of harmonic 1-forms on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}.
Theorem 2 together with Lemma 2.4 and (2.59) give part 1 of Theorem 1:
Corollary 3.1
Theorem 2, the algebraic multiplicities of 0 as a resonance of the operators Pk,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{k,0}$$\end{document} arem0,0(0)=m4,0(0)=1,m1,0(0)=m3,0(0)=2b1(Σ),m2,0(0)=2b1(Σ)+2\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&m_{0,0}(0)=m_{4,0}(0)=1,\quad m_{1,0}(0)=m_{3,0}(0)=2b_1(\Sigma ),\nonumber \\&\quad m_{2,0}(0)=2b_1(\Sigma )+2 \end{aligned}$$\end{document}and the order of vanishing of the Ruelle zeta function ζR\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _{\mathrm R}$$\end{document} at 0 is equal tomR(0)=2m0,0(0)-2m1,0(0)+m2,0(0)=4-2b1(Σ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_{\mathrm R}(0)=2m_{0,0}(0)-2m_{1,0}(0)+m_{2,0}(0)=4-2b_1(\Sigma ). \end{aligned}$$\end{document}
Previously (3.1) was proved in [16, Proposition 7.7] using different methods. Here we give a more refined description: we construct the resonant forms, prove pairing formulas, and study the existence of Jordan blocks. We emphasize that these properties are of crucial importance for the perturbation arguments in § 4 and were not known prior to this work.
This section is structured as follows: in §3.1 we review the geometric features of hyperbolic 3-manifolds used here. In §3.2 we construct the smooth invariant 2-form ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} and study its properties, proving part 1 of Theorem 2. In §3.3 we study the resonant 1-forms and 3-forms, proving parts 2, 3, and 6 of Theorem 2. In §3.4 we study the resonant 2-forms, proving parts 4 and 5 of Theorem 2. Finally, in §3.5 we show that the pushforward operator πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document}, proving part 7 of Theorem 2.
Hyperbolic 3-manifolds
We first review the geometry of hyperbolic 3-manifolds, following [14, §3]. We define a hyperbolic 3-manifold to be a nonempty compact connected oriented 3-dimensional Riemannian manifold Σ\documentclass[12pt]{minimal}
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Geodesic flow
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\begin{document}$$S{\mathbb {H}}^3 \simeq {{\,\mathrm{SO}\,}}_+(1,3)/{{\,\mathrm{SO}\,}}(2)$$\end{document} as a homogeneous space for the SO+(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{SO}\,}}_+(1, 3)$$\end{document}-action, since SO(2)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{SO}\,}}(2)$$\end{document} is the stabilizer of the point (1,0,0,0,0,1,0,0)∈SH3\documentclass[12pt]{minimal}
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\begin{document}$$(1, 0, 0, 0, 0, 1, 0, 0) \in S{\mathbb {H}}^3$$\end{document}. The contact form α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}, defined in (2.11), and the generator X of the geodesic flow areα=-⟨v,dx⟩1,3,X=v·∂x+x·∂v,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha =-\langle v,dx\rangle _{1,3},\quad X=v\cdot \partial _x+x\cdot \partial _v, \end{aligned}$$\end{document}where ‘·\documentclass[12pt]{minimal}
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\begin{document}$$\cdot $$\end{document}’ denotes the (positive definite) Euclidean inner product on R1,3\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^{1,3}$$\end{document}. The geodesic flow is then given byφt(x,v)=(xcosht+vsinht,xsinht+vcosht).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varphi _t(x,v)=(x\cosh t+v\sinh t,x\sinh t+v\cosh t). \end{aligned}$$\end{document}As a corollary, the distance function on H3\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {H}}^3$$\end{document} with respect to the hyperbolic metric is given bycoshdH3(x,y)=⟨x,y⟩1,3for allx,y∈H3.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \cosh d_{{\mathbb {H}}^3}(x,y)=\langle x,y\rangle _{1,3}\quad \text {for all}\quad x,y\in {\mathbb {H}}^3. \end{aligned}$$\end{document}The tangent space T(x,v)(SH3)\documentclass[12pt]{minimal}
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\begin{document}$$T_{(x,v)}(S{\mathbb {H}}^3)$$\end{document} consists of vectors (ξx,ξv)∈R1,3⊕R1,3\documentclass[12pt]{minimal}
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\begin{document}$$(\xi _x,\xi _v)\in {\mathbb {R}}^{1,3}\oplus {\mathbb {R}}^{1,3}$$\end{document} such that⟨x,ξx⟩1,3=⟨v,ξv⟩1,3=⟨x,ξv⟩1,3+⟨v,ξx⟩1,3=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle x,\xi _x\rangle _{1,3}=\langle v,\xi _v\rangle _{1,3}=\langle x,\xi _v\rangle _{1,3}+\langle v,\xi _x\rangle _{1,3}=0. \end{aligned}$$\end{document}The connection map (2.14) is given byK(x,v)(ξx,ξv)=ξv-⟨x,ξv⟩1,3x=ξv+⟨v,ξx⟩1,3x.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {K}}(x,v)(\xi _x,\xi _v)=\xi _v-\langle x,\xi _v\rangle _{1,3} \,x =\xi _v+\langle v,\xi _x\rangle _{1,3} x. \end{aligned}$$\end{document}Here and throughout we note that the addition of points x and vectors ξv\documentclass[12pt]{minimal}
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\begin{document}$$\xi _v$$\end{document} (or ξx\documentclass[12pt]{minimal}
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\begin{document}$$\xi _x$$\end{document}) has to be understood in R1,3\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^{1, 3}$$\end{document}. The horizontal and vertical spaces H(x,v),V(x,v)⊂T(x,v)(SH3)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {H}}(x,v),{\mathbf {V}}(x,v)\subset T_{(x,v)}(S{\mathbb {H}}^3)$$\end{document} are thenH(x,v)={(ξx,ξv)∣⟨x,ξx⟩1,3=0,ξv=-⟨v,ξx⟩1,3x},V(x,v)={(0,ξv)∣⟨x,ξv⟩1,3=⟨v,ξv⟩1,3=0}\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} {\mathbf {H}}(x,v)&=\{(\xi _x,\xi _v)\mid \langle x,\xi _x\rangle _{1,3}=0,\ \xi _v=-\langle v,\xi _x\rangle _{1,3}\, x\},\\ {\mathbf {V}}(x,v)&=\{(0,\xi _v)\mid \langle x,\xi _v\rangle _{1,3}=\langle v,\xi _v\rangle _{1,3}=0\} \end{aligned} \end{aligned}$$\end{document}and the horizontal-vertical splitting map (2.15) takes for ξ=(ξx,ξv)∈T(x,v)(SH3)⊂R1,3⊕R1,3\documentclass[12pt]{minimal}
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\begin{document}$$\xi = (\xi _x,\xi _v) \in T_{(x, v)}(S{\mathbb {H}}^3) \subset {\mathbb {R}}^{1,3}\oplus {\mathbb {R}}^{1,3}$$\end{document} the formξH=ξx,ξV=ξv+⟨v,ξx⟩1,3x.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \xi _H=\xi _x,\quad \xi _V=\xi _v+\langle v,\xi _x\rangle _{1,3}\,x. \end{aligned}$$\end{document}The Sasaki metric (2.17) is for ξ,η∈T(x,v)(SH3)\documentclass[12pt]{minimal}
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\begin{document}$$\xi , \eta \in T_{(x, v)}(S{\mathbb {H}}^3)$$\end{document} given by⟨ξ,η⟩S=-⟨ξx,ηx⟩1,3-⟨ξv,ηv⟩1,3+⟨v,ξx⟩1,3⟨v,ηx⟩1,3.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \xi ,\eta \rangle _S=-\langle \xi _x,\eta _x\rangle _{1,3}-\langle \xi _v,\eta _v\rangle _{1,3}+\langle v,\xi _x\rangle _{1,3}\langle v,\eta _x\rangle _{1,3}. \end{aligned}$$\end{document}The unstable/stable subspaces Eu,Es\documentclass[12pt]{minimal}
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\begin{document}$$E_u,E_s$$\end{document} from (2.2) on SH3\documentclass[12pt]{minimal}
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\begin{document}$$S{\mathbb {H}}^3$$\end{document} are given byEu(x,v)={(w,w)∣w∈R1,3,⟨w,x⟩1,3=⟨w,v⟩1,3=0},Es(x,v)={(w,-w)∣w∈R1,3,⟨w,x⟩1,3=⟨w,v⟩1,3=0}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} E_u(x,v)&=\{(w,w)\mid w\in {\mathbb {R}}^{1,3},\ \langle w,x\rangle _{1,3}=\langle w,v\rangle _{1,3}=0\},\\ E_s(x,v)&=\{(w,-w)\mid w\in {\mathbb {R}}^{1,3},\ \langle w,x\rangle _{1,3}=\langle w,v\rangle _{1,3}=0\}. \end{aligned} \end{aligned}$$\end{document}In terms of the horizontal-vertical splitting (2.15) they can be characterized as follows:Eu={ξV=ξH},Es={ξV=-ξH}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} E_u=\{\xi _V=\xi _H\},\quad E_s=\{\xi _V=-\xi _H\}. \end{aligned}$$\end{document}A distinguished feature of hyperbolic manifolds is that the restriction of the differential of the geodesic flow to the unstable/stable spaces is conformal with respect to the Sasaki metric:|dφt(x,v)ξ|S=et|ξ|S,ξ∈Eu(x,v);e-t|ξ|S,ξ∈Es(x,v).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} |d\varphi _t(x,v)\xi |_S={\left\{ \begin{array}{ll} e^t|\xi |_S,&{} \xi \in E_u(x,v);\\ e^{-t}|\xi |_S,&{} \xi \in E_s(x,v).\end{array}\right. } \end{aligned}$$\end{document}The objects discussed above are invariant under the action of SO+(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{SO}\,}}_+(1,3)$$\end{document} and thus descend naturally to the quotients Σ,SΣ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma ,S\Sigma $$\end{document}.
The frame bundle and canonical vector fields
A convenient tool for computations on M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$M=S\Sigma $$\end{document} is the frame bundleFΣ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}\Sigma $$\end{document}, consisting of quadruples (x,v1,v2,v3)\documentclass[12pt]{minimal}
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\begin{document}$$(x,v_1,v_2,v_3)$$\end{document} where x∈Σ\documentclass[12pt]{minimal}
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\begin{document}$$x\in \Sigma $$\end{document} and v1,v2,v3∈TxΣ\documentclass[12pt]{minimal}
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\begin{document}$$v_1,v_2,v_3\in T_x\Sigma $$\end{document} form a positively oriented orthonormal basis. We haveFΣ=Γ\FH3,FH3≃SO+(1,3),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {F}}\Sigma =\Gamma \backslash {\mathcal {F}}{\mathbb {H}}^3,\quad {\mathcal {F}}{\mathbb {H}}^3\simeq {{\,\mathrm{SO}\,}}_+(1,3), \end{aligned}$$\end{document}where the frame bundle FH3\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}{\mathbb {H}}^3$$\end{document} is identified with the group SO+(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{SO}\,}}_+(1,3)$$\end{document} by the following map (where e0=(1,0,0,0),e1=(0,1,0,0),⋯\documentclass[12pt]{minimal}
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\begin{document}$$e_0=(1,0,0,0),e_1=(0,1,0,0),\dots $$\end{document})γ∈SO+(1,3)↦(γe0,γe1,γe2,γe3).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \gamma \in {{\,\mathrm{SO}\,}}_+(1,3)\mapsto (\gamma e_0,\gamma e_1,\gamma e_2,\gamma e_3). \end{aligned}$$\end{document}Under this identification, the action of SO+(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{SO}\,}}_+(1,3)$$\end{document} on FH3\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}{\mathbb {H}}^3$$\end{document} corresponds to the action of this group on itself by left multiplications. Therefore, SO+(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{SO}\,}}_+(1,3)$$\end{document}-invariant vector fields on FH3\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}{\mathbb {H}}^3$$\end{document} correspond to left-invariant vector fields on the group SO+(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{SO}\,}}_+(1,3)$$\end{document}, that is to elements of its Lie algebra so(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{\mathfrak {so}}\,}}(1,3)$$\end{document}. We define the basis of left-invariant vector fields on SO+(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{SO}\,}}_+(1,3)$$\end{document} corresponding to the following matrices in so(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{\mathfrak {so}}\,}}(1,3)$$\end{document}:X=0100100000000000,R=00000000000100-10,U1+=00-1000-10-11000000,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} X&=\begin{pmatrix} 0 &{} 1 &{} 0 &{} 0\\ 1 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{pmatrix},&R&=\begin{pmatrix} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 1\\ 0 &{} 0 &{} -1 &{} 0\\ \end{pmatrix},&U^+_1&=\begin{pmatrix} 0 &{} 0 &{} -1 &{} 0\\ 0 &{} 0 &{} -1 &{} 0\\ -1 &{} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{pmatrix}, \end{aligned}$$\end{document}U2+=000-1000-10000-1100,U1-=00-100010-1-1000000,U2-=000-100010000-1-100.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} U^+_2&=\begin{pmatrix} 0 &{} 0 &{} 0 &{} -1\\ 0 &{} 0 &{} 0 &{} -1\\ 0 &{} 0 &{} 0 &{} 0\\ -1 &{} 1 &{} 0 &{} 0\\ \end{pmatrix},&U^-_1&=\begin{pmatrix} 0 &{} 0 &{} -1 &{} 0\\ 0 &{} 0 &{} 1 &{} 0\\ -1 &{} -1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{pmatrix},&U^-_2&=\begin{pmatrix} 0 &{} 0 &{} 0 &{} -1\\ 0 &{} 0 &{} 0 &{} 1\\ 0 &{} 0 &{} 0 &{} 0\\ -1 &{} -1 &{} 0 &{} 0\\ \end{pmatrix}. \end{aligned}$$\end{document}Under the identification (3.8), and considering FH3\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}{\mathbb {H}}^3$$\end{document} as a submanifold of (R1,3)4\documentclass[12pt]{minimal}
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\begin{document}$$({\mathbb {R}}^{1,3})^4$$\end{document}, we can write using coordinates (x,v1,v2,v3)∈(R1,3)4\documentclass[12pt]{minimal}
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\begin{document}$$(x, v_1, v_2, v_3) \in ({\mathbb {R}}^{1, 3})^4$$\end{document} and writing ‘·\documentclass[12pt]{minimal}
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\begin{document}$$\cdot $$\end{document}’ for the Euclidean inner productX=v1·∂x+x·∂v1,R=v2·∂v3-v3·∂v2,U1±=-v2·∂x-x·∂v2±(v2·∂v1-v1·∂v2),U2±=-v3·∂x-x·∂v3±(v3·∂v1-v1·∂v3).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} X&=v_1\cdot \partial _x+x\cdot \partial _{v_1},\quad R=v_2\cdot \partial _{v_3}-v_3\cdot \partial _{v_2},\\ U_1^\pm&=-v_2\cdot \partial _x-x\cdot \partial _{v_2}\pm (v_2\cdot \partial _{v_1}-v_1\cdot \partial _{v_2}),\\ U_2^\pm&=-v_3\cdot \partial _x-x\cdot \partial _{v_3}\pm (v_3\cdot \partial _{v_1}-v_1\cdot \partial _{v_3}). \end{aligned} \end{aligned}$$\end{document}Since the vector fields above are invariant under the action of SO+(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{SO}\,}}_+(1,3)$$\end{document}, they descend to the frame bundle of the quotient, FΣ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}\Sigma $$\end{document}.
The commutation relations between these fields are (as can be seen by computing the commutators of the corresponding matrices, or by using the explicit formulas above)[X,Ui±]=±Ui±,[Ui+,Ui-]=2X,[U1±,U2∓]=2R,[X,R]=[U1±,U2±]=0,[R,U1±]=-U2±,[R,U2±]=U1±.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&[X,U_{i}^{\pm }]=\pm U_i^\pm , \quad [U_i^+,U_i^-]=2X, \quad [U_1^\pm ,U_2^\mp ]=2R,\nonumber \\&[X,R]=[U_1^\pm ,U_2^\pm ]=0,\quad [R,U_1^\pm ]=-U_2^\pm , \quad [R,U_2^\pm ]=U_1^\pm . \end{aligned}$$\end{document}The mapπF:(x,v1,v2,v3)∈FΣ↦(x,v1)∈SΣ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _{{\mathcal {F}}}:(x,v_1,v_2,v_3)\in {\mathcal {F}}\Sigma \mapsto (x,v_1)\in S\Sigma \end{aligned}$$\end{document}is a submersion, with one-dimensional fibers whose tangent spaces are spanned by the field R. Thus, if a vector field on FΣ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}\Sigma $$\end{document} commutes with R then this vector field descends to the sphere bundle SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document}. In particular, the vector field X descends to the generator of the geodesic flow (which we also denote by X).
The vector fields Ui±\documentclass[12pt]{minimal}
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\begin{document}$$U_i^\pm $$\end{document} do not commute with R and thus do not descend to SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document}. However, the vector space span(U1+,U2+)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{span}\,}}(U_1^+,U_2^+)$$\end{document} is R-invariant and descends to the stable space Es\documentclass[12pt]{minimal}
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\begin{document}$$E_s$$\end{document} on SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document}. Similarly, the space span(U1-,U2-)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{span}\,}}(U_1^-,U_2^-)$$\end{document} descends to Eu\documentclass[12pt]{minimal}
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\begin{document}$$E_u$$\end{document}. Because of this we think of U1+,U2+\documentclass[12pt]{minimal}
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\begin{document}$$U_1^+,U_2^+$$\end{document} as stable vector fields and U1-,U2-\documentclass[12pt]{minimal}
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\begin{document}$$U_1^-,U_2^-$$\end{document} as unstable vector fields.
Canonical differential forms
We next introduce the frame of canonical differential 1-forms on FΣ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}\Sigma $$\end{document}α,R∗,U1±∗,U2±∗\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha ,\ R^*,\ U_1^{\pm *},\ U_2^{\pm *} \end{aligned}$$\end{document}which is defined as a dual frame for the vector fields X,R,U1∓,U2∓\documentclass[12pt]{minimal}
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\begin{document}$$X,R,U_1^\mp ,U_2^\mp $$\end{document}, in the sense compatible with the definition of the dual stable/unstable bundles (2.4), as follows:⟨α,X⟩=⟨R∗,R⟩=⟨U1±∗,U1∓⟩=⟨U2±∗,U2∓⟩=1\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \alpha ,X\rangle =\langle R^*,R\rangle =\langle U_1^{\pm *},U_1^{\mp }\rangle =\langle U_2^{\pm *},U_2^\mp \rangle =1 \end{aligned}$$\end{document}and all the other pairings between the 1-forms and the vector fields in question are equal to 0. In particular, ⟨Ui±∗,Ui±⟩=0\documentclass[12pt]{minimal}
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\begin{document}$$\langle U_i^{\pm *},U_i^\pm \rangle =0$$\end{document}.
Using the following identity valid for any 1-form β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document} and any two vector fields Y, Zdβ(Y,Z)=Yβ(Z)-Zβ(Y)-β([Y,Z]),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d\beta (Y,Z)=Y\beta (Z)-Z\beta (Y)-\beta ([Y,Z]), \end{aligned}$$\end{document}the commutation relations (3.9), and the duality relations (3.10), we compute the differentials of the canonical forms:dα=2(U1+∗∧U1-∗+U2+∗∧U2-∗),dR∗=2(U2-∗∧U1+∗+U2+∗∧U1-∗),dU1±∗=±α∧U1±∗-R∗∧U2±∗,dU2±∗=±α∧U2±∗+R∗∧U1±∗.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&d\alpha =2(U_1^{+*}\wedge U_1^{-*}+U_2^{+*}\wedge U_2^{-*}),\quad dR^*=2(U_2^{-*}\wedge U_1^{+*}+U_2^{+*}\wedge U_1^{-*}),\nonumber \\&dU^{\pm *}_1=\pm \alpha \wedge U^{\pm *}_1-R^*\wedge U^{\pm *}_2,\quad dU^{\pm *}_2=\pm \alpha \wedge U^{\pm *}_2+R^*\wedge U^{\pm *}_1. \end{aligned}$$\end{document}It follows thatLXUj±∗=±Uj±∗,LRU1±∗=-U2±∗,LRU2±∗=U1±∗.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {L}}_X U^{\pm *}_j=\pm U^{\pm *}_j,\quad {\mathcal {L}}_R U^{\pm *}_1=-U^{\pm *}_2,\quad {\mathcal {L}}_R U^{\pm *}_2=U^{\pm *}_1. \end{aligned}$$\end{document}If ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} is a differential form on FΣ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}\Sigma $$\end{document}, then ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} descends to SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document} (i.e. it is a pullback by πF\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{{\mathcal {F}}}$$\end{document} of a form on SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document}) if and only if ιRω=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _R\omega =0$$\end{document}, LRω=0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_R\omega =0$$\end{document}. In particular the form α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} on FΣ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}\Sigma $$\end{document} descends to the contact form on SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document}, which we also denote by α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}.
Conformal infinity
Following [14, §3.4] we consider the mapsΦ±:SH3→(0,∞),B±:SH3→S2,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Phi _\pm :S{\mathbb {H}}^3\rightarrow (0,\infty ),\quad B_\pm :S{\mathbb {H}}^3\rightarrow {\mathbb {S}}^2, \end{aligned}$$\end{document}where S2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^3$$\end{document}, defined by the identitiesx±v=Φ±(x,v)(1,B±(x,v))for all(x,v)∈SH3.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} x\pm v=\Phi _\pm (x,v)(1,B_\pm (x,v))\quad \text {for all}\quad (x,v)\in S{\mathbb {H}}^3. \end{aligned}$$\end{document}Note that B±(x,v)\documentclass[12pt]{minimal}
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\begin{document}$$t\rightarrow \pm \infty $$\end{document} of the projection to H3\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {H}}^3$$\end{document}. Let(S2×S2)-:={(ν-,ν+)∈S2×S2∣ν-≠ν+}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} ({\mathbb {S}}^2\times {\mathbb {S}}^2)_-:= \{(\nu _-,\nu _+)\in {\mathbb {S}}^2\times {\mathbb {S}}^2\mid \nu _-\ne \nu _+\}. \end{aligned}$$\end{document}In fact, the maps B±\documentclass[12pt]{minimal}
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\begin{document}$$B_\pm $$\end{document} yield the following diffeomorphism of SH3\documentclass[12pt]{minimal}
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\begin{document}$$S{\mathbb {H}}^3$$\end{document} (see [14, (3.24)]):Ξ:SH3∋(y,v)↦(ν-,ν+,t)∈(S2×S2)-×Rwithν±=B±(y,v),t=12log(Φ+(y,v)Φ-(y,v)).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned}&\Xi :S{\mathbb {H}}^3 \ni (y, v) \mapsto (\nu _-, \nu _+, t) \in ({\mathbb {S}}^2\times {\mathbb {S}}^2)_-\times {\mathbb {R}}\\&\quad \text {with}\quad \nu _\pm =B_\pm (y,v),\quad t={1\over 2}\log \Big ({\Phi _+(y,v)\over \Phi _-(y,v)}\Big ). \end{aligned} \end{aligned}$$\end{document}The geometric interpretation of Ξ\documentclass[12pt]{minimal}
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\begin{document}$$e_0$$\end{document} on that geodesic (as can be seen from (5.30) below and noting that Xt=1\documentclass[12pt]{minimal}
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\begin{document}$$Xt=1$$\end{document} by (3.22)).
We have the identity [14, (3.23)]Φ-(x,v)Φ+(x,v)|B-(x,v)-B+(x,v)|2=4,\documentclass[12pt]{minimal}
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\begin{document}$$|\bullet |$$\end{document} denotes the Euclidean distance on R3⊃S2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^3\supset {\mathbb {S}}^2$$\end{document}.
We also introduce the Poisson kernelP(x,ν)=(⟨x,(1,ν)⟩1,3)-1>0,x∈H3,ν∈S2⊂R3.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} P(x,\nu )=\big (\langle x,(1,\nu )\rangle _{1,3}\big )^{-1}>0,\quad x\in {\mathbb {H}}^3,\quad \nu \in {\mathbb {S}}^2\subset {\mathbb {R}}^3. \end{aligned}$$\end{document}The following relations hold [14, (3.21)]:Φ±(x,v)=P(x,B±(x,v)).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Phi _\pm (x, v) = P(x, B_\pm (x,v)). \end{aligned}$$\end{document}If we fix x∈H3\documentclass[12pt]{minimal}
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\begin{document}$$v\mapsto B_\pm (x,v)$$\end{document} are diffeomorphisms from the fiber SxH3\documentclass[12pt]{minimal}
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\begin{document}$$S_x{\mathbb {H}}^3$$\end{document} onto S2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {S}}^2$$\end{document}. The inverse maps are given by ν↦v±(x,ν)\documentclass[12pt]{minimal}
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\begin{document}$$\nu \mapsto v_\pm (x,\nu )$$\end{document} where [14, (3.20)]v±(x,ν)=∓x±P(x,ν)(1,ν)∈SxH3,B±(x,v±(x,ν))=ν.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} v_\pm (x, \nu ) = \mp x \pm P(x, \nu ) (1, \nu ) \in S_x {\mathbb {H}}^3,\quad B_\pm (x,v_\pm (x,\nu ))=\nu . \end{aligned}$$\end{document}The diffeomorphisms v↦B±(x,v)\documentclass[12pt]{minimal}
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\begin{document}$$|\bullet |_{{\mathbb {S}}^2}$$\end{document}: by [14, (3.22)]) we have|∂vB±(x,v)η|S2=|η|gΦ±(x,v)for allη∈Tv(SxH3).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} |\partial _v B_\pm (x,v) \eta |_{{\mathbb {S}}^2}={|\eta |_{g}\over \Phi _\pm (x,v)} \quad \text {for all}\quad \eta \in T_v(S_x{\mathbb {H}}^3). \end{aligned}$$\end{document}Next, we have by (3.3) and (3.5)XΦ±=±Φ±,dΦ-|Eu=dΦ+|Es=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} X\Phi _\pm =\pm \Phi _\pm ,\quad d\Phi _-|_{E_u}=d\Phi _+|_{E_s}=0. \end{aligned}$$\end{document}The maps B±\documentclass[12pt]{minimal}
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\begin{document}$$B_\pm $$\end{document} are submersions with connected fibers, the tangent spaces to which are described in terms of the stable/unstable decomposition (2.2) as follows: for each ν∈S2\documentclass[12pt]{minimal}
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\begin{document}$$\nu \in {\mathbb {S}}^2$$\end{document}T(B+-1(ν))=(E0⊕Es)|B+-1(ν),T(B--1(ν))=(E0⊕Eu)|B--1(ν).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} T (B_+^{-1}(\nu ))=(E_0\oplus E_s)|_{B_+^{-1}(\nu )},\quad T (B_-^{-1}(\nu ))=(E_0\oplus E_u)|_{B_-^{-1}(\nu )}.\nonumber \\ \end{aligned}$$\end{document}This can be checked using (3.5), see [14, (3.25)]. The action of the differential dB+\documentclass[12pt]{minimal}
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\begin{document}$$dB_+$$\end{document} on Eu\documentclass[12pt]{minimal}
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\begin{document}$$dB_-$$\end{document} on Es\documentclass[12pt]{minimal}
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\begin{document}$$E_s$$\end{document}, can be described as follows: for any (x,v)∈SH3\documentclass[12pt]{minimal}
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\begin{document}$$(x,v)\in S{\mathbb {H}}^3$$\end{document} and w∈R1,3\documentclass[12pt]{minimal}
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\begin{document}$$w\in {\mathbb {R}}^{1,3}$$\end{document} such that ⟨x,w⟩1,3=⟨v,w⟩1,3=0\documentclass[12pt]{minimal}
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\begin{document}$$\langle x,w\rangle _{1,3}=\langle v,w\rangle _{1,3}=0$$\end{document},dB±(x,v)(w,±w)=2(w′-w0B±(x,v))Φ±(x,v)wherew=(w0,w′).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} dB_\pm (x,v)(w,\pm w)={2(w'-w_0B_\pm (x,v))\over \Phi _\pm (x,v)}\quad \text {where}\quad w=(w_0,w').\nonumber \\ \end{aligned}$$\end{document}We next briefly discuss the action of the group SO+(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {S}}^2$$\end{document}, referring to [14, §3.5] for details. For any γ∈SO+(1,3)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma \in {{\,\mathrm{SO}\,}}_+(1,3)$$\end{document}, defineNγ:S2→(0,∞),Lγ:S2→S2\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} N_\gamma :{\mathbb {S}}^2\rightarrow (0,\infty ),\quad L_\gamma :{\mathbb {S}}^2\rightarrow {\mathbb {S}}^2 \end{aligned}$$\end{document}by the identity (where on the left is the linear action of γ\documentclass[12pt]{minimal}
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\begin{document}$$(1,\nu )\in {\mathbb {R}}^{1,3}$$\end{document})γ·(1,ν)=Nγ(ν)(1,Lγ(ν))for allν∈S2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \gamma \cdot (1,\nu )=N_\gamma (\nu )(1,L_\gamma (\nu ))\quad \text {for all}\quad \nu \in {\mathbb {S}}^2. \end{aligned}$$\end{document}The maps Lγ\documentclass[12pt]{minimal}
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\begin{document}$$L_\gamma $$\end{document} define an action of SO+(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {S}}^2$$\end{document}. This action is transitive and the stabilizer of e1∈S2\documentclass[12pt]{minimal}
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\begin{document}$$A\in {{\,\mathrm{SO}\,}}_+(1,3)$$\end{document} such that A(1,1,0,0)T=τ(1,1,0,0)T\documentclass[12pt]{minimal}
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\begin{document}$$A(1,1,0,0)^T=\tau (1,1,0,0)^T$$\end{document} for some τ>0\documentclass[12pt]{minimal}
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\begin{document}$$\tau >0$$\end{document}, which may be shown to be isomorphic to the group of similarities of the plane Sim(2)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Sim}(2)$$\end{document}, giving S2≃SO+(1,3)/Sim(2)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {S}}^2 \simeq {{\,\mathrm{SO}\,}}_+(1, 3)/\mathrm {Sim}(2)$$\end{document} the structure of a homogeneous space.
This action is by orientation preserving conformal transformations, more precisely|dLγ(ν)ζ|S2=|ζ|S2Nγ(ν)for all(ν,ζ)∈TS2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} |dL_\gamma (\nu )\zeta |_{{\mathbb {S}}^2}={|\zeta |_{{\mathbb {S}}^2}\over N_\gamma (\nu )}\quad \text {for all}\quad (\nu ,\zeta )\in T{\mathbb {S}}^2. \end{aligned}$$\end{document}Moreover, the maps B±\documentclass[12pt]{minimal}
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\begin{document}$$B_\pm $$\end{document} have the equivariance propertyB±(γ·(x,v))=Lγ(B±(x,v))for all(x,v)∈SH3.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} B_\pm (\gamma \cdot (x,v))=L_\gamma (B_\pm (x,v))\quad \text {for all}\quad (x,v)\in S{\mathbb {H}}^3. \end{aligned}$$\end{document}We finally use the maps B±\documentclass[12pt]{minimal}
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\begin{document}$$B_\pm $$\end{document} to describe a special class of differential forms on SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document} defined as follows (c.f. [14, 44]):
Definition 3.2
We call a k-form u∈D′(SΣ;Ω0k)\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {D}}'(S\Sigma ;\Omega ^k_0)$$\end{document}stable if it is a section of ∧kEs∗⊂Ω0k\documentclass[12pt]{minimal}
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\begin{document}$$\wedge ^k E_s^*\subset \Omega ^k_0$$\end{document} where Es∗⊂T∗(SΣ)\documentclass[12pt]{minimal}
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\begin{document}$$E_s^*\subset T^*(S\Sigma )$$\end{document} is the annihilator of E0⊕Es\documentclass[12pt]{minimal}
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\begin{document}$$E_0\oplus E_s$$\end{document} (see (2.4)). We call uunstable if it is a section of ∧kEu∗\documentclass[12pt]{minimal}
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\begin{document}$$\wedge ^k E_u^*$$\end{document} where Eu∗\documentclass[12pt]{minimal}
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\begin{document}$$E_u^*$$\end{document} is the annihilator of E0⊕Eu\documentclass[12pt]{minimal}
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\begin{document}$$E_0\oplus E_u$$\end{document}.
We call a form utotally (un)stable if both u and du are (un)stable.
The lemma below (see also [44, §§2.3–2.4]) shows that totally (un)stable k-forms on SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document}, Σ=Γ\H3\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma =\Gamma \backslash {\mathbb {H}}^3$$\end{document}, correspond to Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}-invariant k-forms on S2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {S}}^2$$\end{document}. Denote by πΓ:SH3→SΣ\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Gamma :S{\mathbb {H}}^3\rightarrow S\Sigma $$\end{document} the covering map.
Lemma 3.3
Let u∈D′(SΣ;Ω0k)\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {D}}'(S\Sigma ;\Omega ^k_0)$$\end{document} be totally stable. Then the lift πΓ∗u\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Gamma ^* u$$\end{document} has the formπΓ∗u=B+∗wwherew∈D′(S2;Ωk),Lγ∗w=wfor allγ∈Γ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _\Gamma ^*u=B_+^* w\quad \text {where}\quad w\in {\mathcal {D}}'({\mathbb {S}}^2;\Omega ^k),\, \, L_\gamma ^*w=w\quad \text {for all}\quad \gamma \in \Gamma . \end{aligned}$$\end{document}Conversely, each form B+∗w\documentclass[12pt]{minimal}
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\begin{document}$$B_+^*w$$\end{document}, where w satisfies (3.27), is the lift of a totally stable k-form on SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document}. A similar statement holds for totally unstable forms, with B+\documentclass[12pt]{minimal}
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\begin{document}$$B_+$$\end{document} replaced by B-\documentclass[12pt]{minimal}
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\begin{document}$$B_-$$\end{document}.
Proof
We only consider the case of totally stable forms, with totally unstable forms handled similarly. First of all, note that lifts of totally stable k-forms on SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document} are exactly the Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}-invariant totally stable k-forms on SH3\documentclass[12pt]{minimal}
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\begin{document}$$S{\mathbb {H}}^3$$\end{document}. Next, by (3.23), a k-form ζ∈D′(SH3;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$\zeta \in {\mathcal {D}}'(S{\mathbb {H}}^3;\Omega ^k)$$\end{document} is totally stable if and only if ιYζ=0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_Y \zeta =0$$\end{document} for any vector field Y tangent to the fibers of the map B+\documentclass[12pt]{minimal}
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\begin{document}$$B_+$$\end{document}, which is equivalent to saying that ζ=B+∗w\documentclass[12pt]{minimal}
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\begin{document}$$\zeta =B_+^* w$$\end{document} for some w∈D′(S2;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$w\in {\mathcal {D}}'({\mathbb {S}}^2;\Omega ^k)$$\end{document}. Finally, by (3.26), Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}-invariance of ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document} is equivalent to Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}-invariance of w. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Lemma 3.3 implies thatevery totally stableu∈D′(SΣ;Ω0k)lies inDEs∗′(SΣ;Ω0k),every totally unstableu∈D′(SΣ;Ω0k)lies inDEu∗′(SΣ;Ω0k).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \text {every totally stable }u\in {\mathcal {D}}'(S\Sigma ;\Omega ^k_0)&\quad \text {lies in }{\mathcal {D}}'_{E_s^*}(S\Sigma ;\Omega ^k_0),\\ \text {every totally unstable }u\in {\mathcal {D}}'(S\Sigma ;\Omega ^k_0)&\quad \text {lies in }{\mathcal {D}}'_{E_u^*}(S\Sigma ;\Omega ^k_0). \end{aligned} \end{aligned}$$\end{document}Indeed, assume that u is totally stable. Write πΓ∗u=B+∗w\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Gamma }^*u = B_+^*w$$\end{document} for some w∈D′(S2;Ωk)\documentclass[12pt]{minimal}
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\begin{document}$$w \in {\mathcal {D}}'({\mathbb {S}}^2; \Omega ^k)$$\end{document}, then we have WF(πΓ∗u)=πΓ∗WF(u)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{WF}\,}}(\pi _\Gamma ^*u) = \pi _\Gamma ^*{{\,\mathrm{WF}\,}}(u)$$\end{document} (as πΓ\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Gamma $$\end{document} is a local diffeomorphism). From the behavior of wavefront sets under pullbacks [38, Theorem 8.2.4], we know that WF(πΓ∗u)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{WF}\,}}(\pi _\Gamma ^*u)$$\end{document} is contained in the conormal bundle of the fibers of the submersion B+\documentclass[12pt]{minimal}
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\begin{document}$$B_+$$\end{document}. From (3.23) and (2.4) we then have WF(u)⊂Es∗\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{WF}\,}}(u)\subset E_s^*$$\end{document}. A similar argument works for the totally unstable case.
Additional invariant 2-form
The space of smooth flow invariant 2-forms on SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document} is known to be 2-dimensional, see Lemma 3.7 below, [40, Claim 3.3] or [36], thus there exists a smooth invariant 2-form which is not a multiple of dα\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha $$\end{document}. In this section we introduce such a 2-form ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} and study its properties; these are crucial for the study of Pollicott–Ruelle resonances at zero in §§3.3–3.4 below.
A rotation on Eu⊕Es\documentclass[12pt]{minimal}
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\begin{document}$$E_u\oplus E_s$$\end{document}
Let x∈Σ\documentclass[12pt]{minimal}
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\begin{document}$$x\in \Sigma $$\end{document}. For any two v,w∈TxΣ\documentclass[12pt]{minimal}
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\begin{document}$$v,w\in T_x\Sigma $$\end{document}, we may define their cross productv×w∈TxΣ\documentclass[12pt]{minimal}
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\begin{document}$$v\times w\in T_x\Sigma $$\end{document}, which is uniquely determined by the following properties: v×w\documentclass[12pt]{minimal}
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\begin{document}$$v\times w$$\end{document} is g-orthogonal to v and w; the length of v×w\documentclass[12pt]{minimal}
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\begin{document}$$v\times w$$\end{document} is the area of the parallelogram spanned by v, w in TxΣ\documentclass[12pt]{minimal}
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\begin{document}$$T_x\Sigma $$\end{document}; and v,w,v×w\documentclass[12pt]{minimal}
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\begin{document}$$v,w,v\times w$$\end{document} is a positively oriented basis of TxΣ\documentclass[12pt]{minimal}
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\begin{document}$$T_x\Sigma $$\end{document} whenever v×w≠0\documentclass[12pt]{minimal}
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\begin{document}$$v\times w\ne 0$$\end{document}.
For future use we record here an identity true for any v,w1,w2,w3,w4∈TxΣ\documentclass[12pt]{minimal}
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\begin{document}$$v,w_1,w_2,w_3,w_4\in T_x\Sigma $$\end{document} such that |v|g=1\documentclass[12pt]{minimal}
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\begin{document}$$|v|_g=1$$\end{document} and w1,w2,w3,w4\documentclass[12pt]{minimal}
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\begin{document}$$w_1,w_2,w_3,w_4$$\end{document} are g-orthogonal to v:⟨v×w1,w2⟩g⟨v×w3,w4⟩g=⟨w1,w3⟩g⟨w2,w4⟩g-⟨w2,w3⟩g⟨w1,w4⟩g.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle v\times w_1,w_2\rangle _g\langle v\times w_3,w_4\rangle _g= \langle w_1,w_3\rangle _g\langle w_2,w_4\rangle _g-\langle w_2,w_3\rangle _g\langle w_1,w_4\rangle _g.\nonumber \\ \end{aligned}$$\end{document}Using the horizontal/vertical decomposition (2.15), we define the bundle homomorphismI:TSΣ→TSΣ,I(x,v)(ξH,ξV)=(v×ξV,v×ξH).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {I}}:TS\Sigma \rightarrow TS\Sigma ,\quad {\mathcal {I}}(x,v)(\xi _H,\xi _V)=(v\times \xi _V,v\times \xi _H). \end{aligned}$$\end{document}From (2.18) and (3.6) we see that I\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {I}}$$\end{document} preserves the flow/stable/unstable decomposition (2.2). Moreover, it annihilates E0=RX\documentclass[12pt]{minimal}
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\begin{document}$$E_0=\mathbb RX$$\end{document} and it is a rotation by π/2\documentclass[12pt]{minimal}
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\begin{document}$$E_u$$\end{document} and on Es\documentclass[12pt]{minimal}
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\begin{document}$$E_s$$\end{document} (with respect to the Sasaki metric), so in particular it satisfies I2=-Id\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {I}}^2 = -{{\,\mathrm{Id}\,}}$$\end{document} on kerα=Eu⊕Es\documentclass[12pt]{minimal}
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\begin{document}$$\ker \alpha = E_u \oplus E_s$$\end{document}; however, the direction of the rotation is opposite on Eu\documentclass[12pt]{minimal}
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\begin{document}$$E_u$$\end{document} and on Es\documentclass[12pt]{minimal}
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\begin{document}$$E_s$$\end{document} if we identify them by (3.5).
The map I\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {I}}$$\end{document} is invariant under the geodesic flow φt=etX\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _t=e^{tX}$$\end{document}:LXI=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {L}}_X{\mathcal {I}}=0. \end{aligned}$$\end{document}This follows from the conformal property of the geodesic flow (3.7) and the description of the action of I\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {I}}$$\end{document} on E0,Eu,Es\documentclass[12pt]{minimal}
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\begin{document}$$E_0,E_u,E_s$$\end{document} in the previous paragraph.
For any point (x,v1,v2,v3)\documentclass[12pt]{minimal}
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\begin{document}$$(x,v_1,v_2,v_3)$$\end{document} in the frame bundle FΣ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}\Sigma $$\end{document}, we have (using the horizontal/vertical decomposition)I(x,v1)(v2,±v2)=±(v3,±v3),I(x,v1)(v3,±v3)=∓(v2,±v2).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {I}}(x,v_1)(v_2,\pm v_2)=\pm (v_3,\pm v_3),\quad {\mathcal {I}}(x,v_1)(v_3,\pm v_3)=\mp (v_2,\pm v_2).\nonumber \\ \end{aligned}$$\end{document}It follows that (see §3.1.2 for the definition of the vector fields Ui±\documentclass[12pt]{minimal}
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\begin{document}$$U_i^\pm $$\end{document} on FΣ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}\Sigma $$\end{document})I(x,v1)(dπFU1±(x,v1,v2,v3))=∓dπFU2±(x,v1,v2,v3),I(x,v1)(dπFU2±(x,v1,v2,v3))=±dπFU1±(x,v1,v2,v3).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} {\mathcal {I}}(x,v_1)(d\pi _{{\mathcal {F}}}U^\pm _1(x,v_1,v_2,v_3))&=\mp d\pi _{{\mathcal {F}}}U^\pm _2(x,v_1,v_2,v_3),\\ {\mathcal {I}}(x,v_1)(d\pi _{{\mathcal {F}}}U^\pm _2(x,v_1,v_2,v_3))&=\pm d\pi _{{\mathcal {F}}}U^\pm _1(x,v_1,v_2,v_3). \end{aligned} \end{aligned}$$\end{document}
Relation to conformal infinity
The homomorphism I\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {I}}$$\end{document} lifts to TSH3\documentclass[12pt]{minimal}
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\begin{document}$$B_\pm :S{\mathbb {H}}^3\rightarrow {\mathbb {S}}^2$$\end{document} are the maps defined in (3.14) and ‘×\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} dB_\pm (x,v)({\mathcal {I}}(x,v)\xi )=B_\pm (x,v)\times dB_\pm (x,v)(\xi ). \end{aligned}$$\end{document}To see this, we use (3.23), and the fact that I\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^{1,3}$$\end{document}. In the latter case (3.34) is verified directly using (3.24) and (3.32).
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\begin{document}$$\begin{aligned} \langle (\star w)(\nu ),\zeta \rangle =-\langle w(\nu ),\nu \times \zeta \rangle . \end{aligned}$$\end{document}From (3.34) we get the following relation of I\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (B_\pm ^*w)\circ {\mathcal {I}}=-B_\pm ^*(\star w), \end{aligned}$$\end{document}where for any 1-form β\documentclass[12pt]{minimal}
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\begin{document}$$S{\mathbb {H}}^3$$\end{document} the 1-form β∘I\documentclass[12pt]{minimal}
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\begin{document}$$S{\mathbb {H}}^3$$\end{document} is defined by⟨(β∘I)(x,v),ξ⟩=⟨β(x,v),I(x,v)ξ⟩.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle (\beta \circ {\mathcal {I}})(x,v),\xi \rangle =\langle \beta (x,v),{\mathcal {I}}(x,v)\xi \rangle . \end{aligned}$$\end{document}
The new invariant 2-form
We next define the 2-form ψ∈C∞(SΣ;Ω2)\documentclass[12pt]{minimal}
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\begin{document}$$\psi \in C^\infty (S\Sigma ;\Omega ^2)$$\end{document} as follows: for all (x,v)∈SΣ\documentclass[12pt]{minimal}
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\begin{document}$$(x,v)\in S\Sigma $$\end{document} and ξ,η∈T(x,v)SΣ\documentclass[12pt]{minimal}
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\begin{document}$$\xi ,\eta \in T_{(x,v)}S\Sigma $$\end{document},ψ(x,v)(ξ,η)=dα(x,v)(I(x,v)ξ,η).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \psi (x,v)(\xi ,\eta )=d\alpha (x,v)({\mathcal {I}}(x,v)\xi ,\eta ). \end{aligned}$$\end{document}To see that ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} is indeed an antisymmetric form, we may use (2.16) and (3.30) to write it in terms of the horizontal/vertical decomposition of ξ,η\documentclass[12pt]{minimal}
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\begin{document}$$\xi ,\eta $$\end{document}:ψ(x,v)(ξ,η)=⟨v×ξH,ηH⟩g-⟨v×ξV,ηV⟩g.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \psi (x,v)(\xi ,\eta )=\langle v\times \xi _H,\eta _H\rangle _g-\langle v\times \xi _V,\eta _V\rangle _g. \end{aligned}$$\end{document}Using (3.12), (3.33) we may also compute the lift of ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} to the frame bundle FΣ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}\Sigma $$\end{document}, which we still denote by ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document}:ψ=2(U1+∗∧U2-∗+U1-∗∧U2+∗).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \psi =2(U_1^{+*}\wedge U_2^{-*}+U_1^{-*}\wedge U_2^{+*}). \end{aligned}$$\end{document}We haveιXψ=0,LXψ=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \iota _X\psi =0,\quad {\mathcal {L}}_X\psi =0. \end{aligned}$$\end{document}The first of these statements is checked directly using (2.18). The second statement can be verified using (3.13) and (3.39), or using that LXI=0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X {\mathcal {I}}=0$$\end{document} and LXdα=0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_Xd\alpha =0$$\end{document}.
We now establish several properties of the form ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document}. We will use the following corollaries of (2.16), (3.38):dα|H×H=0,dα|V×V=0,ψ|H×V=0\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d\alpha |_{{\mathbf {H}}\times {\mathbf {H}}}=0,\quad d\alpha |_{{\mathbf {V}}\times {\mathbf {V}}}=0,\quad \psi |_{{\mathbf {H}}\times {\mathbf {V}}}=0 \end{aligned}$$\end{document}where the horizontal/vertical spaces H,V\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {H}},{\mathbf {V}}$$\end{document} are defined in §2.2.1.
By (3.40) we have ιXdψ=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X d\psi =0$$\end{document}. Therefore, dψ(x,v)(ξ1,ξ2,ξ3)=0\documentclass[12pt]{minimal}
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\begin{document}$$d\psi (x,v)(\xi _1,\xi _2,\xi _3)=0$$\end{document} for ξ1,ξ2,ξ3∈T(x,v)SΣ\documentclass[12pt]{minimal}
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\begin{document}$$\xi _1,\xi _2,\xi _3\in T_{(x,v)}S\Sigma $$\end{document} such that one of these vectors lies in E0\documentclass[12pt]{minimal}
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\begin{document}$$E_0$$\end{document}. Next, LXdψ=0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X d\psi =0$$\end{document}, that is dψ\documentclass[12pt]{minimal}
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\begin{document}$$d\psi $$\end{document} is invariant under the geodesic flow. Using this invariance for time t→±∞\documentclass[12pt]{minimal}
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\begin{document}$$t\rightarrow \pm \infty $$\end{document} together with (3.7) and the fact that 3 is an odd number, we see that dψ(x,v)(ξ1,ξ2,ξ3)=0\documentclass[12pt]{minimal}
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\begin{document}$$d\psi (x,v)(\xi _1,\xi _2,\xi _3)=0$$\end{document} also when each of the vectors ξ1,ξ2,ξ3\documentclass[12pt]{minimal}
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\begin{document}$$\xi _1,\xi _2,\xi _3$$\end{document} lies in either Eu(x,v)\documentclass[12pt]{minimal}
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\begin{document}$$E_u(x,v)$$\end{document} or Es(x,v)\documentclass[12pt]{minimal}
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\begin{document}$$E_s(x,v)$$\end{document}. It follows that (3.42) holds.
To check (3.43), we first note that ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} of both sides is zero. Thus it suffices to check thatψ∧ψ(x,v)(ξ1,ξ2,ξ3,ξ4)=dα∧dα(x,v)(ξ1,ξ2,ξ3,ξ4)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \psi \wedge \psi (x,v)(\xi _1,\xi _2,\xi _3,\xi _4)=d\alpha \wedge d\alpha (x,v)(\xi _1,\xi _2,\xi _3,\xi _4) \end{aligned}$$\end{document}for some choice of basis ξ1,ξ2,ξ3,ξ4∈T(x,v)SΣ\documentclass[12pt]{minimal}
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\begin{document}$$\xi _1,\xi _2,\xi _3,\xi _4\in T_{(x,v)}S\Sigma $$\end{document} of the kernel of α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}. We takeξ1=(w1,0),ξ2=(w2,0),ξ3=(0,w3),ξ4=(0,w4)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \xi _1=(w_1,0),\quad \xi _2=(w_2,0),\quad \xi _3=(0,w_3),\quad \xi _4=(0,w_4) \end{aligned}$$\end{document}under the horizontal/vertical decomposition (2.15), where each wj∈TxΣ\documentclass[12pt]{minimal}
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\begin{document}$$w_j\in T_x\Sigma $$\end{document} is orthogonal to v. By (2.16), (3.38)ψ∧ψ(x,v)(ξ1,ξ2,ξ3,ξ4)=-2⟨v×w1,w2⟩g⟨v×w3,w4⟩g,dα∧dα(x,v)(ξ1,ξ2,ξ3,ξ4)=2(⟨w2,w3⟩g⟨w1,w4⟩g-⟨w1,w3⟩g⟨w2,w4⟩g)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \psi \wedge \psi (x,v)(\xi _1,\xi _2,\xi _3,\xi _4)&=-2\langle v\times w_1,w_2\rangle _g\langle v\times w_3,w_4\rangle _g,\\ d\alpha \wedge d\alpha (x,v)(\xi _1,\xi _2,\xi _3,\xi _4)&=2(\langle w_2,w_3\rangle _g\langle w_1,w_4\rangle _g\\&\quad -\langle w_1,w_3\rangle _g\langle w_2,w_4\rangle _g) \end{aligned} \end{aligned}$$\end{document}and (3.45) follows from (3.29).
Finally, to show (3.44) it suffices to prove that dα∧ψ=0\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge \psi =0$$\end{document}. To show this we may argue similarly to the proof of (3.43) above, using (3.41).
Alternatively, (3.42)–(3.44) can be checked by lifting to the frame bundle FΣ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}\Sigma $$\end{document} and using (3.12) and (3.39). □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
The next lemma studies the relation of ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} to the de Rham cohomology of M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$M=S\Sigma $$\end{document}; in particular, its first item and (3.40) give the first item of Theorem 2. Recall the pullback and pushforward operators πΣ∗,πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Sigma ^*,\pi _{\Sigma *}^{}$$\end{document} defined in §2.2.2 and denote by dvolg\documentclass[12pt]{minimal}
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\begin{document}$$d{{\,\mathrm{vol}\,}}_g$$\end{document} the volume 3-form on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} induced by g and the choice of orientation.
1. Let (x,v)∈SΣ\documentclass[12pt]{minimal}
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\begin{document}$$(x,v)\in S\Sigma $$\end{document} and v2,v3\documentclass[12pt]{minimal}
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\begin{document}$$v_2,v_3$$\end{document} be a positively oriented g-orthonormal basis of the tangent space to the fiber Tv(SxΣ)\documentclass[12pt]{minimal}
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\begin{document}$$T_v(S_x\Sigma )$$\end{document}. We consider v2,v3\documentclass[12pt]{minimal}
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\begin{document}$$v_2,v_3$$\end{document} as vertical vectors in T(x,v)SΣ\documentclass[12pt]{minimal}
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\begin{document}$$T_{(x,v)}S\Sigma $$\end{document}, as well as vectors in TxΣ\documentclass[12pt]{minimal}
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\begin{document}$$T_x\Sigma $$\end{document}. The triple v,v2,v3\documentclass[12pt]{minimal}
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\begin{document}$$v,v_2,v_3$$\end{document} is a positively oriented g-orthonormal basis of TxΣ\documentclass[12pt]{minimal}
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\begin{document}$$T_x\Sigma $$\end{document}, so by (3.38)ψ(x,v)(v2,v3)=-⟨v×v2,v3⟩g=-1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \psi (x,v)(v_2,v_3)=-\langle v\times v_2,v_3\rangle _g=-1. \end{aligned}$$\end{document}Thus the restriction of ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} to each fiber SxΣ\documentclass[12pt]{minimal}
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\begin{document}$$S_x\Sigma $$\end{document} is -1\documentclass[12pt]{minimal}
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\begin{document}$$-1$$\end{document} times the standard volume form on SxΣ≃S2\documentclass[12pt]{minimal}
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\begin{document}$$S_x\Sigma \simeq {\mathbb {S}}^2$$\end{document}, which implies that πΣ∗(ψ)=-4π\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}(\psi )=-4\pi $$\end{document}. It now follows from (2.22) that [ψ]H2≠0\documentclass[12pt]{minimal}
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\begin{document}$$[\psi ]_{H^2} \ne 0$$\end{document}.
2. Fix x∈Σ\documentclass[12pt]{minimal}
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\begin{document}$$v_1\in T_x\Sigma $$\end{document}. Let v∈SxΣ\documentclass[12pt]{minimal}
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\begin{document}$$v\in S_x\Sigma $$\end{document} and v2,v3\documentclass[12pt]{minimal}
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\begin{document}$$v_2,v_3$$\end{document} be a positively oriented g-orthonormal basis of the tangent space Tv(SxΣ)\documentclass[12pt]{minimal}
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\begin{document}$$T_v(S_x\Sigma )$$\end{document} as in part 1 of this proof. Let ξ1=(v1,0)\documentclass[12pt]{minimal}
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\begin{document}$$\xi _1=(v_1,0)$$\end{document} be the horizontal lift of v1\documentclass[12pt]{minimal}
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\begin{document}$$v_1$$\end{document} to T(x,v)(SΣ)\documentclass[12pt]{minimal}
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\begin{document}$$T_{(x,v)}(S\Sigma )$$\end{document}. By (2.16) and (3.38) we computeα∧ψ(x,v)(ξ1,v2,v3)=-⟨v1,v⟩g⟨v×v2,v3⟩g=-⟨v1,v⟩g.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha \wedge \psi (x,v)(\xi _1,v_2,v_3)=-\langle v_1,v\rangle _g\langle v\times v_2,v_3\rangle _g=-\langle v_1,v\rangle _g. \end{aligned}$$\end{document}Since v↦⟨v1,v⟩g\documentclass[12pt]{minimal}
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\begin{document}$$v\mapsto \langle v_1,v\rangle _g$$\end{document} is an odd function on SxΣ\documentclass[12pt]{minimal}
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\begin{document}$$S_x\Sigma $$\end{document}, we have(πΣ∗(α∧ψ))(x)(v1)=∫SxΣ-⟨v1,v⟩gdvol(v)=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (\pi _{\Sigma *}^{}(\alpha \wedge \psi ))(x)(v_1)=\int _{S_x\Sigma } -\langle v_1,v\rangle _g \,d{{\,\mathrm{vol}\,}}(v)=0. \end{aligned}$$\end{document}3. If ξ1,ξ2,ξ3∈T(x,v)(SΣ)\documentclass[12pt]{minimal}
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\begin{document}$$\xi _1,\xi _2,\xi _3\in T_{(x,v)}(S\Sigma )$$\end{document} and ξ2,ξ3\documentclass[12pt]{minimal}
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\begin{document}$$\xi _2,\xi _3$$\end{document} are vertical, then by (2.16) we haveα∧dα(x,v)(ξ1,ξ2,ξ3)=0\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha \wedge d\alpha (x,v)(\xi _1,\xi _2,\xi _3)=0 \end{aligned}$$\end{document}which implies that πΣ∗(α∧dα)=0\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}(\alpha \wedge d\alpha )=0$$\end{document}.
4. Let x∈Σ\documentclass[12pt]{minimal}
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\begin{document}$$T_x\Sigma $$\end{document}. Let ξ=X(x,v),ξ2,ξ3\documentclass[12pt]{minimal}
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\begin{document}$$v,v_2,v_3$$\end{document} to T(x,v)SΣ\documentclass[12pt]{minimal}
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\begin{document}$$T_{(x,v)}S\Sigma $$\end{document}; we treat v2,v3\documentclass[12pt]{minimal}
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\begin{document}$$T_{(x,v)}S\Sigma $$\end{document}. Using (2.16) and (3.38), we computeα∧dα∧dα(x,v)(ξ,ξ2,ξ3,v2,v3)=-2=2ψ∧πΣ∗(dvolg)(x,v)(ξ,ξ2,ξ3,v2,v3).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha \wedge d\alpha&\wedge&d\alpha (x,v)(\xi ,\xi _2,\xi _3,v_2,v_3)\\= & {} -2 =2\psi \wedge \pi _\Sigma ^*(d{{\,\mathrm{vol}\,}}_g)(x,v)(\xi ,\xi _2,\xi _3,v_2,v_3). \end{aligned}$$\end{document}5. Using the exact sequence (2.27) and the fact that πΣ∗(α∧ψ)=0\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}(\alpha \wedge \psi )=0$$\end{document}, we see that[α∧ψ]H3=c[πΣ∗(dvolg)]H3\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}{}[\alpha \wedge \psi ]_{H^3}=c[\pi _{\Sigma }^*(d{{\,\mathrm{vol}\,}}_g)]_{H^3} \end{aligned}$$\end{document}for some constant c. To determine c, note that α∧ψ∧ψ\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \wedge \psi \wedge \psi $$\end{document} has the same integral over SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document} as cψ∧πΣ∗(dvolg)\documentclass[12pt]{minimal}
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\begin{document}$$c\psi \wedge \pi _{\Sigma }^*(d{{\,\mathrm{vol}\,}}_g)$$\end{document}. Since α∧ψ∧ψ=α∧dα∧dα=2ψ∧πΣ∗(dvolg)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \wedge \psi \wedge \psi =\alpha \wedge d\alpha \wedge d\alpha =2\psi \wedge \pi _\Sigma ^*(d{{\,\mathrm{vol}\,}}_g)$$\end{document}, we get c=2\documentclass[12pt]{minimal}
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\begin{document}$$c=2$$\end{document}. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
We also have the following identity relating the operators dα∧\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge $$\end{document} and ψ∧\documentclass[12pt]{minimal}
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\begin{document}$$\psi \wedge $$\end{document} on 1-forms in Ω01\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^1_0$$\end{document}:
Lemma 3.6
For any 1-form β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document} on SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document} such that ιXβ=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X\beta =0$$\end{document}, we havedα∧β=ψ∧(β∘I),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d\alpha \wedge \beta =\psi \wedge (\beta \circ {\mathcal {I}}), \end{aligned}$$\end{document}where the 1-form β∘I\documentclass[12pt]{minimal}
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\begin{document}$$\beta \circ {\mathcal {I}}$$\end{document} is defined by (3.36).
Proof
It is easy to see that ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} of both sides of (3.46) is equal to 0. It is thus enough to check thatdα∧β(x,v)(ξ1,ξ2,ξ3)=ψ∧(β∘I)(x,v)(ξ1,ξ2,ξ3)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d\alpha \wedge \beta (x,v)(\xi _1,\xi _2,\xi _3)=\psi \wedge (\beta \circ {\mathcal {I}})(x,v)(\xi _1,\xi _2,\xi _3) \end{aligned}$$\end{document}for any three vectors ξ1,ξ2,ξ3\documentclass[12pt]{minimal}
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\begin{document}$$\xi _1,\xi _2,\xi _3$$\end{document}, each of which is either horizontal or vertical under the decomposition (2.15). Moreover, we may assume that the horizontal components of these vectors lie in the orthogonal complement {v}⊥\documentclass[12pt]{minimal}
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\begin{document}$$\{v\}^\perp $$\end{document} to v in TxΣ\documentclass[12pt]{minimal}
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\begin{document}$$T_x\Sigma $$\end{document}. It suffices to consider the following two cases:
Case 1:β(x,v)(ξ)=⟨ξH,w4⟩g\documentclass[12pt]{minimal}
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\begin{document}$$\beta (x,v)(\xi )=\langle \xi _H,w_4\rangle _g$$\end{document} for some w4∈{v}⊥\documentclass[12pt]{minimal}
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\begin{document}$$w_4\in \{v\}^\perp $$\end{document}. By (3.30) and (3.41), both sides of (3.47) are equal to 0 unless two of ξ1,ξ2,ξ3\documentclass[12pt]{minimal}
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\begin{document}$$\xi _1,\xi _2,\xi _3$$\end{document} are horizontal and one is vertical; we writeξ1=(w1,0),ξ2=(w2,0),ξ3=(0,w3),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \xi _1=(w_1,0),\quad \xi _2=(w_2,0),\quad \xi _3=(0,w_3), \end{aligned}$$\end{document}where wj∈{v}⊥\documentclass[12pt]{minimal}
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\begin{document}$$w_j\in \{v\}^\perp $$\end{document}. We compute using (2.16), (3.30), and (3.38)dα∧β(x,v)(ξ1,ξ2,ξ3)=⟨w1,w3⟩g⟨w2,w4⟩g-⟨w2,w3⟩g⟨w1,w4⟩g,ψ∧(β∘I)(x,v)(ξ1,ξ2,ξ3)=⟨v×w1,w2⟩g⟨v×w3,w4⟩g\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} d\alpha \wedge \beta (x,v)(\xi _1,\xi _2,\xi _3)&=\langle w_1,w_3\rangle _g\langle w_2,w_4\rangle _g-\langle w_2,w_3\rangle _g \langle w_1,w_4\rangle _g,\\ \psi \wedge (\beta \circ {\mathcal {I}})(x,v)(\xi _1,\xi _2,\xi _3)&=\langle v\times w_1,w_2\rangle _g\langle v\times w_3,w_4\rangle _g \end{aligned} \end{aligned}$$\end{document}and (3.47) follows from (3.29).
Case 2:β(x,v)(ξ)=⟨ξV,w4⟩g\documentclass[12pt]{minimal}
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\begin{document}$$\beta (x,v)(\xi )=\langle \xi _V,w_4\rangle _g$$\end{document} for some w4∈{v}⊥\documentclass[12pt]{minimal}
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\begin{document}$$w_4\in \{v\}^\perp $$\end{document}. By (3.30) and (3.41), both sides of (3.47) are equal to 0 unless two of ξ1,ξ2,ξ3\documentclass[12pt]{minimal}
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\begin{document}$$\xi _1,\xi _2,\xi _3$$\end{document} are vertical and one is horizontal; we writeξ1=(0,w1),ξ2=(0,w2),ξ3=(w3,0),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \xi _1=(0,w_1),\quad \xi _2=(0,w_2),\quad \xi _3=(w_3,0), \end{aligned}$$\end{document}where wj∈{v}⊥\documentclass[12pt]{minimal}
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\begin{document}$$w_j\in \{v\}^\perp $$\end{document}. We compute using (2.16), (3.30), and (3.38)dα∧β(x,v)(ξ1,ξ2,ξ3)=⟨w2,w3⟩g⟨w1,w4⟩g-⟨w1,w3⟩g⟨w2,w4⟩g,ψ∧(β∘I)(x,v)(ξ1,ξ2,ξ3)=-⟨v×w1,w2⟩g⟨v×w3,w4⟩g\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} d\alpha \wedge \beta (x,v)(\xi _1,\xi _2,\xi _3)&=\langle w_2,w_3\rangle _g \langle w_1,w_4\rangle _g-\langle w_1,w_3\rangle _g\langle w_2,w_4\rangle _g,\\ \psi \wedge (\beta \circ {\mathcal {I}})(x,v)(\xi _1,\xi _2,\xi _3)&=-\langle v\times w_1,w_2\rangle _g\langle v\times w_3,w_4\rangle _g \end{aligned} \end{aligned}$$\end{document}and (3.47) again follows from (3.29).
Alternatively, we may lift both sides of (3.46) to the frame bundle FΣ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}\Sigma $$\end{document}: it suffices to consider the cases when β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document} is replaced by one of the forms Ui±∗\documentclass[12pt]{minimal}
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\begin{document}$$U^{\pm *}_i$$\end{document}, in which case (3.46) is checked by a direct calculation using (3.12), (3.33), and (3.39). □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Characterization of all smooth flow-invariant 2-forms
We finally give
Lemma 3.7
Assume that u∈C∞(SΣ;Ω2)\documentclass[12pt]{minimal}
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\begin{document}$$u\in C^\infty (S\Sigma ;\Omega ^2)$$\end{document} satisfies LXu=0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_Xu=0$$\end{document}. Then u is a linear combination of dα\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha $$\end{document} and ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document}.
Proof
Without loss of generality we assume that u is real valued. Since dα∧ψ=0\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge \psi =0$$\end{document} and ψ∧ψ=dα∧dα\documentclass[12pt]{minimal}
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\begin{document}$$\psi \wedge \psi =d\alpha \wedge d\alpha $$\end{document} by (3.43)–(3.44), we may subtract from u a linear combination of dα\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} to make∫Mα∧dα∧u=∫Mα∧ψ∧u=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _M \alpha \wedge d\alpha \wedge u=\int _M \alpha \wedge \psi \wedge u=0. \end{aligned}$$\end{document}We will show that under the condition (3.48) we have u=0\documentclass[12pt]{minimal}
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\begin{document}$$u=0$$\end{document}.
Next, ιXu∈C∞(SΣ;Ω01)\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X u\in C^\infty (S\Sigma ;\Omega ^1_0)$$\end{document} and LXιXu=0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X\iota _X u=0$$\end{document}, so by (2.3) (similarly to the last step of the proof of Lemma 2.10) we get ιXu=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X u = 0$$\end{document}. Also by (2.3) we obtain u|Eu×Eu=0\documentclass[12pt]{minimal}
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\begin{document}$$u|_{E_u \times E_u} = 0$$\end{document} and u|Es×Es=0\documentclass[12pt]{minimal}
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\begin{document}$$u|_{E_s \times E_s} = 0$$\end{document}. Therefore, it is enough to show that u|Es×Eu=0\documentclass[12pt]{minimal}
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\begin{document}$$u|_{E_s\times E_u}=0$$\end{document}.
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\begin{document}$$d\alpha $$\end{document} is nondegenerate on Es×Eu\documentclass[12pt]{minimal}
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\begin{document}$$E_s\times E_u$$\end{document} (as follows for instance from (2.16) and (3.6)), there exists unique smooth bundle homomorphism A:Es→Es\documentclass[12pt]{minimal}
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\begin{document}$$A:E_s\rightarrow E_s$$\end{document} such thatu(x,v)(ξ,η)=dα(A(x,v)ξ,η)for all(x,v)∈SΣ,ξ∈Es(x,v),η∈Eu(x,v).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u(x,v)(\xi ,\eta )=d\alpha (A(x,v)\xi ,\eta )\quad \text {for all}\quad (x,v)\in & {} S\Sigma ,\ \xi \in E_s(x,v),\\ \eta\in & {} E_u(x,v). \end{aligned}$$\end{document}It remains to show that A=0\documentclass[12pt]{minimal}
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\begin{document}$$A=0$$\end{document}.
Take any (x,v)∈SΣ\documentclass[12pt]{minimal}
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\begin{document}$$(x,v)\in S\Sigma $$\end{document}, assume that v,w1,w2\documentclass[12pt]{minimal}
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\begin{document}$$T_x\Sigma $$\end{document}, and define using the horizontal/vertical decomposition and (3.6)ξj=(wj,-wj)∈Es(x,v),ηj=(wj,wj)∈Eu(x,v),j=1,2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \xi _j=(w_j,-w_j)\in E_s(x,v),\quad \eta _j=(w_j,w_j)\in E_u(x,v),\quad j=1,2. \end{aligned}$$\end{document}Applying (3.49) to the vectors X(x,v),ξ1,ξ2,η1,η2\documentclass[12pt]{minimal}
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\begin{document}$$X(x,v),\xi _1,\xi _2,\eta _1,\eta _2$$\end{document} and using (2.16), (3.32), and (3.37), we gettrA(x,v)=0,A(x,v)T=A(x,v),detA(x,v)=c,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{tr}\,}}A(x,v)=0,\quad A(x,v)^T=A(x,v),\quad \det A(x,v)=c, \end{aligned}$$\end{document}where the transpose is with respect to the restriction of the Sasaki metric to Es(x,v)\documentclass[12pt]{minimal}
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\begin{document}$$E_s(x,v)$$\end{document}.
If c=0\documentclass[12pt]{minimal}
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\begin{document}$$c=0$$\end{document}, then (3.50) implies that A=0\documentclass[12pt]{minimal}
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\begin{document}$$A=0$$\end{document}. Assume that c≠0\documentclass[12pt]{minimal}
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\begin{document}$$c\ne 0$$\end{document}, then by (3.50) we have c<0\documentclass[12pt]{minimal}
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\begin{document}$$c<0$$\end{document} and A has eigenvalues ±-c\documentclass[12pt]{minimal}
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\begin{document}$$E_s(x,v)$$\end{document} depending continuously on (x, v). This is impossible since by restricting to a single fiber SxΣ⊂SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S_x\Sigma \subset S\Sigma $$\end{document} and projecting Es\documentclass[12pt]{minimal}
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\begin{document}$$E_s$$\end{document} onto the vertical space V\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {V}}$$\end{document} we would obtain a continuous one-dimensional subbundle of the tangent space to the 2-sphere. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Resonant 1-forms and 3-forms
In this section we apply the properties of the 2-form ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} defined in (3.37) to determine the precise structure of resonant 1-forms on M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$M=S\Sigma $$\end{document}. Let us introduce some notation for (co-)resonant 1-forms (see (3.36) for the definition of u∘I\documentclass[12pt]{minimal}
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\begin{document}$$u\circ {\mathcal {I}}$$\end{document})C(∗):=Res0(∗)1∩kerd,Cψ(∗):={u∘I∣u∈C(∗)},\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {C}}_{(*)} := {{\,\mathrm{Res}\,}}_{0(*)}^1 \cap \ker d, \quad {\mathcal {C}}_{\psi (*)} := \{u\circ {\mathcal {I}}\mid u\in {\mathcal {C}}_{(*)}\}, \end{aligned}$$\end{document}where the subscript (∗)\documentclass[12pt]{minimal}
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\begin{document}$$(*)$$\end{document} means we either suppress the star or we include it, respectively corresponding to resonances or co-resonances; we apply this convention to other notions appearing in this section. We remark that the use of subscript ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} in Cψ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}_\psi $$\end{document} is motivated by the property dα∧Cψ=ψ∧C\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge {\mathcal {C}}_\psi =\psi \wedge {\mathcal {C}}$$\end{document} demonstrated in (3.58) below; in fact we initially used this relation as the definition of Cψ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}_\psi $$\end{document}, before coming to the interpretation via the map I\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {I}}$$\end{document}.
Since I\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {I}}$$\end{document} is invariant under the geodesic flow by (3.31) and annihilates X, we haveCψ(∗)⊂Res0(∗)1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {C}}_{\psi (*)}\subset {{\,\mathrm{Res}\,}}_{0(*)}^1. \end{aligned}$$\end{document}By Lemma 2.8 and (2.28) we havedimC(∗)=dimCψ(∗)=b1(Σ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \dim {\mathcal {C}}_{(*)}=\dim {\mathcal {C}}_{\psi (*)} = b_1(\Sigma ). \end{aligned}$$\end{document}We next show that all resonant 1-forms lie in the direct sum C⊕Cψ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}\oplus {\mathcal {C}}_\psi $$\end{document}. This is done in Lemma 3.9 below but first we need
Lemma 3.8
Assume that u∈Res01\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^1_0$$\end{document}. Then u is totally unstable in the sense of Definition 3.2. Similarly, if u∈Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^1_{0*}$$\end{document}, then u is totally stable.
Remark
Lemma 3.8 was previously proved by Küster–Weich [44, §2.6].
Proof
We consider the case u∈Res01\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^1_0$$\end{document}, with the case u∈Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^1_{0*}$$\end{document} handled in the same way.
We first show that u is unstable in the sense of Definition 3.2. For that it is enough to prove that u(Y)=0\documentclass[12pt]{minimal}
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\begin{document}$$u(Y)=0$$\end{document} for any Y∈C∞(M;E0⊕Eu)\documentclass[12pt]{minimal}
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\begin{document}$$Y\in C^\infty (M;E_0\oplus E_u)$$\end{document}. Since ιXu=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X u=0$$\end{document}, we may assume that Y∈C∞(M;Eu)\documentclass[12pt]{minimal}
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\begin{document}$$Y\in C^\infty (M;E_u)$$\end{document}. By the integral formula (2.29) for the Pollicott–Ruelle resolvent Rk,0(λ)\documentclass[12pt]{minimal}
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\begin{document}$$R_{k,0}(\lambda )$$\end{document}, we have for Imλ≫1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Im}\,}}\lambda \gg 1$$\end{document} and any w∈C∞(M;Ω01)\documentclass[12pt]{minimal}
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\begin{document}$$w\in C^\infty (M;\Omega ^1_0)$$\end{document}, ρ∈M\documentclass[12pt]{minimal}
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\begin{document}$$\rho \in M$$\end{document}⟨R1,0(λ)w,Y⟩(ρ)=i∫0∞eiλt⟨w(φ-t(ρ)),dφ-t(ρ)Y(ρ)⟩dt.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle R_{1,0}(\lambda )w,Y\rangle (\rho )=i\int _0^\infty e^{i\lambda t}\langle w(\varphi _{-t}(\rho )),d\varphi _{-t}(\rho )Y(\rho )\rangle \,dt. \end{aligned}$$\end{document}Since Y is a section of the unstable bundle, by (3.7) we have |⟨w(φ-t(ρ))\documentclass[12pt]{minimal}
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\begin{document}$$|\langle w(\varphi _{-t}(\rho ))$$\end{document}, dφ-t(ρ)Y(ρ)⟩|≤Ce-t\documentclass[12pt]{minimal}
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\begin{document}$$d\varphi _{-t}(\rho )Y(\rho )\rangle |\le Ce^{-t}$$\end{document} for some constant C and all t≥0\documentclass[12pt]{minimal}
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\begin{document}$$t\ge 0$$\end{document}, ρ∈M\documentclass[12pt]{minimal}
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\begin{document}$$\rho \in M$$\end{document}. Therefore, the integral above converges uniformly in ρ\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Im}\,}}\lambda >-1$$\end{document}, which implies that λ↦⟨R1,0(λ)w,Y⟩\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \mapsto \langle R_{1,0}(\lambda )w,Y\rangle $$\end{document} is holomorphic in Imλ>-1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Im}\,}}\lambda >-1$$\end{document}. If Π1,0\documentclass[12pt]{minimal}
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\begin{document}$$\Pi _{1,0}$$\end{document} is the projector appearing in the Laurent expansion of R1,0(λ)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_0$$\end{document} is contained in the range of Π1,0\documentclass[12pt]{minimal}
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\begin{document}$$u(Y)=0$$\end{document} as needed.
We now analyze du. First of all, ιXdu=0\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^1_0$$\end{document}. Next, we have du|Eu×Eu=0\documentclass[12pt]{minimal}
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\begin{document}$$du|_{E_u\times E_u}=0$$\end{document}. This can be seen by following the argument above, or using that u(Y)=0\documentclass[12pt]{minimal}
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\begin{document}$$Y\in C^\infty (M;E_0\oplus E_u)$$\end{document}, the identity (3.11), and the fact that the class C∞(M;E0⊕Eu)\documentclass[12pt]{minimal}
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\begin{document}$$C^\infty (M;E_0\oplus E_u)$$\end{document} is closed under Lie brackets (as follows from (3.23)).
It remains to show that du|Eu×Es=0\documentclass[12pt]{minimal}
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\begin{document}$$du|_{E_u\times E_s}=0$$\end{document}. Let ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document} be the restriction of du to Eu×Es\documentclass[12pt]{minimal}
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\begin{document}$$E_u\times E_s$$\end{document}, considered as a section in DEu∗′(M;Es∗⊗Eu∗)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'_{E_u^*}(M;E_s^*\otimes E_u^*)$$\end{document}. (Here Es∗,Eu∗\documentclass[12pt]{minimal}
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\begin{document}$$E_s^*,E_u^*$$\end{document} are dual to Eu,Es\documentclass[12pt]{minimal}
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\begin{document}$$E_u,E_s$$\end{document} as in (2.4).) We endow Es∗⊗Eu∗\documentclass[12pt]{minimal}
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\begin{document}$$E_s^*\otimes E_u^*$$\end{document} with the inner product which is the tensor product of the dual Sasaski metrics on Es∗\documentclass[12pt]{minimal}
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\begin{document}$$E_s^*$$\end{document} and Eu∗\documentclass[12pt]{minimal}
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\begin{document}$$E_u^*$$\end{document}. The operatorP:=-iLX:C∞(M;Es∗⊗Eu∗)→C∞(M;Es∗⊗Eu∗)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} P:=-i{\mathcal {L}}_X:C^\infty (M;E_s^*\otimes E_u^*)\rightarrow C^\infty (M;E_s^*\otimes E_u^*) \end{aligned}$$\end{document}is formally self-adjoint as follows from (3.7), and Pζ=0\documentclass[12pt]{minimal}
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\begin{document}$$P\zeta =0$$\end{document}. Then by [21, Lemma 2.3] the section ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document} is in C∞\documentclass[12pt]{minimal}
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\begin{document}$$C^\infty $$\end{document}.
Let us now consider ζ=du|Eu×Es\documentclass[12pt]{minimal}
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\begin{document}$$\zeta =du|_{E_u\times E_s}$$\end{document} as a smooth 2-form on M (i.e. ιXζ=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X \zeta =0$$\end{document}, ζ|Eu×Eu=ζ|Es×Es=0\documentclass[12pt]{minimal}
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\begin{document}$$\zeta |_{E_u\times E_u}=\zeta |_{E_s\times E_s}=0$$\end{document}, and ζ|Eu×Es=du|Eu×Es\documentclass[12pt]{minimal}
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\begin{document}$$\zeta |_{E_u\times E_s}=du|_{E_u\times E_s}$$\end{document}), then LXζ=0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X\zeta =0$$\end{document} and by Lemma 3.7 we see that ζ=adα+bψ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta =a\,d\alpha +b\,\psi $$\end{document} for some constants a, b. We claim that a=b=0\documentclass[12pt]{minimal}
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\begin{document}$$a=b=0$$\end{document}. This follows from (3.43)–(3.44) and the identities∫Mα∧dα∧ζ=∫Mα∧dα∧du=0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _M \alpha \wedge d\alpha \wedge \zeta&=\int _M \alpha \wedge d\alpha \wedge du=0, \end{aligned}$$\end{document}∫Mα∧ψ∧ζ=∫Mα∧ψ∧du=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _M \alpha \wedge \psi \wedge \zeta&=\int _M \alpha \wedge \psi \wedge du=0. \end{aligned}$$\end{document}Here the first identity in each line follows from the fact that dα|Eu×Eu=ψ|Eu×Eu=0\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha |_{E_u\times E_u}=\psi |_{E_u\times E_u}=0$$\end{document} (which can be verified using (2.16), (3.6), and (3.37)). More precisely, it suffices to observe that α∧dα∧(du-ζ)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \wedge d\alpha \wedge (du - \zeta )$$\end{document} and α∧dψ∧(du-ζ)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \wedge d\psi \wedge (du - \zeta )$$\end{document} are pointwise zero, as du-ζ\documentclass[12pt]{minimal}
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\begin{document}$$du - \zeta $$\end{document} is supported on Es×Es\documentclass[12pt]{minimal}
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\begin{document}$$E_s \times E_s$$\end{document} by definition. The second identity in each line follows by integration by parts and the fact that dα∧dα∧u=dα∧ψ∧u=0\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge d\alpha \wedge u=d\alpha \wedge \psi \wedge u=0$$\end{document} (as ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} of both of these 5-forms is equal to 0). Now, a=b=0\documentclass[12pt]{minimal}
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\begin{document}$$a=b=0$$\end{document} implies that ζ=0\documentclass[12pt]{minimal}
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\begin{document}$$\zeta =0$$\end{document}, that is du|Eu×Es=0\documentclass[12pt]{minimal}
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\begin{document}$$du|_{E_u\times E_s}=0$$\end{document} as needed. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
We finally discuss the properties of the maps π3(∗):Res0(∗)3→H3(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{3(*)}:{{\,\mathrm{Res}\,}}^3_{0(*)}\rightarrow H^3(M;{\mathbb {C}})$$\end{document}; as explained at the top of § 3.3, recall that the subscript (∗)\documentclass[12pt]{minimal}
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\begin{document}$$(*)$$\end{document} denotes the corresponding resonance or co-resonance space, so we can include both in the discussion. Recall that all forms in Res0(∗)3\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^3_{0(*)}$$\end{document} are closed by Lemma 2.9 and Res0(∗)3=dα∧Res0(∗)1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^3_{0(*)}=d\alpha \wedge {{\,\mathrm{Res}\,}}^1_{0(*)}$$\end{document} by (2.45), (2.49). Moreover, by Lemma 3.6 and the definition of Cψ(∗)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}_{\psi (*)}$$\end{document}dα∧Cψ(∗)=ψ∧C(∗).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d\alpha \wedge {\mathcal {C}}_{\psi (*)}=\psi \wedge {\mathcal {C}}_{(*)}. \end{aligned}$$\end{document}We have π3(∗)(dα∧C(∗))=0\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{3(*)}(d\alpha \wedge {\mathcal {C}}_{(*)})=0$$\end{document}. Assume now that u∈Cψ\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {C}}_\psi $$\end{document}, then u∘I∈C\documentclass[12pt]{minimal}
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\begin{document}$$u\circ {\mathcal {I}}\in {\mathcal {C}}$$\end{document}, and by Lemma 2.5 and (2.25) we may writeu∘I=πΣ∗w+dffor somew∈C∞(Σ;Ω1),dw=0,f∈DEu∗′(M;C).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u\circ {\mathcal {I}}=\pi _\Sigma ^* w+df\quad \text {for some}\quad w\in & {} C^\infty (\Sigma ;\Omega ^1),\ dw=0,\\&f\in {\mathcal {D}}'_{E_u^*}(M;{\mathbb {C}}). \end{aligned}$$\end{document}Wedging with ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document}, taking πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}$$\end{document}, and using (2.22)–(2.23), part 1 of Lemma 3.5, and Lemma 3.6 we getπΣ∗π3(dα∧u)=πΣ∗(ψ∧πΣ∗w)=-4πw,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _{\Sigma *}^{}\pi _3(d\alpha \wedge u) = \pi _{\Sigma *}^{} (\psi \wedge \pi _{\Sigma }^*w) = -4\pi w, \end{aligned}$$\end{document}which (together with a similar argument for coresonant states) immediately shows thatπΣ∗π3(∗):dα∧Cψ(∗)→H1(Σ;C)is an isomorphism.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _{\Sigma *}^{}\pi _{3(*)}:d\alpha \wedge {\mathcal {C}}_{\psi (*)}\rightarrow H^1(\Sigma ;{\mathbb {C}})\text { is an isomorphism}. \end{aligned}$$\end{document}This implies thatkerπ3(∗)=dα∧C(∗)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \ker \pi _{3(*)}=d\alpha \wedge {\mathcal {C}}_{(*)} \end{aligned}$$\end{document}and so by (2.27) the range of π3(∗)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{3(*)}$$\end{document} is a codimension 1 subspace of H3(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$H^3(M;{\mathbb {C}})$$\end{document} which does not contain [πΣ∗dvolg]H3\documentclass[12pt]{minimal}
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\begin{document}$$[\pi _\Sigma ^*d{{\,\mathrm{vol}\,}}_g]_{H^3}$$\end{document}.
Summarizing the contents of § 3.3, we note that the second item of Theorem 2 follows from (3.58), Lemma 3.9, Lemma 2.8, and (2.28), the third item by Lemma 3.10, and the sixth item by the discussion in the preceding paragraph.
We consider the case of resonant 2-forms, with the case of coresonant 2-forms handled similarly. We first show that d(Res02)=0\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}^2_0)=0$$\end{document}, arguing similarly to the proof of Lemma 2.11. Take ζ∈Res02\documentclass[12pt]{minimal}
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\begin{document}$$\zeta \in {{\,\mathrm{Res}\,}}^2_0$$\end{document}, then by the definition (2.61) of π3\documentclass[12pt]{minimal}
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\begin{document}$$\pi _3$$\end{document} we have dζ∈kerπ3\documentclass[12pt]{minimal}
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\begin{document}$$d\zeta \in \ker \pi _3$$\end{document}. Thus by (3.60), dζ=dα∧u\documentclass[12pt]{minimal}
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\begin{document}$$d\zeta =d\alpha \wedge u$$\end{document} for some u∈C\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {C}}$$\end{document}. Take arbitrary u∗∈C∗\documentclass[12pt]{minimal}
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\begin{document}$$u_*\in {\mathcal {C}}_*$$\end{document}, then precisely as in (2.65)⟨⟨u,dα∧u∗⟩⟩=∫Mα∧dζ∧u∗=∫Mdα∧ζ∧u∗=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \!\langle u,d\alpha \wedge u_*\rangle \!\rangle =\int _M\alpha \wedge d\zeta \wedge u_*=\int _M d\alpha \wedge \zeta \wedge u_*=0. \end{aligned}$$\end{document}Now Lemma 2.10 implies that u=0\documentclass[12pt]{minimal}
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\begin{document}$$u=0$$\end{document} and thus dζ=0\documentclass[12pt]{minimal}
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\begin{document}$$d\zeta =0$$\end{document} as needed.
Next, if u∈Cψ\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {C}}_\psi $$\end{document}, then using the same argument of integration by parts as in (3.52) yields∫Mα∧dα∧du=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _M \alpha \wedge d\alpha \wedge du=0. \end{aligned}$$\end{document}Therefore, du cannot be a nonzero multiple of dα\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha $$\end{document}, which means that Cdα∩dCψ={0}\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb Cd\alpha \cap d{\mathcal {C}}_\psi =\{0\}$$\end{document}. We have dα∈kerπ2\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \in \ker \pi _2$$\end{document} and by Lemma 2.6 we have dCψ⊂kerπ2\documentclass[12pt]{minimal}
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\begin{document}$$d{\mathcal {C}}_\psi \subset \ker \pi _2$$\end{document} as well.
It remains to show that kerπ2⊂Cdα⊕dCψ\documentclass[12pt]{minimal}
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\begin{document}$$\ker \pi _2\subset \mathbb Cd\alpha \oplus d{\mathcal {C}}_\psi $$\end{document}. By Lemma 2.6, kerπ2\documentclass[12pt]{minimal}
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\begin{document}$$\ker \pi _2$$\end{document} is contained in d(Res1,∞)\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}^{1,\infty })$$\end{document}. By (2.43) and Lemmas 2.4, 3.9, and 3.10, we have Res1,∞=Cα⊕C⊕Cψ\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{1,\infty }={\mathbb {C}}\alpha \oplus {\mathcal {C}}\oplus {\mathcal {C}}_\psi $$\end{document}. Then d(Res1,∞)=Cdα⊕dCψ\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}^{1,\infty })=\mathbb Cd\alpha \oplus d{\mathcal {C}}_\psi $$\end{document}, which finishes the proof. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
The proof of Lemma 3.12 uses the 2-form ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} which is only available in the case of constant curvature. By contrast, Lemma 3.12 is false if Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$$b_1(\Sigma )>0$$\end{document}; in fact the equation (3.61) then has a solution w∈DEu∗′(M;Ω01)\documentclass[12pt]{minimal}
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\begin{document}$$w\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^1_0)$$\end{document} for any closed η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document}. Indeed, in this case ⟨⟨ιXπΣ∗η,dα∧u∗⟩⟩=∫MπΣ∗η∧dα∧u∗=0\documentclass[12pt]{minimal}
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\begin{document}$$\langle \!\langle \iota _X\pi _\Sigma ^*\eta ,d\alpha \wedge u_*\rangle \!\rangle =\int _M\pi _\Sigma ^*\eta \wedge d\alpha \wedge u_*=0$$\end{document} for any u∗∈Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$$u_*\in {{\,\mathrm{Res}\,}}^1_{0*}$$\end{document} by integration by parts, and the existence of w now follows from Lemma 2.1.
Proof
Put ζ:=πΣ∗η+dw\documentclass[12pt]{minimal}
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\begin{document}$$\zeta :=\pi _\Sigma ^*\eta +dw$$\end{document}, then ιXζ=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X\zeta =0$$\end{document}. Take arbitrary closed η∗∈C∞(Σ;Ω1)\documentclass[12pt]{minimal}
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\begin{document}$$\eta _*\in C^\infty (\Sigma ;\Omega ^1)$$\end{document} and put u∗:=π1∗-1([πΣ∗η∗]H1)∈C∗\documentclass[12pt]{minimal}
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\begin{document}$$u_*=\pi _\Sigma ^*\eta _*+dw_*$$\end{document} for some w∗∈DEs∗′(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$w_*\in {\mathcal {D}}'_{E_s^*}(M;{\mathbb {C}})$$\end{document}. We compute0=∫Mψ∧ζ∧u∗=∫Mψ∧πΣ∗η∧πΣ∗η∗=-∫Σ(πΣ∗ψ)η∧η∗=4π∫Ση∧η∗.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} 0&=\int _M \psi \wedge \zeta \wedge u_*=\int _M \psi \wedge \pi _\Sigma ^*\eta \wedge \pi _\Sigma ^* \eta _*\\&=-\int _\Sigma (\pi _{\Sigma *}^{}\psi ) \eta \wedge \eta _*=4\pi \int _\Sigma \eta \wedge \eta _*. \end{aligned} \end{aligned}$$\end{document}Here the first equality follows since the 5-form under the integral lies in the kernel of ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document}. The second equality follows by integration by parts, using that ψ,η,η∗\documentclass[12pt]{minimal}
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\begin{document}$$\psi ,\eta ,\eta _*$$\end{document} are closed. The third equality follows from (2.20) and (2.23). The fourth equality follows from part 1 of Lemma 3.5.
We see that η∧η∗\documentclass[12pt]{minimal}
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\begin{document}$$\eta \wedge \eta _*$$\end{document} integrates to 0 on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} for any closed smooth 1-form η∗\documentclass[12pt]{minimal}
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\begin{document}$$\eta _*$$\end{document}. This implies that η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document} is exact; indeed, we can reduce to the case when η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document} is harmonic and take η∗\documentclass[12pt]{minimal}
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\begin{document}$$\eta _*$$\end{document} to be the Hodge star of η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document} (we note that this final argument is just a form of Poincaré duality). □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
We now describe the space of resonant 2-forms (recalling the convention (∗)\documentclass[12pt]{minimal}
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\begin{document}$$(*)$$\end{document} at the top of § 3.3):
Lemma 3.13
The range of π2(∗)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{2(*)}$$\end{document} is equal to C[ψ]H2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}[\psi ]_{H^2}$$\end{document}, and Res0(∗)2=Cdα⊕Cψ⊕dCψ(∗)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^2_{0(*)}={\mathbb {C}} d\alpha \oplus {\mathbb {C}}\psi \oplus d{\mathcal {C}}_{\psi (*)}$$\end{document}. In particular, dimRes0(∗)2=b1(Σ)+2\documentclass[12pt]{minimal}
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\begin{document}$$\dim {{\,\mathrm{Res}\,}}^2_{0(*)}=b_1(\Sigma )+2$$\end{document}.
Proof
We consider the case of resonant 2-forms, with the case of coresonant 2-forms handled similarly. First of all, ψ∈Res02\documentclass[12pt]{minimal}
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\begin{document}$$\psi \in {{\,\mathrm{Res}\,}}^2_0$$\end{document}, thus [ψ]H2=π2(ψ)\documentclass[12pt]{minimal}
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\begin{document}$$[\psi ]_{H^2}=\pi _2(\psi )$$\end{document} is in the range of π2\documentclass[12pt]{minimal}
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\begin{document}$$\pi _2$$\end{document}. Next, by (2.26) and part 1 of Lemma 3.5 we haveH2(M;C)=πΣ∗H2(Σ;C)⊕C[ψ]H2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} H^2(M;{\mathbb {C}})=\pi _\Sigma ^*H^2(\Sigma ;{\mathbb {C}})\oplus {\mathbb {C}}[\psi ]_{H^2}. \end{aligned}$$\end{document}To show that the range of π2\documentclass[12pt]{minimal}
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\begin{document}$$\pi _2$$\end{document} is equal to C[ψ]H2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}[\psi ]_{H^2}$$\end{document}, it remains to prove that the intersection of this range with πΣ∗H2(Σ;C)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Sigma ^*H^2(\Sigma ;{\mathbb {C}})$$\end{document} is trivial. Take u∈Res02\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^2_0$$\end{document} and assume that π2(u)=[πΣ∗η]H2\documentclass[12pt]{minimal}
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\begin{document}$$\pi _2(u)=[\pi _\Sigma ^* \eta ]_{H^2}$$\end{document} for some η∈C∞(Σ;Ω2)\documentclass[12pt]{minimal}
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\begin{document}$$\eta \in C^\infty (\Sigma ;\Omega ^2)$$\end{document}, dη=0\documentclass[12pt]{minimal}
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\begin{document}$$d\eta =0$$\end{document}. Then u=πΣ∗η+dw\documentclass[12pt]{minimal}
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\begin{document}$$u=\pi _\Sigma ^*\eta +dw$$\end{document} for some w∈DEu∗′(M;Ω1)\documentclass[12pt]{minimal}
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\begin{document}$$w\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^1)$$\end{document}. Since ιXu=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _Xu=0$$\end{document}, Lemma 3.12 implies that η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document} is exact, that is π2(u)=0\documentclass[12pt]{minimal}
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\begin{document}$$\pi _2(u)=0$$\end{document} as needed.
Finally, the statement that Res02=Cdα⊕Cψ⊕dCψ\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^2_0=\mathbb Cd\alpha \oplus {\mathbb {C}}\psi \oplus d{\mathcal {C}}_\psi $$\end{document} follows from the first statement of this lemma together with Lemma 3.11. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
The next lemma describes the space of generalized resonant states Res02,2\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{2,2}_0$$\end{document} (see (2.39) and §2.3.3). It implies in particular that the operator P2,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{2,0}$$\end{document} does not satisfy the semisimplicity condition (2.41), assuming that b1(Σ)>0\documentclass[12pt]{minimal}
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\begin{document}$$b_1(\Sigma )>0$$\end{document}:
1. The identities (3.62) follow immediately from (3.43) and (3.44). We next show (3.63), with (3.64) proved similarly. Let ζ=du\documentclass[12pt]{minimal}
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\begin{document}$$\zeta =du$$\end{document} where u∈Cψ\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {C}}_\psi $$\end{document}. We compute⟨⟨ζ,ζ∗⟩⟩=∫Mdα∧u∧ζ∗=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \!\langle \zeta ,\zeta _*\rangle \!\rangle =\int _M d\alpha \wedge u\wedge \zeta _*=0. \end{aligned}$$\end{document}Here in the first equality we integrate by parts and use that dζ∗=0\documentclass[12pt]{minimal}
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\begin{document}$$d\zeta _*=0$$\end{document} by Lemma 3.11. The second equality follows from the fact that ιX(dα∧u∧ζ∗)=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X(d\alpha \wedge u\wedge \zeta _*)=0$$\end{document}.
2. We consider generalized resonant states, with generalized coresonant states handled similarly. First, assume that ζ∈Res02,2\documentclass[12pt]{minimal}
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\begin{document}$$\zeta \in {{\,\mathrm{Res}\,}}^{2,2}_0$$\end{document}, then LXζ∈Res02\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X\zeta \in {{\,\mathrm{Res}\,}}^2_0$$\end{document}. Moreover, since the transpose of LX\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X$$\end{document} is equal to -LX\documentclass[12pt]{minimal}
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\begin{document}$$-{\mathcal {L}}_X$$\end{document} (see §2.3.4) we compute⟨⟨LXζ,ζ∗⟩⟩=-⟨⟨ζ,LXζ∗⟩⟩=0for allζ∗∈Res0∗2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \!\langle {\mathcal {L}}_X\zeta ,\zeta _*\rangle \!\rangle =-\langle \!\langle \zeta ,{\mathcal {L}}_X\zeta _*\rangle \!\rangle =0\quad \text {for all}\quad \zeta _*\in {{\,\mathrm{Res}\,}}^2_{0*}. \end{aligned}$$\end{document}Using this for ζ∗=dα\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _*=d\alpha $$\end{document} and ζ∗=ψ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _*=\psi $$\end{document} together with (3.62)–(3.63), we see that LXζ∈dCψ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X\zeta \in d{\mathcal {C}}_\psi $$\end{document}. That is, the range of the map (3.65) is contained in dCψ\documentclass[12pt]{minimal}
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\begin{document}$$d{\mathcal {C}}_\psi $$\end{document}.
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\begin{document}$$\eta \in d{\mathcal {C}}_\psi $$\end{document}. By (3.63), we have ⟨⟨η,ζ∗⟩⟩=0\documentclass[12pt]{minimal}
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\begin{document}$$\langle \!\langle \eta ,\zeta _*\rangle \!\rangle =0$$\end{document} for all ζ∗∈Res0∗2\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _*\in {{\,\mathrm{Res}\,}}^2_{0*}$$\end{document}. Then by Lemma 2.1 there exists ζ∈DEu∗′(M;Ω02)\documentclass[12pt]{minimal}
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\begin{document}$$\zeta \in {\mathcal {D}}'_{E_u^*}(M;\Omega ^2_0)$$\end{document} such that LXζ=η\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X\zeta =\eta $$\end{document}. Since η∈Res02\documentclass[12pt]{minimal}
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\begin{document}$$\eta \in {{\,\mathrm{Res}\,}}^2_0$$\end{document}, we see that ζ∈Res02,2\documentclass[12pt]{minimal}
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\begin{document}$$\zeta \in {{\,\mathrm{Res}\,}}^{2,2}_0$$\end{document}. This shows that the range of the map (3.65) contains dCψ\documentclass[12pt]{minimal}
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\begin{document}$$d{\mathcal {C}}_\psi $$\end{document}.
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\begin{document}$$\dim {{\,\mathrm{Res}\,}}^{2,2}_{0}=2b_1(\Sigma )+2$$\end{document} follows from Lemma 3.13 and the fact that the kernel of the map (3.65) is given by Res02\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^2_0$$\end{document}. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
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\begin{document}$$P_{2,0}$$\end{document} at 0:
Lemma 3.15
We have Res0(∗)2,∞=Res0(∗)2,2\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{2,\infty }_{0(*)}={{\,\mathrm{Res}\,}}^{2,2}_{0(*)}$$\end{document}.
Proof
We consider the case of generalized resonant states, with generalized coresonant states handled similarly. It suffices to prove that Res02,3⊂Res02,2\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{2,3}_0\subset {{\,\mathrm{Res}\,}}^{2,2}_0$$\end{document}. Take η∈Res02,3\documentclass[12pt]{minimal}
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\begin{document}$$\eta \in {{\,\mathrm{Res}\,}}^{2,3}_0$$\end{document} and put ζ:=LXη∈Res02,2\documentclass[12pt]{minimal}
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\begin{document}$$\zeta :={\mathcal {L}}_X\eta \in {{\,\mathrm{Res}\,}}^{2,2}_0$$\end{document}. Exactly as in (3.66), the pairing of ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document} with any element of Res0∗2\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^2_{0*}$$\end{document} is equal to 0. In particular⟨⟨ζ,du∗⟩⟩=0for allu∗∈Res0∗1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \!\langle \zeta ,du_*\rangle \!\rangle =0\quad \text {for all}\quad u_*\in {{\,\mathrm{Res}\,}}^1_{0*}. \end{aligned}$$\end{document}By part 2 of Lemma 3.14, we have LXζ=du\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X\zeta =du$$\end{document} for some u∈Cψ\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {C}}_\psi $$\end{document}. Putω:=d(ζ+α∧u)∈DEu∗′(M;Ω3).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \omega :=d(\zeta +\alpha \wedge u)\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^3). \end{aligned}$$\end{document}Then ιXω=ιXdζ-du=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X\omega =\iota _Xd\zeta -du=0$$\end{document}. Since ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} is exact we have LXω=0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X \omega = 0$$\end{document} and moreover that ω∈kerπ3⊂Res03\documentclass[12pt]{minimal}
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\begin{document}$$\omega \in \ker \pi _3\subset {{\,\mathrm{Res}\,}}^3_0$$\end{document}. By (3.60), we then have ω∈dα∧C\documentclass[12pt]{minimal}
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\begin{document}$$\omega \in d\alpha \wedge {\mathcal {C}}$$\end{document}.
We now compute0=⟨⟨ζ,du∗⟩⟩=-∫Mα∧dζ∧u∗=⟨⟨u,dα∧u∗⟩⟩-⟨⟨ω,u∗⟩⟩.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} 0&=\langle \!\langle \zeta ,du_*\rangle \!\rangle =-\int _M\alpha \wedge d\zeta \wedge u_* =\langle \!\langle u,d\alpha \wedge u_*\rangle \!\rangle -\langle \!\langle \omega ,u_*\rangle \!\rangle . \end{aligned} \end{aligned}$$\end{document}Here in the second equality we integrated by parts and used that the 5-form dα∧ζ∧u∗\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge \zeta \wedge u_*$$\end{document} lies in the kernel of ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} and thus equals 0. Using the identities (3.55)–(3.57) and Lemma 2.10, recalling that u∈Cψ\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {C}}_\psi $$\end{document}, ω∈dα∧C\documentclass[12pt]{minimal}
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\begin{document}$$u_*$$\end{document} can be chosen as an arbitrary element of C∗\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}_*$$\end{document} or Cψ∗\documentclass[12pt]{minimal}
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\begin{document}$$u=0$$\end{document} and ω=0\documentclass[12pt]{minimal}
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\begin{document}$$\omega =0$$\end{document}. Just using that u=0\documentclass[12pt]{minimal}
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\begin{document}$$u = 0$$\end{document} implies LX2η=LXζ=0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}_X^2\eta ={\mathcal {L}}_X\zeta =0$$\end{document}, that is η∈Res02,2\documentclass[12pt]{minimal}
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\begin{document}$$\eta \in {{\,\mathrm{Res}\,}}^{2,2}_0$$\end{document} as needed. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Relation to harmonic forms
In this section we show that pushforwards of elements of Res03=dα∧(C⊕Cψ)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^3_0=d\alpha \wedge ({\mathcal {C}}\oplus {\mathcal {C}}_\psi )$$\end{document} to the base Σ\documentclass[12pt]{minimal}
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\begin{document}$$dh = 0$$\end{document} and d⋆h=0\documentclass[12pt]{minimal}
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\begin{document}$$d\star h = 0$$\end{document}, where ⋆\documentclass[12pt]{minimal}
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\begin{document}$$\star $$\end{document} is the Hodge star on (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document}. We will denote the set of such forms by H1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}^1(\Sigma )$$\end{document}. We start with the following identity:
We first show (3.67). Take arbitrary (x,v)∈M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$(x,v)\in M=S\Sigma $$\end{document} and assume that v,w1,w2\documentclass[12pt]{minimal}
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\begin{document}$$v,w_1,w_2$$\end{document} is a positively oriented g-orthonormal basis of TxΣ\documentclass[12pt]{minimal}
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\begin{document}$$T_x\Sigma $$\end{document}. It suffices to prove that(ψ∧u∧πΣ∗(⋆β))(x,v)(X,ξ1,ξ2,ξ3,ξ4)=-(α∧dα∧u∧πΣ∗β)(x,v)(X,ξ1,ξ2,ξ3,ξ4)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (\psi \wedge u&\wedge&\pi _\Sigma ^*(\star \beta ))(x,v)(X,\xi _1,\xi _2,\xi _3,\xi _4)\nonumber \\= & {} -(\alpha \wedge d\alpha \wedge u\wedge \pi _\Sigma ^*\beta )(x,v)(X,\xi _1,\xi _2,\xi _3,\xi _4) \end{aligned}$$\end{document}where we write in terms of the horizontal/vertical decomposition (2.15)X=(v,0),ξ1=(w1,0),ξ2=(w2,0),ξ3=(0,w1),ξ4=(0,w2).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} X=(v,0),\quad \xi _1=(w_1,0),\quad \xi _2=(w_2,0),\quad \xi _3=(0,w_1),\quad \xi _4=(0,w_2). \end{aligned}$$\end{document}Using (3.38), (3.41), the fact that dπΣ(x,v)(ξH,ξV)=ξH\documentclass[12pt]{minimal}
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\begin{document}$$d\pi _\Sigma (x,v)(\xi _H,\xi _V)=\xi _H$$\end{document}, the condition ιXu=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X u = 0$$\end{document}, and the identities(⋆β)(x)(v,w1)=β(x)(w2),(⋆β)(x)(v,w2)=-β(x)(w1)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (\star \beta )(x)(v,w_1)=\beta (x)(w_2),\quad (\star \beta )(x)(v,w_2)=-\beta (x)(w_1) \end{aligned}$$\end{document}we see that the left-hand side of (3.69) is equal to-u(x,v)(ξ1)β(x)(w1)-u(x,v)(ξ2)β(x)(w2).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} -u(x,v)(\xi _1)\beta (x)(w_1)-u(x,v)(\xi _2)\beta (x)(w_2). \end{aligned}$$\end{document}Using (2.16), we next see that the right-hand side of (3.69) is equal tou(x,v)(ξ3)β(x)(w1)+u(x,v)(ξ4)β(x)(w2).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u(x,v)(\xi _3)\beta (x)(w_1)+u(x,v)(\xi _4)\beta (x)(w_2). \end{aligned}$$\end{document}It remains to note that by (3.6) the vectors ξ1+ξ3\documentclass[12pt]{minimal}
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\begin{document}$$\xi _1+\xi _3$$\end{document} and ξ2+ξ4\documentclass[12pt]{minimal}
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\begin{document}$$E_u(x,v)$$\end{document} and thus u(x,v)(ξ1+ξ3)=u(x,v)(ξ2+ξ4)=0\documentclass[12pt]{minimal}
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\begin{document}$$u(x,v)(\xi _1+\xi _3)=u(x,v)(\xi _2+\xi _4)=0$$\end{document} since u is unstable.
The identity (3.68) is verified by a similar calculation, or simply by applying (3.67) to u∘I\documentclass[12pt]{minimal}
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\begin{document}$$u\circ {\mathcal {I}}$$\end{document} and using Lemma 3.6 and the fact that u∘I∘I=-u\documentclass[12pt]{minimal}
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\begin{document}$$u\circ {\mathcal {I}}\circ {\mathcal {I}}=-u$$\end{document}. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
We can now prove item 7 of Theorem 2:
Lemma 3.17
The map πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}$$\end{document} annihilates dα∧C(∗)\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge {\mathcal {C}}_{(*)}$$\end{document} and it is an isomorphism from dα∧Cψ(∗)\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge {\mathcal {C}}_{\psi (*)}$$\end{document} onto the space H1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}^1(\Sigma )$$\end{document}. In particular, by Lemma 3.9 we have πΣ∗:dα∧Res0(∗)1→H1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}:d\alpha \wedge {{\,\mathrm{Res}\,}}^1_{0(*)}\rightarrow {\mathcal {H}}^1(\Sigma )$$\end{document}.
Proof
We consider the case of resonant 3-forms, with coresonant 3-forms handled similarly (using a version of Lemma 3.16 for stable 1-forms). We first show that for any u∈C\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {C}}$$\end{document}, the push-forwards to Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} of dα∧u\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge u$$\end{document} and ψ∧u\documentclass[12pt]{minimal}
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\begin{document}$$\psi \wedge u$$\end{document} are coclosed, that isd⋆πΣ∗(dα∧u)=0,d⋆πΣ∗(ψ∧u)=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d\star \pi _{\Sigma *}^{}(d\alpha \wedge u)=0,\quad d\star \pi _{\Sigma *}^{}(\psi \wedge u)=0. \end{aligned}$$\end{document}To show the first equality in (3.70), it suffices to prove that∫ΣπΣ∗(dα∧u)∧⋆df=0for allf∈C∞(Σ;C).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _\Sigma \pi _{\Sigma *}^{}(d\alpha \wedge u)\wedge \star df=0\quad \text {for all}\quad f\in C^\infty (\Sigma ;{\mathbb {C}}). \end{aligned}$$\end{document}Using (2.20) and (2.23), we compute this integral as-∫Mdα∧u∧πΣ∗(⋆df)=-∫Mα∧ψ∧u∧d(πΣ∗f)=∫MπΣ∗fdα∧ψ∧u=0\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} -\int _M d\alpha \wedge u\wedge \pi _\Sigma ^*(\star df)= & {} -\int _M \alpha \wedge \psi \wedge u\wedge d(\pi _\Sigma ^*f)\nonumber \\= & {} \int _M \pi _\Sigma ^*f d\alpha \wedge \psi \wedge u=0 \end{aligned}$$\end{document}Here in the first equality we used (3.68), where u is unstable by Lemma 3.8. In the second equality we integrated by parts and used that dψ=0\documentclass[12pt]{minimal}
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\begin{document}$$d\psi =0$$\end{document} and du=0\documentclass[12pt]{minimal}
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\begin{document}$$du=0$$\end{document}. In the third equality we used that ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} of the 5-form under the integral is equal to 0. The second equality in (3.70) is proved similarly, using (3.67) instead of (3.68).
Next, by (2.22), since all forms in dα∧C\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge {\mathcal {C}}$$\end{document} are exact, their pushforwards to Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} are exact as well. Since these pushforwards are also coclosed, we get πΣ∗(dα∧C)=0\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}(d\alpha \wedge {\mathcal {C}})=0$$\end{document}. Similarly, all forms in dα∧Cψ=ψ∧C\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge {\mathcal {C}}_\psi =\psi \wedge {\mathcal {C}}$$\end{document} are closed, so their pushforwards are closed as well; since these pushforwards are also coclosed, we get πΣ∗(dα∧Cψ)⊂H1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}(d\alpha \wedge {\mathcal {C}}_\psi )\subset {\mathcal {H}}^1(\Sigma )$$\end{document}.
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\begin{document}$$\square $$\end{document}
We finally remark that for any 1-form u∈D′(M;Ω1)\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {D}}'(M;\Omega ^1)$$\end{document} we haveπΣ∗(α∧u)=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _{\Sigma *}^{}(\alpha \wedge u)=0. \end{aligned}$$\end{document}Indeed, by (2.16) we see that α\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {D}}'(M;\Omega ^1)$$\end{document}πΣ∗(dα∧u)=πΣ∗(α∧du).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _{\Sigma *}^{}(d\alpha \wedge u)=\pi _{\Sigma *}^{}(\alpha \wedge du). \end{aligned}$$\end{document}
Contact perturbations of geodesic flows on hyperbolic 3-manifolds
Let M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _0$$\end{document} be the contact form on M corresponding to the geodesic flow on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}, see §§2.2,3.1. In this section we study Pollicott–Ruelle resonances at λ=0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _0$$\end{document}. Ultimately, we will study perturbations of the metric, but via perturbations of the contact form. In particular, we give the proof of Theorem 1 in §4.4 below, relying on Theorem 5 (in §5) and Proposition A.1 proved later.
Letατ∈C∞(M;T∗M),τ∈(-ε,ε)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha _\tau \in C^\infty (M;T^*M),\quad \tau \in (-\varepsilon ,\varepsilon ) \end{aligned}$$\end{document}be a family of 1-forms depending smoothly on τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}. We may shrink ε>0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _\tau $$\end{document} is a contact form on M and the corresponding Reeb vector fieldXτ∈C∞(M;TM)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} X_\tau \in C^\infty (M;TM) \end{aligned}$$\end{document}is Anosov; the latter follows from stability of the Anosov condition under perturbations (see for instance [23, Corollary 5.1.12] or [41, Corollary 6.4.7] for the related case of Anosov diffeomorphisms).
We will use first variation methods, introducing the 1-formβ:=∂τατ|τ=0∈C∞(M;Ω1).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \beta :=\partial _\tau \alpha _\tau |_{\tau =0}\in C^\infty (M;\Omega ^1). \end{aligned}$$\end{document}We use the subscript or superscript (τ)\documentclass[12pt]{minimal}
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\begin{document}$$(\tau )$$\end{document} to refer to the objects corresponding to the contact manifold (M,ατ)\documentclass[12pt]{minimal}
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\begin{document}$$(M,\alpha _\tau )$$\end{document} and the flow φt(τ):=etXτ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _t^{(\tau )}:=e^{tX_\tau }$$\end{document}. For example, we use the operators (see §2.3)Pk(τ)=-iLXτ,Pk,0(τ),Rk(τ)(λ),Πk(τ):=Πk(τ)(0),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} P_k^{(\tau )}=-i{\mathcal {L}}_{X_\tau },\quad P_{k,0}^{(\tau )},\quad R_k^{(\tau )}(\lambda ),\quad \Pi _k^{(\tau )}:=\Pi _k^{(\tau )}(0), \end{aligned}$$\end{document}the spaces of (generalized) resonant states at λ=0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda =0$$\end{document}Res(τ)k,ℓ,Res0(τ)k,ℓ,Res(τ)k,Res0(τ)k,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{Res}\,}}^{k,\ell }_{(\tau )},\quad {{\,\mathrm{Res}\,}}^{k,\ell }_{0(\tau )},\quad {{\,\mathrm{Res}\,}}^{k}_{(\tau )},\quad {{\,\mathrm{Res}\,}}^{k}_{0(\tau )}, \end{aligned}$$\end{document}and the algebraic multiplicities of 0 as a resonance of the operators Pk(τ)\documentclass[12pt]{minimal}
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\begin{document}$$P_k^{(\tau )}$$\end{document}, Pk,0(τ)\documentclass[12pt]{minimal}
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\begin{document}$$P_{k,0}^{(\tau )}$$\end{document}mk(τ)(0),mk,0(τ)(0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_k^{(\tau )}(0),\quad m_{k,0}^{(\tau )}(0). \end{aligned}$$\end{document}When we omit τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} this means that we are considering the unperturbed hyperbolic case τ=0\documentclass[12pt]{minimal}
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\begin{document}$$\tau =0$$\end{document}, that isα:=α0,Pk:=Pk(0),Rk:=Rk(0),Resk,ℓ:=Res(0)k,ℓ,Πk:=Πk(0),⋯\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha :=\alpha _0,\quad P_k:=P_k^{(0)},\quad R_k:=R_k^{(0)},\quad {{\,\mathrm{Res}\,}}^{k,\ell }:={{\,\mathrm{Res}\,}}^{k,\ell }_{(0)},\quad \Pi _k:=\Pi _k^{(0)},\dots \nonumber \\ \end{aligned}$$\end{document}The first result of this section, proved in §4.1 below, is the following theorem. (Here the maps πk(τ):Res0(τ)k∩kerd→Hk(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k^{(\tau )}:{{\,\mathrm{Res}\,}}^{k}_{0(\tau )}\cap \ker d\rightarrow H^k(M;{\mathbb {C}})$$\end{document} are defined in (2.61).)
Theorem 3
Let the assumptions above in this section hold. Assume moreover the following nondegeneracy condition:⟨⟨ιXβ∙,∙⟩⟩defines a nondegenerate pairing ond(Res01)×d(Res0∗1).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \!\langle \iota _X\beta \,\bullet ,\bullet \rangle \!\rangle \quad \text {defines a nondegenerate pairing on}\quad d({{\,\mathrm{Res}\,}}^1_0)\times d({{\,\mathrm{Res}\,}}^1_{0*}).\nonumber \\ \end{aligned}$$\end{document}Then there exists ε0>0\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon _0>0$$\end{document} such that for all τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} with 0<|τ|<ε0\documentclass[12pt]{minimal}
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\begin{document}$$0<|\tau |<\varepsilon _0$$\end{document} we have:
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\begin{document}$$d({{\,\mathrm{Res}\,}}^{1}_{0(\tau )})=0$$\end{document} and thus by Lemma 2.8 and (2.28) we have dimRes0(τ)1=b1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$\dim {{\,\mathrm{Res}\,}}^{1}_{0(\tau )}=b_1(\Sigma )$$\end{document}.
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\begin{document}$$\pi _3^{(\tau )}$$\end{document} is equal to 0.
The semisimplicity condition (2.41) holds at λ0=0\documentclass[12pt]{minimal}
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Theorem 3 together with Lemma 2.4 and (2.59) give the following
Corollary 4.1
Under the assumptions of Theorem 3 we have for 0<|τ|<ε0\documentclass[12pt]{minimal}
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\begin{document}$$0<|\tau |<\varepsilon _0$$\end{document}m0,0(τ)(0)=m4,0(τ)(0)=1,m1,0(τ)(0)=m3,0(τ)(0)=b1(Σ),m2,0(τ)(0)=b1(Σ)+2\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_{0,0}^{(\tau )}(0)=m_{4,0}^{(\tau )}(0)=1,~ m_{1,0}^{(\tau )}(0)=m_{3,0}^{(\tau )}(0)=b_1(\Sigma ),~ m_{2,0}^{(\tau )}(0)=b_1(\Sigma )+2 \end{aligned}$$\end{document}and the order of vanishing of the Ruelle zeta function ζR\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _{\mathrm R}$$\end{document} at 0 ismR(0)=2m0,0(τ)(0)-2m1,0(τ)(0)+m2,0(τ)(0)=4-b1(Σ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_{\mathrm R}(0)=2m_{0,0}^{(\tau )}(0)-2m_{1,0}^{(\tau )}(0)+m_{2,0}^{(\tau )}(0)=4-b_1(\Sigma ). \end{aligned}$$\end{document}
Corollary 4.1 is in contrast with the hyperbolic case τ=0\documentclass[12pt]{minimal}
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\begin{document}$$4-2b_1(\Sigma )$$\end{document}.
To give an application of Theorem 3 which is simpler to prove than Theorem 1, we show in §§4.2–4.3 below that the nondegeneracy condition (4.2) holds for a large set of conformal perturbations of the contact form α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}:1
Theorem 4
Let M=SΣ\documentclass[12pt]{minimal}
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Then there exists an open dense subset of Cc∞(U;R)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {a}}$$\end{document} in this set, the 1-form β:=aα\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {a}}$$\end{document} the contact flow on M corresponding to the contact form ατ:=eτaα\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _\tau :=e^{\tau {\mathbf {a}}}\alpha $$\end{document} satisfies the conclusions of Theorem 3, in particular the Ruelle zeta function has order of vanishing 4-b1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$4-b_1(\Sigma )$$\end{document} at 0.
Proof of Theorem 3
We first prove an identity relating the action of the vector fieldY:=∂τXτ|τ=0∈C∞(M;TM)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Y:=\partial _\tau X_\tau |_{\tau =0}\in C^\infty (M;TM) \end{aligned}$$\end{document}on resonant and coresonant 1-forms to the bilinear form featured in (4.2). It reformulates the pairing (4.2) and will subsequently (see Lemma 4.4) be used to show that the non-closed 1-forms may be perturbed away.
1. To show the first equality in (4.4), we note that by the decomposition (2.44) and Lemma 2.4 we have for all w∈DEu∗′(M;Ω1)\documentclass[12pt]{minimal}
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\begin{document}$$w\in {\mathcal {D}}'_{E_u^*}(M;\Omega ^1)$$\end{document}Π1w=Π1,0(w-(ιXw)α)+1volα(M)(∫MιXwdvolα)α.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Pi _1w=\Pi _{1,0}(w-(\iota _X w)\alpha )+{1\over {{\,\mathrm{vol}\,}}_\alpha (M)}\bigg (\int _M \iota _X w\,d{{\,\mathrm{vol}\,}}_\alpha \bigg )\alpha . \end{aligned}$$\end{document}We now compute∫Mα∧dα∧(Π1LYΠ1u)∧u∗=⟨⟨Π1,0(LYu-(ιXLYu)α),dα∧u∗⟩⟩=⟨⟨LYu-(ιXLYu)α,dα∧u∗⟩⟩=∫Mα∧dα∧LYu∧u∗.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \int _M \alpha \wedge d\alpha \wedge (\Pi _1{\mathcal {L}}_Y\Pi _1u)\wedge u_*&=\langle \!\langle \Pi _{1,0}({\mathcal {L}}_Y u-(\iota _X {\mathcal {L}}_Y u)\alpha ),d\alpha \wedge u_*\rangle \!\rangle \\&=\langle \!\langle {\mathcal {L}}_Y u-(\iota _X{\mathcal {L}}_Yu)\alpha ,d\alpha \wedge u_*\rangle \!\rangle \\&=\int _M\alpha \wedge d\alpha \wedge {\mathcal {L}}_Y u\wedge u_*. \end{aligned} \end{aligned}$$\end{document}Here in the first equality we used that u∈Res01\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^1_0$$\end{document} and thus Π1u=u\documentclass[12pt]{minimal}
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\begin{document}$$\Pi _1 u=u$$\end{document}. In the second equality we used that dα∧u∗∈Res0∗3\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge u_*\in {{\,\mathrm{Res}\,}}^3_{0*}$$\end{document} and thus (Π1,0)T(dα∧u∗)=dα∧u∗\documentclass[12pt]{minimal}
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\begin{document}$$(\Pi _{1,0})^T (d\alpha \wedge u_*)=d\alpha \wedge u_*$$\end{document} (see §2.3.4). This proves the first equality in (4.4).
2. We now show the second equality in (4.4). Differentiating the relations ιXτατ=1\documentclass[12pt]{minimal}
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\begin{document}$$\iota _{X_\tau } \alpha _\tau = 1$$\end{document} and ιXτdατ=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _{X_\tau } d\alpha _{\tau } = 0$$\end{document} (see (2.1)) at τ=0\documentclass[12pt]{minimal}
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\begin{document}$$\tau =0$$\end{document}, we getιYα=-ιXβ,ιYdα=-ιXdβ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \iota _Y \alpha = - \iota _X \beta , \quad \iota _Y d\alpha = -\iota _X d\beta . \end{aligned}$$\end{document}Note also thatα∧dα∧du=α∧dα∧du∗=0\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha \wedge d\alpha \wedge du=\alpha \wedge d\alpha \wedge du_*=0 \end{aligned}$$\end{document}as follows from Lemma 2.4 as the 5-forms above are in Res00dvolα\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^0_0d{{\,\mathrm{vol}\,}}_\alpha $$\end{document}, respectively Res0∗0dvolα\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^0_{0*}d{{\,\mathrm{vol}\,}}_\alpha $$\end{document}, and integrate to 0 on M using integration by parts (since the 5-forms dα∧dα∧u\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge d\alpha \wedge u$$\end{document}, dα∧dα∧u∗\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge d\alpha \wedge u_*$$\end{document} lie in the kernel of ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} and thus are equal to 0).
We have∫Mα∧dα∧LYu∧u∗=∫Mα∧dα∧ιYdu∧u∗+∫Mα∧dα∧dιYu∧u∗.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _M\alpha \wedge d\alpha \wedge {\mathcal {L}}_Y u\wedge u_*&= \int _M \alpha \wedge d\alpha \wedge \iota _Y du\wedge u_*\nonumber \\&+ \int _M \alpha \wedge d\alpha \wedge d\iota _Y u\wedge u_*. \end{aligned}$$\end{document}We first compute∫Mα∧dα∧ιYdu∧u∗=-∫Mα∧ιYdα∧du∧u∗-∫M(ιYu∗)α∧dα∧du=∫Mα∧ιXdβ∧du∧u∗=∫Mdβ∧du∧u∗=∫Mβ∧du∧du∗=∫M(ιXβ)α∧du∧du∗.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \int _M \alpha \wedge d\alpha \wedge \iota _Y du\wedge u_*&= -\int _M \alpha \wedge \iota _Y d\alpha \wedge du\wedge u_*-\int _M(\iota _Y u_*)\alpha \wedge d\alpha \wedge du\\&=\int _M \alpha \wedge \iota _X d\beta \wedge du\wedge u_* =\int _M d\beta \wedge du\wedge u_*\\&=\int _M \beta \wedge du\wedge du_*=\int _M (\iota _X\beta )\alpha \wedge du\wedge du_*. \end{aligned} \end{aligned}$$\end{document} Here in the first equality we used that the 5-form dα∧du∧u∗\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge du\wedge u_*$$\end{document} lies in the kernel of ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} and is thus equal to 0, implying ιY(dα∧du∧u∗)=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _Y(d\alpha \wedge du\wedge u_*)=0$$\end{document}. In the second equality we used the identities (4.5) and (4.6). In the third equality we used that α∧ιXdβ∧du∧u∗=dβ∧du∧u∗\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \wedge \iota _X d\beta \wedge du\wedge u_* = d\beta \wedge du \wedge u_*$$\end{document} as the difference of the two forms belongs to kerιX\documentclass[12pt]{minimal}
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\begin{document}$$\ker \iota _X$$\end{document}, by ιXdu=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _Xdu=0$$\end{document} and ιXu∗=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X u_*=0$$\end{document}. In the fourth equality we integrated by parts, and in the fifth equality we used that ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} of the integrated 5-forms are equal.
We next compute∫Mα∧dα∧dιYu∧u∗=∫MιYu(dα∧dα∧u∗-α∧dα∧du∗)=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _M \alpha \wedge d\alpha \wedge d\iota _Y u\wedge u_*=\int _M \iota _Y u(d\alpha \wedge d\alpha \wedge u_*-\alpha \wedge d\alpha \wedge du_*)=0.\nonumber \\ \end{aligned}$$\end{document}Here in the first equality we integrated by parts and in the second one we used (4.6) and the fact that dα∧dα∧u∗=0\documentclass[12pt]{minimal}
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\begin{document}$$d\alpha \wedge d\alpha \wedge u_*=0$$\end{document} (as ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} of this 5-form is equal to 0).
Plugging (4.8)–(4.9) into (4.7), we get the second equality in (4.4). □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
The pairing in (4.4) controls how the resonance at 0 for the operator P1,0(τ)\documentclass[12pt]{minimal}
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\begin{document}$$P_{1,0}^{(\tau )}$$\end{document} moves as we perturb τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} from 0, and the nondegeneracy condition (4.2) roughly speaking means that the multiplicity of 0 as a resonance of P1,0(τ)\documentclass[12pt]{minimal}
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\begin{document}$$P_{1,0}^{(\tau )}$$\end{document} drops by dimd(Res01)=b1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$\dim d({{\,\mathrm{Res}\,}}^1_0)=b_1(\Sigma )$$\end{document}. This observation is made precise in Lemma 4.4 below, but first we need to review perturbation theory of Pollicott–Ruelle resonances. It will be more convenient for us to use the operators Pk(τ)\documentclass[12pt]{minimal}
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\begin{document}$$P_k^{(\tau )}$$\end{document} rather than Pk,0(τ)\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}-dependent space of k-forms annihilated by ιXτ\documentclass[12pt]{minimal}
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\begin{document}$$\iota _{X_\tau }$$\end{document}. In the rest of this section we assume that ε0>0\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon _0>0$$\end{document} is chosen small, with the precise value varying from line to line.
We will use the perturbation theory developed in [7]. For an alternative approach, see [16, §6]. Since we are interested in the resonance at 0, we may restrict ourselves to the strip {Imλ>-1}\documentclass[12pt]{minimal}
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\begin{document}$$\{{{\,\mathrm{Im}\,}}\lambda >-1\}$$\end{document}. Following the notation of [12, §6.1], we consider the τ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {H}}_{rG,s}(M;\Omega ^k):=e^{-r{{\,\mathrm{Op}\,}}(G)}H^s(M;\Omega ^k),\quad r\ge 0,\quad s\in {\mathbb {R}}. \end{aligned}$$\end{document}Here Op\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Op}\,}}$$\end{document} is a quantization procedure on M, G(ρ,ξ)=m(ρ,ξ)log(1+|ξ|)\documentclass[12pt]{minimal}
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\begin{document}$$\tau \in (-\varepsilon _0,\varepsilon _0)$$\end{document}. The space Hs\documentclass[12pt]{minimal}
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\begin{document}$$H^s$$\end{document} is the usual Sobolev space of order s. Denote the domain of Pk(τ)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}_{rG,s}$$\end{document} byDrG,s(τ)(M;Ωk):={u∈HrG,s(M;Ωk)∣Pk(τ)u∈HrG,s(M;Ωk)}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {D}}^{(\tau )}_{rG,s}(M;\Omega ^k):=\{u\in {\mathcal {H}}_{rG,s}(M;\Omega ^k)\mid P_k^{(\tau )}u\in {\mathcal {H}}_{rG,s}(M;\Omega ^k)\}. \end{aligned}$$\end{document}The following lemma summarizes the perturbation theory used here. For details see for example [7, Theorem 1 and Corollary 2] or [12, Lemma 6.1 and §6.2].
Lemma 4.3
There exists a constant C0\documentclass[12pt]{minimal}
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\begin{document}$$r>C_0+|s|$$\end{document} and τ∈(-ε0,ε0)\documentclass[12pt]{minimal}
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\begin{document}$$\tau \in (-\varepsilon _0,\varepsilon _0)$$\end{document}, the operatorPk(τ)-λ:DrG,s(τ)(M;Ωk)→HrG,s(M;Ωk),Imλ>-1\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} P_k^{(\tau )}-\lambda :{\mathcal {D}}^{(\tau )}_{rG,s}(M;\Omega ^k)\rightarrow {\mathcal {H}}_{rG,s}(M;\Omega ^k),\quad {{\,\mathrm{Im}\,}}\lambda >-1 \end{aligned}$$\end{document}is Fredholm and its inverse (assuming λ\documentclass[12pt]{minimal}
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\begin{document}$$R_k^{(\tau )}(\lambda )$$\end{document}. Moreover, the set of pairs (τ,λ)\documentclass[12pt]{minimal}
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\begin{document}$$P_k^{(\tau )}$$\end{document} is closed and the resolvent Rk(τ)(λ):HrG,s→HrG,s\documentclass[12pt]{minimal}
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\begin{document}$$R_k^{(\tau )}(\lambda ):{\mathcal {H}}_{rG,s}\rightarrow {\mathcal {H}}_{rG,s}$$\end{document} is bounded locally uniformly in τ,λ\documentclass[12pt]{minimal}
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\begin{document}$$\tau ,\lambda $$\end{document} outside of this set.
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\begin{document}$$R_k^{(\tau )}(\lambda )$$\end{document} is the inverse of Pk(τ)-λ\documentclass[12pt]{minimal}
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\begin{document}$$P_k^{(\tau )}-\lambda $$\end{document} on anisotropic Sobolev spaces, we have the resolvent identity for all τ,τ′∈(-ε0,ε0)\documentclass[12pt]{minimal}
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\begin{document}$$\tau ,\tau '\in (-\varepsilon _0,\varepsilon _0)$$\end{document}Rk(τ)(λ)-Rk(τ′)(λ)=Rk(τ)(λ)(Pk(τ′)-Pk(τ))Rk(τ′)(λ),Imλ>-1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} R_k^{(\tau )}(\lambda )-R_k^{(\tau ')}(\lambda )=R_k^{(\tau )}(\lambda )(P_k^{(\tau ')}-P_k^{(\tau )})R_k^{(\tau ')}(\lambda ),\quad {{\,\mathrm{Im}\,}}\lambda >-1.\nonumber \\ \end{aligned}$$\end{document}Here the right-hand side is well-defined since for r>C0+|s|+1\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}_{rG,s}$$\end{document} to HrG,s-1\documentclass[12pt]{minimal}
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\begin{document}$$R_k^{(\tau )}(\lambda )$$\end{document} maps HrG,s-1\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}_{rG,s-1}$$\end{document} to itself. Using (4.12) we see that for r>C0+|s|+1\documentclass[12pt]{minimal}
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\begin{document}$$r>C_0+|s|+1$$\end{document} the family Rk(τ)(λ):HrG,s→HrG,s-1\documentclass[12pt]{minimal}
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\begin{document}$$R_k^{(\tau )}(\lambda ):{\mathcal {H}}_{rG,s}\rightarrow {\mathcal {H}}_{rG,s-1}$$\end{document} is locally Lipschitz continuous in τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}. Next, recalling (4.3) and that Pk(τ)=-iLXτ\documentclass[12pt]{minimal}
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\begin{document}$$P_k^{(\tau )}=-i{\mathcal {L}}_{X_\tau }$$\end{document}, we have by (4.12)∂τRk(τ)(λ)|τ=0=iRk(λ)LYRk(λ)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \partial _\tau R_k^{(\tau )}(\lambda )|_{\tau =0}=iR_k(\lambda ){\mathcal {L}}_Y R_k(\lambda ) \end{aligned}$$\end{document}as operators HrG,s→HrG,s-2\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}_{rG,s}\rightarrow {\mathcal {H}}_{rG,s-2}$$\end{document} when r>C0+|s|+2\documentclass[12pt]{minimal}
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\begin{document}$$r>C_0+|s|+2$$\end{document}.
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\begin{document}$$\gamma $$\end{document} in the complex plane which encloses 0 but no other resonances of the unperturbed operators Pk=Pk(0)\documentclass[12pt]{minimal}
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\begin{document}$$P_k^{(\tau )}$$\end{document} lie on the contour γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document}, so we may define the operatorsΠ~k(τ):=-12πi∮γRk(τ)(λ)dλ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\widetilde{\Pi }}_k^{(\tau )}:=-{1\over 2\pi i}\oint _\gamma R_k^{(\tau )}(\lambda )\,d\lambda . \end{aligned}$$\end{document}Unlike the spectral projectors Πk(τ)\documentclass[12pt]{minimal}
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\begin{document}$$\Pi _k^{(\tau )}$$\end{document} corresponding to the resonance at 0, the operators Π~k(τ)\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{\Pi }}_k^{(\tau )}$$\end{document} depend continuously on τ\documentclass[12pt]{minimal}
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\begin{document}$$R_k^{(\tau )}(\lambda )$$\end{document} is continuous in τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}. Moreover, the rank of Π~k(τ)\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{\Pi }}_k^{(\tau )}$$\end{document} is constant in τ∈(-ε0,ε0)\documentclass[12pt]{minimal}
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\begin{document}$$\tau \in (-\varepsilon _0,\varepsilon _0)$$\end{document}, see [12, Lemma 6.2]. By (2.36) we haveΠ~k(0)=Πk:=Πk(0)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\widetilde{\Pi }}_k^{(0)}=\Pi _k:=\Pi _k(0) \end{aligned}$$\end{document}so the rank of Π~k(τ)\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{\Pi }}_k^{(\tau )}$$\end{document} can be computed using the algebraic multiplicities of 0 as a resonance in the unperturbed case τ=0\documentclass[12pt]{minimal}
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\begin{document}$$\tau =0$$\end{document} (using (2.43)):rankΠ~k(τ)=mk(0)=mk,0(0)+mk-1,0(0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{rank}\,}}{\widetilde{\Pi }}_{k}^{(\tau )}=m_k(0)=m_{k,0}(0)+m_{k-1,0}(0). \end{aligned}$$\end{document}By (2.36), we also haveΠ~k(τ)=∑λ∈ΥτkΠk(τ)(λ)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\widetilde{\Pi }}_k^{(\tau )}=\sum _{\lambda \in \Upsilon ^k_\tau } \Pi _k^{(\tau )}(\lambda ) \end{aligned}$$\end{document}where Υτk\documentclass[12pt]{minimal}
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\begin{document}$$\Upsilon ^k_\tau $$\end{document} is the set of resonances of the operator Pk(τ)\documentclass[12pt]{minimal}
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\begin{document}$$P_k^{(\tau )}$$\end{document} which are enclosed by the contour γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document}. Note that by (4.15) and (2.42)Π~k(τ)Πk(τ)(λ)=Πk(τ)(λ)for allλ∈Υτk\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\widetilde{\Pi }}_k^{(\tau )}\Pi _k^{(\tau )}(\lambda )=\Pi _k^{(\tau )}(\lambda )\quad \text {for all}\quad \lambda \in \Upsilon ^k_\tau \end{aligned}$$\end{document}and the range of Π~k(τ)\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{\Pi }}_k^{(\tau )}$$\end{document} is the direct sum of the ranges Res(τ)k,∞(λ)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^{k,\infty }_{(\tau )}(\lambda )$$\end{document} of Πk(τ)(λ)\documentclass[12pt]{minimal}
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\begin{document}$$\Pi _k^{(\tau )}(\lambda )$$\end{document} over λ∈Υτk\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \in \Upsilon ^k_\tau $$\end{document}. In particular, using (2.43) we getrankΠ~k(τ)=∑λ∈Υτk(mk,0(τ)(λ)+mk-1,0(τ)(λ)).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{rank}\,}}{\widetilde{\Pi }}_k^{(\tau )}=\sum _{\lambda \in \Upsilon ^k_\tau } \big (m_{k,0}^{(\tau )}(\lambda )+m_{k-1,0}^{(\tau )}(\lambda )\big ). \end{aligned}$$\end{document}Together with (4.14) and induction on k this implies for |τ|<ε0\documentclass[12pt]{minimal}
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\begin{document}$$|\tau |<\varepsilon _0$$\end{document}∑λ∈Υτkmk,0(τ)(λ)=mk,0(0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sum _{\lambda \in \Upsilon ^k_\tau } m_{k,0}^{(\tau )}(\lambda )=m_{k,0}(0). \end{aligned}$$\end{document}We are now ready to show that under the condition (4.2) the space Res0(τ)1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_{0(\tau )}$$\end{document} of resonant 1-forms at 0 for the perturbed operator P1,0(τ)\documentclass[12pt]{minimal}
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\begin{document}$$P^{(\tau )}_{1,0}$$\end{document}, τ≠0\documentclass[12pt]{minimal}
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\begin{document}$$\tau \ne 0$$\end{document}, consists of closed forms:
Lemma 4.4
Under the assumptions of Theorem 3, there exists ε0>0\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon _0>0$$\end{document} such that for 0<|τ|<ε0\documentclass[12pt]{minimal}
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\begin{document}$$0<|\tau |<\varepsilon _0$$\end{document} we have d(Res0(τ)1)=0\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}^1_{0(\tau )})=0$$\end{document}.
Proof
1. Define the operatorZ(τ):=P1(τ)Π~1(τ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Z(\tau ):=P_1^{(\tau )}{\widetilde{\Pi }}_1^{(\tau )}. \end{aligned}$$\end{document}Roughly speaking this operator contains information about the nonzero resonances of P1(τ)\documentclass[12pt]{minimal}
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\begin{document}$$P_1^{(\tau )}$$\end{document} enclosed by γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document}; in particular, each of the corresponding spaces of generalized resonant states is in the range of Z(τ)\documentclass[12pt]{minimal}
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\begin{document}$$Z(\tau )$$\end{document} as can be seen from (4.16).
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\begin{document}$$\tau =0$$\end{document}, the semisimplicity condition (2.41) holds for the operator P1\documentclass[12pt]{minimal}
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\begin{document}$$P_1$$\end{document} at λ=0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda =0$$\end{document}, as follows from Lemmas 2.4 and 3.10 together with (2.43). Therefore, the range of Π~1(0)=Π1\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{\Pi }}_1^{(0)}=\Pi _1$$\end{document} is contained in Res1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1$$\end{document}, implying thatZ(0)=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Z(0)=0. \end{aligned}$$\end{document}By (4.14), the rank of Π~1(τ)\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{\Pi }}_1^{(\tau )}$$\end{document} can be computed using the algebraic multiplicities of 0 as a resonance in the hyperbolic case τ=0\documentclass[12pt]{minimal}
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\begin{document}$$\tau =0$$\end{document}, which are known by (3.1):rankΠ~1(τ)=2b1(Σ)+1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{rank}\,}}{\widetilde{\Pi }}_1^{(\tau )}=2b_1(\Sigma )+1. \end{aligned}$$\end{document}The intersection of the range of Π~1(τ)\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{\Pi }}_1^{(\tau )}$$\end{document} with the kernel of P1(τ)\documentclass[12pt]{minimal}
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\begin{document}$$P_1^{(\tau )}$$\end{document} is equal to Res(τ)1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_{(\tau )}$$\end{document}. By (2.43) and Lemma 2.4 we have Res(τ)1=Res0(τ)1⊕Cατ\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_{(\tau )}={{\,\mathrm{Res}\,}}^1_{0(\tau )}\oplus {\mathbb {C}}\alpha _\tau $$\end{document}. Next, by Lemma 2.8 and (2.28) we have dimRes0(τ)1=b1(Σ)+dimd(Res0(τ)1)\documentclass[12pt]{minimal}
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\begin{document}$$\dim {{\,\mathrm{Res}\,}}^1_{0(\tau )}=b_1(\Sigma )+\dim d({{\,\mathrm{Res}\,}}^1_{0(\tau )})$$\end{document}. ThereforedimRes(τ)1=b1(Σ)+1+dimd(Res0(τ)1).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \dim {{\,\mathrm{Res}\,}}^1_{(\tau )}=b_1(\Sigma )+1+\dim d({{\,\mathrm{Res}\,}}^1_{0(\tau )}). \end{aligned}$$\end{document}By the Rank-Nullity Theorem and (4.20) we then haverankZ(τ)=b1(Σ)-dimd(Res0(τ)1).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{rank}\,}}Z(\tau )= b_1(\Sigma )-\dim d({{\,\mathrm{Res}\,}}^1_{0(\tau )}). \end{aligned}$$\end{document}2. Since (P1(τ)-λ)R1(τ)(λ)\documentclass[12pt]{minimal}
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\begin{document}$$(P_1^{(\tau )}-\lambda ) R_1^{(\tau )}(\lambda )$$\end{document} is the identity operator, we have for all τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}Z(τ)=-12πi∮γλR1(τ)(λ)dλ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Z(\tau )=-{1\over 2\pi i}\oint _\gamma \lambda R_1^{(\tau )}(\lambda )\,d\lambda . \end{aligned}$$\end{document}Using (4.13) we now compute the derivative∂τZ(0)=-12π∮γλR1(λ)LYR1(λ)dλ=-iΠ1LYΠ1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \partial _\tau Z(0) =-{1\over 2\pi }\oint _\gamma \lambda R_1(\lambda ){\mathcal {L}}_Y R_1(\lambda )\,d\lambda =-i\Pi _1{\mathcal {L}}_Y\Pi _1. \end{aligned}$$\end{document}Here in the second equality we used the Laurent expansion (2.36) for R1(λ)\documentclass[12pt]{minimal}
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\begin{document}$$R_1(\lambda )$$\end{document} at λ0=0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0=0$$\end{document} (recalling that J1(0)=1\documentclass[12pt]{minimal}
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\begin{document}$$J_1(0)=1$$\end{document} by semisimplicity).
By Lemma 4.2, for any u∈Res01\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^1_0$$\end{document}, u∗∈Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$$u_*\in {{\,\mathrm{Res}\,}}^1_{0*}$$\end{document} we have∫Mα∧dα∧(∂τZ(0)u)∧u∗=-i⟨⟨(ιXβ)du,du∗⟩⟩.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _M \alpha \wedge d\alpha \wedge \big (\partial _\tau Z(0)u\big )\wedge u_*=-i\langle \!\langle (\iota _X \beta )du,du_*\rangle \!\rangle . \end{aligned}$$\end{document}By the nondegeneracy assumption (4.2) the bilinear form (4.22) is nondegenerate on u∈Cψ\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathcal {C}}_\psi $$\end{document}, u∗∈Cψ∗\documentclass[12pt]{minimal}
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\begin{document}$$u_*\in {\mathcal {C}}_{\psi *}$$\end{document}. This implies thatrank∂τZ(0)≥dimCψ=b1(Σ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{rank}\,}}\partial _{\tau }Z(0)\ge \dim {\mathcal {C}}_\psi =b_1(\Sigma ). \end{aligned}$$\end{document}Together (4.19) and (4.23) show that for 0<|τ|<ε0\documentclass[12pt]{minimal}
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\begin{document}$$0<|\tau |<\varepsilon _0$$\end{document}rankZ(τ)≥b1(Σ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{rank}\,}}Z(\tau )\ge b_1(\Sigma ). \end{aligned}$$\end{document}Then by (4.21) we have dimd(Res0(τ)1)=0\documentclass[12pt]{minimal}
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\begin{document}$$\dim d({{\,\mathrm{Res}\,}}^1_{0(\tau )})=0$$\end{document} for 0<|τ|<ε0\documentclass[12pt]{minimal}
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\begin{document}$$0< |\tau | < \varepsilon _0$$\end{document} which finishes the proof. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Remark
Lemma 4.4 holds more generally whenever P1,0\documentclass[12pt]{minimal}
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\begin{document}$$P_{1, 0}$$\end{document} is semisimple. If for all contact perturbations (ατ)τ\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha _\tau )_\tau $$\end{document} we would have that (4.2) is trivial, this would imply that du∧du∗=0\documentclass[12pt]{minimal}
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\begin{document}$$du \wedge du_* = 0$$\end{document} for all u∈Res01\documentclass[12pt]{minimal}
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\begin{document}$$u \in {{\,\mathrm{Res}\,}}_0^1$$\end{document} and u∗∈Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$$u_* \in {{\,\mathrm{Res}\,}}_{0*}^1$$\end{document}. When (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma , g)$$\end{document} is hyperbolic, we will show in § 4.2 that this is impossible, while for general (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma , g)$$\end{document} proving such a statement seems out of reach for now.
Together with Lemma 2.4, Lemma 2.9, Lemma 2.11, and (2.28) Lemma 4.4 gives all the conclusions of Theorem 3 except semisimplicity on 2-forms. In particular we have for 0<|τ|<ε0\documentclass[12pt]{minimal}
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\begin{document}$$0<|\tau |<\varepsilon _0$$\end{document} (using (2.43))dimRes0(τ)2=b1(Σ)+2,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \dim {{\,\mathrm{Res}\,}}^2_{0(\tau )}&=b_1(\Sigma )+2, \end{aligned}$$\end{document}d(Res(τ)1,∞)=Cdατ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d({{\,\mathrm{Res}\,}}^{1,\infty }_{(\tau )})&=\mathbb Cd\alpha _\tau . \end{aligned}$$\end{document}To finish the proof of Theorem 3 it remains to establish semisimplicity on 2-forms:
Lemma 4.5
Under the assumptions of Theorem 3, there exists ε0>0\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon _0>0$$\end{document} such that for 0<|τ|<ε0\documentclass[12pt]{minimal}
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\begin{document}$$0<|\tau |<\varepsilon _0$$\end{document} the semisimplicity condition (2.41) holds at λ0=0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0=0$$\end{document} for the operator P2,0(τ)\documentclass[12pt]{minimal}
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\begin{document}$$P^{(\tau )}_{2,0}$$\end{document}.
Proof
We first claim that for 0<|τ|<ε0\documentclass[12pt]{minimal}
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\begin{document}$$0<|\tau |<\varepsilon _0$$\end{document}rank(ατ∧(Π~2(τ)-Π2(τ)))≥rank(ατ∧d(Π~1(τ)-Π1(τ)))≥b1(Σ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{rank}\,}}\big (\alpha _\tau \wedge ({\widetilde{\Pi }}_2^{(\tau )}-\Pi _2^{(\tau )})\big )\ge {{\,\mathrm{rank}\,}}\big (\alpha _\tau \wedge d({\widetilde{\Pi }}_1^{(\tau )}-\Pi _1^{(\tau )})\big )\ge b_1(\Sigma ).\nonumber \\ \end{aligned}$$\end{document}Indeed, by (2.37) and (4.15) we have d(Π~1(τ)-Π1(τ))=(Π~2(τ)-Π2(τ))d\documentclass[12pt]{minimal}
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\begin{document}$$d({\widetilde{\Pi }}_1^{(\tau )}-\Pi _1^{(\tau )})=({\widetilde{\Pi }}_2^{(\tau )}-\Pi _2^{(\tau )})d$$\end{document} which implies the first inequality in (4.26). Next, we have rank(α∧dΠ~1(0))=b1(Σ)+1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{rank}\,}}(\alpha \wedge d{\widetilde{\Pi }}_1^{(0)})=b_1(\Sigma )+1$$\end{document} as the range of dΠ~1(0)\documentclass[12pt]{minimal}
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\begin{document}$$d{\widetilde{\Pi }}_1^{(0)}$$\end{document} is equal to dRes1=Cdα⊕dCψ\documentclass[12pt]{minimal}
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\begin{document}$$d{{\,\mathrm{Res}\,}}^1=\mathbb Cd\alpha \oplus d{\mathcal {C}}_\psi $$\end{document}. Since Π~1(τ)\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{\Pi }}_1^{(\tau )}$$\end{document} depends continuously on τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}, we see that rank(ατ∧dΠ~1(τ))≥b1(Σ)+1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{rank}\,}}(\alpha _\tau \wedge d{\widetilde{\Pi }}_1^{(\tau )})\ge b_1(\Sigma )+1$$\end{document} for all small enough τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}. On the other hand, for τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} small but nonzero we have rankdΠ1(τ)=1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{rank}\,}}d\Pi _1^{(\tau )}=1$$\end{document} by (4.25). Together these imply the second inequality in (4.26).
Now, by (4.15) and (2.43) the range of ατ∧(Π~2(τ)-Π2(τ))\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _\tau \wedge ({\widetilde{\Pi }}_2^{(\tau )}-\Pi _2^{(\tau )})$$\end{document} is contained in the sum of the spaces ατ∧Res0(τ)2,∞(λ)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _\tau \wedge {{\,\mathrm{Res}\,}}^{2,\infty }_{0(\tau )}(\lambda )$$\end{document} over λ∈Υτ2\{0}\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \in \Upsilon ^2_\tau {\setminus } \{0\}$$\end{document}. Therefore (4.26) implies that for 0<|τ|<ε0\documentclass[12pt]{minimal}
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\begin{document}$$0<|\tau |<\varepsilon _0$$\end{document}∑λ∈Υτ2\{0}m2,0(τ)(λ)≥b1(Σ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sum _{\lambda \in \Upsilon ^2_\tau {\setminus } \{0\}} m_{2,0}^{(\tau )}(\lambda )\ge b_1(\Sigma ). \end{aligned}$$\end{document}From (4.18) and (3.1) we see that∑λ∈Υτ2m2,0(τ)(λ)=m2,0(0)=2b1(Σ)+2\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sum _{\lambda \in \Upsilon ^2_\tau }m_{2,0}^{(\tau )}(\lambda )=m_{2,0}(0)=2b_1(\Sigma )+2 \end{aligned}$$\end{document}therefore by (4.27) we have m2,0(τ)(0)≤b1(Σ)+2\documentclass[12pt]{minimal}
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\begin{document}$$m_{2,0}^{(\tau )}(0)\le b_1(\Sigma )+2$$\end{document}. Since dimRes0(τ)2=b1(Σ)+2\documentclass[12pt]{minimal}
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\begin{document}$$\dim {{\,\mathrm{Res}\,}}^2_{0(\tau )}=b_1(\Sigma )+2$$\end{document} by (4.24), we showed that the algebraic and geometric multiplicities for 0 as a resonance of P2,0(τ)\documentclass[12pt]{minimal}
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\begin{document}$$P^{(\tau )}_{2,0}$$\end{document} coincide, finishing the proof. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
The full support property
In this section, we prove a full support statement which will be used in the proof of Theorem 4. In fact, we recall that we need to prove the nondegeneracy assumption (4.2), that is, that ⟨⟨ιXβ∙,∙⟩⟩\documentclass[12pt]{minimal}
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\begin{document}$$\langle \!\langle {\iota _X \beta \bullet , \bullet }\rangle \!\rangle $$\end{document} is nondegenerate on dRes01×dRes0∗1\documentclass[12pt]{minimal}
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\begin{document}$$d{{\,\mathrm{Res}\,}}_0^1 \times d{{\,\mathrm{Res}\,}}_{0*}^1$$\end{document}, and the support properties of elements of dRes0(∗)1\documentclass[12pt]{minimal}
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\begin{document}$$d{{\,\mathrm{Res}\,}}_{0(*)}^1$$\end{document} will be useful. In §§4.2–4.4 we assume that M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$M=S\Sigma $$\end{document} where (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document} is a hyperbolic 3-manifold and the contact form α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} and the spaces of (co-)resonant states at zero Res01\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^1_0$$\end{document}, Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document}.
We consider the case of du, with du∗\documentclass[12pt]{minimal}
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\begin{document}$$du_*$$\end{document} studied similarly. From Lemma 3.8 we know that u is a totally unstable 1-form, which implies that du is a section of Eu∗∧Eu∗\documentclass[12pt]{minimal}
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\begin{document}$$E_u^*\wedge E_u^*$$\end{document}. The latter is a one-dimensional vector bundle over M and ω-\documentclass[12pt]{minimal}
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\begin{document}$$du=f_-\omega _-$$\end{document} for some f-∈DEu∗′(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$f_-\in {\mathcal {D}}'_{E_u^*}(M;{\mathbb {C}})$$\end{document}. Using (4.30) we compute0=d(f-ω-)=(df--2f-α)∧ω-.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} 0=d(f_-\omega _-)=(df_- - 2f_-\alpha )\wedge \omega _-. \end{aligned}$$\end{document}Taking ιX\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X$$\end{document} and ιU-\documentclass[12pt]{minimal}
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\begin{document}$$\iota _{U_-}$$\end{document} of this identity and using that ιXω-=ιU-ω-=ιU-α=0\documentclass[12pt]{minimal}
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\begin{document}$$\iota _X\omega _-=\iota _{U_-}\omega _-=\iota _{U_-}\alpha =0$$\end{document} (recalling the definitions of U1±∗,U2±∗\documentclass[12pt]{minimal}
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\begin{document}$$U_1^{\pm *}, U_2^{\pm *}$$\end{document} in (3.10) and below), we get (4.33).
Finally, (4.32) follows from (4.31) and the following identity which can be verified using either (4.28) and (2.16) or (4.29) and (3.12):α∧ω-∧ω+=-18dvolα.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha \wedge \omega _-\wedge \omega _+=-\textstyle {1\over 8}d{{\,\mathrm{vol}\,}}_\alpha . \end{aligned}$$\end{document}□\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
We can now finish the proof of Proposition 4.6. Given (4.32) it suffices to prove that, assuming that f-≠0\documentclass[12pt]{minimal}
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\begin{document}$$f_-\ne 0$$\end{document} and f+≠0\documentclass[12pt]{minimal}
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\begin{document}$$f_+\ne 0$$\end{document},supp(f-f+)=M.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{supp}\,}}(f_-f_+)=M. \end{aligned}$$\end{document}Let πΓ:SH3→SΣ=M\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Gamma :S{\mathbb {H}}^3\rightarrow S\Sigma =M$$\end{document} be the covering map corresponding to (3.2) and Φ±,B±\documentclass[12pt]{minimal}
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\begin{document}$$\Phi _\pm ,B_\pm $$\end{document} be defined in (3.14). Then by (3.22) and (4.33) we have for any U-∈C∞(SH3;Eu)\documentclass[12pt]{minimal}
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\begin{document}$$U_-\in C^\infty (S{\mathbb {H}}^3;E_u)$$\end{document}, U+∈C∞(SH3;Es)\documentclass[12pt]{minimal}
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\begin{document}$$U_+\in C^\infty (S{\mathbb {H}}^3;E_s)$$\end{document}X(Φ±2(f±∘πΓ))=U±(Φ±2(f±∘πΓ))=0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} X(\Phi _\pm ^{2}(f_\pm \circ \pi _\Gamma ))=U_\pm (\Phi _\pm ^{2}(f_\pm \circ \pi _\Gamma ))=0, \end{aligned}$$\end{document}that is Φ+2(f+∘πΓ)\documentclass[12pt]{minimal}
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\begin{document}$$\Phi _+^2(f_+\circ \pi _\Gamma )$$\end{document} is totally stable and Φ-2(f-∘πΓ)\documentclass[12pt]{minimal}
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\begin{document}$$\Phi _-^2(f_-\circ \pi _\Gamma )$$\end{document} is totally unstable in the sense of Definition 3.2. Similarly to Lemma 3.3 we can then describe the lifts of f±\documentclass[12pt]{minimal}
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\begin{document}$$f_\pm $$\end{document} to SH3\documentclass[12pt]{minimal}
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\begin{document}$$S{\mathbb {H}}^3$$\end{document} in terms of some distributions g±\documentclass[12pt]{minimal}
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\begin{document}$$g_\pm $$\end{document} on the conformal infinity S2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {S}}^2$$\end{document}:f±∘πΓ=Φ±-2(g±∘B±)for someg±∈D′(S2;C).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} f_\pm \circ \pi _\Gamma =\Phi _\pm ^{-2}(g_\pm \circ B_\pm )\quad \text {for some}\quad g_\pm \in {\mathcal {D}}'({\mathbb {S}}^2;{\mathbb {C}}). \end{aligned}$$\end{document}Since f±\documentclass[12pt]{minimal}
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\begin{document}$$f_\pm $$\end{document} are resonant states of X, a result of Weich [54, Theorem 1] shows that suppf+=suppf-=M\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{supp}\,}}f_+={{\,\mathrm{supp}\,}}f_-= M$$\end{document}, which from (4.35) and the facts that Φ±>0\documentclass[12pt]{minimal}
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\begin{document}$$\Phi _\pm > 0$$\end{document}, and that B±\documentclass[12pt]{minimal}
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\begin{document}$$B_\pm $$\end{document} are submersions which map SH3\documentclass[12pt]{minimal}
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\begin{document}$$S{\mathbb {H}}^3$$\end{document} onto S2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {S}}^2$$\end{document}, implies thatsuppg+=suppg-=S2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{supp}\,}}g_+={{\,\mathrm{supp}\,}}g_-={\mathbb {S}}^2. \end{aligned}$$\end{document}We will now use the coordinates (ν-,ν+,t)∈(S2×S2)-×R\documentclass[12pt]{minimal}
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\begin{document}$$(\nu _-, \nu _+, t) \in ({\mathbb {S}}^2 \times {\mathbb {S}}^2)_- \times {\mathbb {R}}$$\end{document} on SH3\documentclass[12pt]{minimal}
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\begin{document}$$S{\mathbb {H}}^3$$\end{document} introduced in (3.16). Then by (4.35) and (3.17) we can write in these coordinates(f-f+)∘πΓ=116|ν--ν+|4g-(ν-)g+(ν+).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (f_-f_+)\circ \pi _\Gamma =\textstyle {1\over 16}|\nu _--\nu _+|^4 g_-(\nu _-)g_+(\nu _+). \end{aligned}$$\end{document}By (4.36), we see that the support of the tensor product g-⊗g+(ν-,ν+)=g-(ν-)g+(ν+)\documentclass[12pt]{minimal}
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\begin{document}$$g_-\otimes g_+(\nu _-,\nu _+)=g_-(\nu _-)g_+(\nu _+)$$\end{document} is equal to the entire S2×S2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {S}}^2\times {\mathbb {S}}^2$$\end{document}, which implies that supp(f-f+)∘πΓ=SH3\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{supp}\,}}(f_-f_+)\circ \pi _\Gamma =S{\mathbb {H}}^3$$\end{document} and thus supp(f-f+)=M\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{supp}\,}}(f_-f_+)=M$$\end{document}. This shows (4.34) and finishes the proof.
Proof of Theorem 4
We first remark that in the special case dimd(Res01)=b1(Σ)=1\documentclass[12pt]{minimal}
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\begin{document}$$\dim d({{\,\mathrm{Res}\,}}^1_0)=b_1(\Sigma )=1$$\end{document}, it is straightforward to see that Proposition 4.6 implies the following simplified version of Theorem 4: for each nonempty open set U⊂M\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {U}}\subset M$$\end{document} there exists a∈C∞(M;R)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {a}}\in C^\infty (M;{\mathbb {R}})$$\end{document} with suppa⊂U\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{supp}\,}}{\mathbf {a}}\subset {\mathscr {U}}$$\end{document} and such that β:=aα\documentclass[12pt]{minimal}
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\begin{document}$$\beta :={\mathbf {a}}\alpha $$\end{document} satisfies (4.2). Indeed, it suffices to fix any nonzero du∈d(Res01)\documentclass[12pt]{minimal}
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\begin{document}$$du\in d({{\,\mathrm{Res}\,}}^1_0)$$\end{document}, du∗∈d(Res0∗1)\documentclass[12pt]{minimal}
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\begin{document}$$du_*\in d({{\,\mathrm{Res}\,}}^1_{0*})$$\end{document}, and choose a\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {a}}$$\end{document} such that ∫Maα∧du∧du∗≠0\documentclass[12pt]{minimal}
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\begin{document}$$\int _M {\mathbf {a}} \alpha \wedge du\wedge du_*\ne 0$$\end{document}. We note that there are examples of hyperbolic 3-manifolds with b1(Σ)=1\documentclass[12pt]{minimal}
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\begin{document}$$b_1(\Sigma ) = 1$$\end{document}, see for instance [24, Theorem 13.4].
For the general case, we will use the following basic fact from linear algebra:
Lemma 4.8
Denote by ⊗2Cn\documentclass[12pt]{minimal}
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\begin{document}$$\otimes ^2 {\mathbb {C}}^n$$\end{document} the space of complex n×n\documentclass[12pt]{minimal}
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\begin{document}$$n\times n$$\end{document} matrices. Assume that V⊂⊗2Cn\documentclass[12pt]{minimal}
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\begin{document}$$V\subset \otimes ^2{\mathbb {C}}^n$$\end{document} is a subspace such that for each v1,v2∈Cn\{0}\documentclass[12pt]{minimal}
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\begin{document}$$v_1,v_2\in {\mathbb {C}}^n{\setminus }\{0\}$$\end{document} there exists B∈V\documentclass[12pt]{minimal}
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\begin{document}$$B\in V$$\end{document} such that ⟨Bv1,v2⟩≠0\documentclass[12pt]{minimal}
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\begin{document}$$\langle Bv_1,v_2\rangle \ne 0$$\end{document}. (Here ⟨∙,∙⟩\documentclass[12pt]{minimal}
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\begin{document}$$\langle \bullet ,\bullet \rangle $$\end{document} denotes the canonical bilinear inner product on Cn\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}^n$$\end{document}.) Then the set of invertible matrices in V is dense.
Proof
Let O\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {O}}$$\end{document} be a nonempty open subset of V. We need to show that O\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {O}}$$\end{document} contains an invertible matrix. Assume that there are no invertible matrices in O\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {O}}$$\end{document}. Let A be a matrix of maximal rank in O\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {O}}$$\end{document}, then k:=rankA<n\documentclass[12pt]{minimal}
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\begin{document}$$k:={{\,\mathrm{rank}\,}}A<n$$\end{document} since A cannot be invertible. There exist bases e1,⋯,en\documentclass[12pt]{minimal}
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\begin{document}$$e_1,\dots ,e_n$$\end{document} and e1∗,⋯,en∗\documentclass[12pt]{minimal}
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\begin{document}$$e_1^*,\dots ,e_n^*$$\end{document} of Cn\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}^n$$\end{document} such that⟨Aej,eℓ∗⟩=1ifj=ℓ≤k;0otherwise.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle Ae_j,e^*_{\ell }\rangle ={\left\{ \begin{array}{ll} 1&{}\text {if }j=\ell \le k;\\ 0&{}\text {otherwise.} \end{array}\right. } \end{aligned}$$\end{document}By the assumption of the lemma, there exists B∈V\documentclass[12pt]{minimal}
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\begin{document}$$B\in V$$\end{document} such that ⟨Bek+1,ek+1∗⟩≠0\documentclass[12pt]{minimal}
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\begin{document}$$\langle Be_{k+1},e^*_{k+1}\rangle \ne 0$$\end{document}. Consider the matrix At=A+tB\documentclass[12pt]{minimal}
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\begin{document}$$A_t=A+tB$$\end{document} which lies in O\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {O}}$$\end{document} for sufficiently small t, and let b(t) be the determinant of the matrix (⟨Atej,eℓ∗⟩)j,ℓ=1k+1\documentclass[12pt]{minimal}
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\begin{document}$$(\langle A_t e_j,e^*_\ell \rangle )_{j,\ell =1}^{k+1}$$\end{document}. Then b(0)=0\documentclass[12pt]{minimal}
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\begin{document}$$b(0)=0$$\end{document} and b′(0)=⟨Bek+1,ek+1∗⟩≠0\documentclass[12pt]{minimal}
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\begin{document}$$b'(0)=\langle Be_{k+1},e^*_{k+1}\rangle \ne 0$$\end{document}. Therefore, for small enough t≠0\documentclass[12pt]{minimal}
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\begin{document}$$t\ne 0$$\end{document} we have b(t)≠0\documentclass[12pt]{minimal}
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\begin{document}$$b(t)\ne 0$$\end{document}, which means that rankAt≥k+1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{rank}\,}}A_t\ge k+1$$\end{document}. This contradicts the fact that k was the maximal rank of any matrix in O\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {O}}$$\end{document}. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
We are now ready to give the proof of Theorem 4. For a∈C∞(M;R)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {a}}\in C^\infty (M;{\mathbb {R}})$$\end{document}, define the bilinear formSa:d(Res01)×d(Res0∗1)→C,Sa(du,du∗)=∫Maα∧du∧du∗.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} S_{{\mathbf {a}}}:d({{\,\mathrm{Res}\,}}^1_0)\times d({{\,\mathrm{Res}\,}}^1_{0*})\rightarrow {\mathbb {C}},\quad S_{{\mathbf {a}}}(du,du_*)=\int _M{\mathbf {a}} \alpha \wedge du\wedge du_*. \end{aligned}$$\end{document}To prove Theorem 4, it then suffices to show that the set of a∈Cc∞(U;R)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {a}}\in C^\infty _{\mathrm {c}}({\mathscr {U}};{\mathbb {R}})$$\end{document} such that Sa\documentclass[12pt]{minimal}
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\begin{document}$$S_{{\mathbf {a}}}$$\end{document} is nondegenerate is open and dense. Since nondegeneracy is an open condition, this set is automatically open. To show that it is dense, consider the finite dimensional vector spaceV:={Sa∣a∈Cc∞(U;R)}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} V:=\{S_{{\mathbf {a}}}\mid {\mathbf {a}}\in C^\infty _{\mathrm {c}}({\mathscr {U}};{\mathbb {R}})\}. \end{aligned}$$\end{document}Choosing bases of the b1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$b_1(\Sigma )$$\end{document}-dimensional spaces d(Res01)\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}^1_0)$$\end{document} and d(Res0∗1)\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}^1_{0*})$$\end{document}, we can identify V with a subspace of ⊗2Cb1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$\otimes ^2{\mathbb {C}}^{b_1(\Sigma )}$$\end{document}. Let du∈d(Res01)\documentclass[12pt]{minimal}
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\begin{document}$$du\in d({{\,\mathrm{Res}\,}}^1_0)$$\end{document}, du∗∈d(Res0∗1)\documentclass[12pt]{minimal}
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\begin{document}$$du_*\in d({{\,\mathrm{Res}\,}}^1_{0*})$$\end{document} be nonzero, then by Proposition 4.6 we have supp(α∧du∧du∗)=M\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{supp}\,}}(\alpha \wedge du\wedge du_*)=M$$\end{document}, so there exists a∈Cc∞(U;R)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {a}}\in C^\infty _{\mathrm {c}}({\mathscr {U}};{\mathbb {R}})$$\end{document} such that Sa(du,du∗)≠0\documentclass[12pt]{minimal}
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\begin{document}$$S_{{\mathbf {a}}}(du,du_*)\ne 0$$\end{document}. Then by Lemma 4.8 the set of nondegenerate bilinear forms in V is dense.
Let U\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {U}}$$\end{document} be a nonempty open subset of Cc∞(U;R)\documentclass[12pt]{minimal}
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\begin{document}$$C^\infty _{\mathrm {c}}({\mathscr {U}};{\mathbb {R}})$$\end{document}. Then {Sa∣a∈U}\documentclass[12pt]{minimal}
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\begin{document}$$\{S_{{\mathbf {a}}}\mid {\mathbf {a}}\in {\mathbf {U}}\}$$\end{document} is a nonempty open subset of V. Thus there exists a∈U\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {a}}\in {\mathbf {U}}$$\end{document} such that Sa\documentclass[12pt]{minimal}
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\begin{document}$$S_{{\mathbf {a}}}$$\end{document} is nondegenerate, which finishes the proof.
Proof of Theorem 1
We now give the proof of part 2 of Theorem 1, relying on Theorem 5 (in §5) and Proposition A.1 below, combined together in Corollary 5.1. (Part 1 of Theorem 1 was proved in Corollary 3.1 above.)
We start by computing how a general metric perturbation affects the contact form for the geodesic flow. Let (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document} be any compact 3-dimensional Riemannian manifold and the contact form α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} and the generator X of the geodesic flow on SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document} be defined as in §2.2. Letgτ,τ∈(-ε,ε)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} g_\tau ,\quad \tau \in (-\varepsilon ,\varepsilon ) \end{aligned}$$\end{document}be a family of Riemannian metrics on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} depending smoothly on τ\documentclass[12pt]{minimal}
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\begin{document}$$g_0=g$$\end{document}. The associated geodesic flows act on the τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}-dependent sphere bundlesS(τ)Σ={(x,v)∈TΣ:|v|gτ=1}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} S^{(\tau )}\Sigma =\{(x,v)\in T\Sigma :|v|_{g_\tau }=1\}. \end{aligned}$$\end{document}To bring these geodesic flows to SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S\Sigma $$\end{document}, we use the diffeomorphismsΦτ:SΣ→S(τ)Σ,Φτ(x,v)=(x,v|v|gτ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Phi _\tau :S\Sigma \rightarrow S^{(\tau )}\Sigma ,\quad \Phi _\tau (x,v)=\bigg (x,{v\over |v|_{g_\tau }}\bigg ). \end{aligned}$$\end{document}Denote by ατ\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _\tau $$\end{document} the contact form on S(τ)Σ\documentclass[12pt]{minimal}
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\begin{document}$$S^{(\tau )}\Sigma $$\end{document} corresponding to gτ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\widetilde{\alpha }}_\tau :=\Phi _\tau ^*\alpha _\tau \end{aligned}$$\end{document}is a contact 1-form on SΣ\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g_\tau )$$\end{document} pulled back by Φτ\documentclass[12pt]{minimal}
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\begin{document}$$\Phi _\tau $$\end{document}.
Let πΣ(τ):S(τ)Σ→Σ\documentclass[12pt]{minimal}
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\begin{document}$$\pi ^{(\tau )}_\Sigma :S^{(\tau )}\Sigma \rightarrow \Sigma $$\end{document} be the projection map. Using (2.11) and the fact that πΣ(τ)∘Φτ\documentclass[12pt]{minimal}
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\begin{document}$$\xi \in T_{(x,v)}(S\Sigma )$$\end{document}⟨α~τ(x,v),ξ⟩=⟨v,dπΣ(x,v)ξ⟩gτ|v|gτ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle {\widetilde{\alpha }}_\tau (x,v),\xi \rangle ={\langle v,d\pi _\Sigma (x,v)\xi \rangle _{g_\tau }\over |v|_{g_\tau }}. \end{aligned}$$\end{document}Recalling dπΣ(x,v)X(x,v)=v\documentclass[12pt]{minimal}
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\begin{document}$$d\pi _{\Sigma }(x, v)X(x, v) = v$$\end{document} (see (2.18)) and using g0(v,v)=1\documentclass[12pt]{minimal}
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\begin{document}$$g_0(v, v) = 1$$\end{document}, it follows thatιX∂τα~τ|τ=0(x,v)=∂τgτ(v,v)|τ=0-12g0(v,v)·∂τgτ(v,v)|τ=0=∂τ|v|gτ|τ=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \iota _X\partial _\tau {\widetilde{\alpha }}_\tau |_{\tau =0}(x,v)= & {} \partial _\tau g_\tau (v, v)|_{\tau = 0} - \frac{1}{2} g_0(v, v) \cdot \partial _\tau g_\tau (v, v)|_{\tau = 0}\nonumber \\= & {} \partial _\tau |v|_{g_\tau }|_{\tau =0}. \end{aligned}$$\end{document}In particular, if the metric gτ\documentclass[12pt]{minimal}
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\begin{document}$$g_\tau =e^{-2\tau {\mathbf {b}}}g$$\end{document}, where b∈C∞(Σ;R)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \iota _X\partial _\tau {\widetilde{\alpha }}_\tau |_{\tau =0}(x,v)=-{\mathbf {b}}\circ \pi _\Sigma . \end{aligned}$$\end{document}We are now ready to prove Theorem 1. Assume that (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$g_\tau :=e^{-2\tau {\mathbf {b}}}g$$\end{document}. By Theorem 3 applied to the family of contact forms α~τ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (du,du_*)\mapsto \int _M ({\mathbf {b}}\circ \pi _\Sigma )\alpha \wedge du\wedge du_* \end{aligned}$$\end{document}is nondegenerate on d(Res01)×d(Res0∗1)\documentclass[12pt]{minimal}
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\begin{document}$$d({{\,\mathrm{Res}\,}}^1_0)\times d({{\,\mathrm{Res}\,}}^1_{0*})$$\end{document}.
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\begin{document}$${{\,\mathrm{Res}\,}}^1_0$$\end{document} is preserved by complex conjugation as follows from its definition (2.60); here we use that for any u we have WF(u¯)={(ρ,-ξ)∣(ρ,ξ)∈WF(u)}\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\widetilde{S}}_{{\mathbf {b}}}(du,du'):=\int _M ({\mathbf {b}}\circ \pi _\Sigma ) \alpha \wedge du\wedge {\mathcal {J}}^*(du') \end{aligned}$$\end{document}is nondegenerate on d(Res0R1)×d(Res0R1)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {b}}\circ \pi _\Sigma $$\end{document} is J\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {J}}$$\end{document}-invariant, J∗α=-α\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {J}}^*\alpha =-\alpha $$\end{document}, and J\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {J}}$$\end{document} is an orientation reversing diffeomorphism on M, we see that S~b\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{S}}_{{\mathbf {b}}}$$\end{document} is a symmetric bilinear form. Unlike in the contact perturbation case in § 4.3, we will not be able to produce for every pair (du,du′)∈d(Res0R1)×d(Res0R1)\documentclass[12pt]{minimal}
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\begin{document}$$(du, du') \in d({{\,\mathrm{Res}\,}}^1_{0{\mathbb {R}}})\times d({{\,\mathrm{Res}\,}}^1_{0{\mathbb {R}}})$$\end{document} an element b∈C∞(Σ;R)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {b}} \in C^\infty (\Sigma ; {\mathbb {R}})$$\end{document} such that S~b(du,du′)≠0\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{S}}_{{\mathbf {b}}}(du,du') \ne 0$$\end{document}. Instead, we will only produce b\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {b}}$$\end{document} such that S~b(du,du)≠0\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{S}}_{{\mathbf {b}}}(du,du) \ne 0$$\end{document}. Hence, we will need the following variant of Lemma 4.8 for symmetric matrices:
Lemma 4.9
Denote by ⊗S2Rn\documentclass[12pt]{minimal}
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\begin{document}$$\otimes ^2_S {\mathbb {R}}^n$$\end{document} the space of real symmetric n×n\documentclass[12pt]{minimal}
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\begin{document}$$n\times n$$\end{document} matrices. Assume that V⊂⊗S2Rn\documentclass[12pt]{minimal}
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\begin{document}$$V\subset \otimes ^2_S{\mathbb {R}}^n$$\end{document} is a subspace such that for each w∈Rn\{0}\documentclass[12pt]{minimal}
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\begin{document}$$w\in {\mathbb {R}}^n{\setminus }\{0\}$$\end{document} there exists B∈V\documentclass[12pt]{minimal}
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\begin{document}$$B\in V$$\end{document} such that ⟨Bw,w⟩≠0\documentclass[12pt]{minimal}
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\begin{document}$$\langle Bw,w\rangle \ne 0$$\end{document}. Then the set of invertible matrices in V is dense.
Proof
Similarly to the proof of Lemma 4.8, assume that O\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {O}}$$\end{document} is a nonempty open subset of V which does not contain any invertible matrices and A is a matrix in O\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {O}}$$\end{document} of maximal rank k<n\documentclass[12pt]{minimal}
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\begin{document}$$k<n$$\end{document}. Since A is symmetric, it can be diagonalized, i.e. there exists an orthonormal basis e1,⋯,en\documentclass[12pt]{minimal}
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\begin{document}$$e_1,\dots ,e_n$$\end{document} of Rn\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^n$$\end{document} such that Aej=λjej\documentclass[12pt]{minimal}
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\begin{document}$$Ae_j=\lambda _j e_j$$\end{document} where λj\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _j$$\end{document} are real and, since rankA=k\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{rank}\,}}A=k$$\end{document}, we may assume that λ1,⋯,λk≠0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _1,\dots ,\lambda _k\ne 0$$\end{document} and λk+1=⋯=λn=0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{k+1}=\cdots =\lambda _n=0$$\end{document}.
By the assumption of the lemma, there exists B∈V\documentclass[12pt]{minimal}
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\begin{document}$$B\in V$$\end{document} such that ⟨Bek+1,ek+1⟩≠0\documentclass[12pt]{minimal}
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\begin{document}$$\langle Be_{k+1},e_{k+1}\rangle \ne 0$$\end{document}. Consider the matrix At=A+tB\documentclass[12pt]{minimal}
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\begin{document}$$A_t=A+tB$$\end{document} which lies in O\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {O}}$$\end{document} for sufficiently small t, and let b(t) be the determinant of the matrix (⟨Atei,ej⟩)i,j=1k+1\documentclass[12pt]{minimal}
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\begin{document}$$(\langle A_t e_i,e_j\rangle )_{i,j=1}^{k+1}$$\end{document}. Then b(0)=0\documentclass[12pt]{minimal}
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\begin{document}$$b(0)=0$$\end{document} and b′(0)=λ1⋯λk⟨Bek+1,ek+1⟩≠0\documentclass[12pt]{minimal}
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\begin{document}$$b'(0)=\lambda _1\cdots \lambda _k\langle Be_{k+1},e_{k+1}\rangle \ne 0$$\end{document}. Therefore, for small enough t≠0\documentclass[12pt]{minimal}
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\begin{document}$$t\ne 0$$\end{document} we have b(t)≠0\documentclass[12pt]{minimal}
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\begin{document}$$b(t)\ne 0$$\end{document}, which means that rankAt≥k+1\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{rank}\,}}A_t\ge k+1$$\end{document}. This contradicts the fact that k was the maximal rank of any matrix in O\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {O}}$$\end{document}. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Now to show Theorem 1 it remains to follow the argument at the end of §4.3, with Lemma 4.8 replaced by Lemma 4.9 and using the following
Proposition 4.10
Assume that u∈Res0R1\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^1_{0{\mathbb {R}}}$$\end{document} and du≠0\documentclass[12pt]{minimal}
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\begin{document}$$du\ne 0$$\end{document}. Then there exists b∈C∞(Σ;R)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {b}}\in C^\infty (\Sigma ;{\mathbb {R}})$$\end{document} such that S~b(du,du)≠0\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{S}}_{{\mathbf {b}}}(du,du)\ne 0$$\end{document}.
Proof
Using the pushforward map πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}$$\end{document} defined in (2.19) we compute by (2.20) and (2.23)S~b(du,du)=-∫ΣbπΣ∗(α∧du∧J∗(du)).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\widetilde{S}}_{{\mathbf {b}}}(du,du)=-\int _\Sigma {\mathbf {b}}\pi _{\Sigma *}^{}(\alpha \wedge du\wedge {\mathcal {J}}^*(du)). \end{aligned}$$\end{document}By Corollary 5.1 below we have πΣ∗(α∧du∧J∗(du))≠0\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}(\alpha \wedge du\wedge {\mathcal {J}}^*(du))\ne 0$$\end{document} which finishes the proof. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
The pushforward identity
In this section we prove an identity, Theorem 5, used in Proposition 4.10 above which is a key component in the proof of our main Theorem 1.
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\begin{document}$$(\Sigma ,g)$$\end{document} is a compact hyperbolic 3-manifold as defined in §3.1 and write Σ=Γ\H3\documentclass[12pt]{minimal}
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\begin{document}$$s>2$$\end{document}, define the operatorQs:Cc∞(H3)→C∞(H3),Qsf(x):=∫H3(coshdH3(x,y))-sf(y)dvolg(y).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Q_s&:C^\infty _{\mathrm {c}}({\mathbb {H}}^3)\rightarrow C^\infty ({\mathbb {H}}^3),\quad Q_s f(x)\nonumber \\&:=\int _{{\mathbb {H}}^3} \big (\cosh d_{{\mathbb {H}}^3}(x,y)\big )^{-s}\,f(y)\,d{{\,\mathrm{vol}\,}}_g(y). \end{aligned}$$\end{document}As shown in §5.1.2 below, the operator Qs\documentclass[12pt]{minimal}
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\begin{document}$$Q_s$$\end{document} can be extended to Γ\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {H}}^3$$\end{document} and it is smoothing, so it descends to an operatorQs:D′(Σ;C)→C∞(Σ;C).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Q_s:{\mathcal {D}}'(\Sigma ;{\mathbb {C}})\rightarrow C^\infty (\Sigma ;{\mathbb {C}}). \end{aligned}$$\end{document}Let Δg\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document}. Recall the pushforward map on forms πΣ∗\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{\Sigma *}^{}$$\end{document} defined in (2.19) and the spaces of (co-)resonant k-forms Res0k,Res0∗k\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Res}\,}}^k_0,{{\,\mathrm{Res}\,}}^k_{0*}$$\end{document} on M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document}, see §§2.2–2.3.
By (4.39) and since Q4\documentclass[12pt]{minimal}
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\begin{document}$$Q_4$$\end{document} is self-adjoint we can rewrite (5.5) as follows: for each b∈D′(Σ)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {b}}\in {\mathcal {D}}'(\Sigma )$$\end{document},16∫ΣbΔg(σ-·σ+)dvolg=∫SΣ(πΣ∗Q4b)α∧du∧du∗.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {1\over 6}\int _\Sigma {\mathbf {b}}\,\Delta _g(\sigma _-\cdot \sigma _+)\,d{{\,\mathrm{vol}\,}}_g =\int _{S\Sigma }(\pi _\Sigma ^*Q_4{\mathbf {b}})\alpha \wedge du\wedge du_*. \end{aligned}$$\end{document}One can think of the right-hand side of (5.6) as the integral of πΣ∗Q4b\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Sigma ^*Q_4{\mathbf {b}}$$\end{document} against a Patterson–Sullivan distributionα∧du∧du∗\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \wedge du\wedge du_*$$\end{document} (note that this distribution is invariant under the geodesic flow) and the left-hand side of (5.6) as a topological quantity because it features harmonic 1-forms. Then (5.6) bears some similarity to the result of Anantharaman–Zelditch [2, Theorem 1.1] for the symbol a:=πΣ∗b\documentclass[12pt]{minimal}
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\begin{document}$$a:=\pi _\Sigma ^*{\mathbf {b}}$$\end{document}; the latter is in the setting when Σ\documentclass[12pt]{minimal}
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\begin{document}$$L_r$$\end{document} used in [2] is different in nature from the operator Q4\documentclass[12pt]{minimal}
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\begin{document}$$Q_4$$\end{document} featured in (5.6): for our application is crucial that the right-hand side of (5.6) depends only on the pushforward of α∧du∧du∗\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \wedge du\wedge du_*$$\end{document} to Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} and that does not seem to typically be the case for the right-hand side of [2, Theorem 1.1]. See also the work of Hansen–Hilgert–Schröder [37] giving an asymptotic statement for higher dimensional situations.
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\begin{document}$${\mathbf {b}}\equiv 1$$\end{document} (which is trivial in our situation because both sides are equal to 0) also has some similarity to the pairing formulas of Dyatlov–Faure–Guillarmou [14, Lemma 5.10] and Guillarmou–Hilgert–Weich [32, Theorem 5]. In this vague analogy between Theorem 5 and the results of [2, 14, 32] our setting would correspond to an exceptional value of the spectral parameter: comparing (5.32) with [2, (1.3)] gives the value s=-2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {s}}=-2$$\end{document} (in the notation of [2]).
Together with Proposition A.1, Theorem 5 gives the following statement which is used in the proof of Proposition 4.10. Recall the map J(x,v)=(x,-v)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {J}}(x,v)=(x,-v)$$\end{document} defined in (2.12).
Put u∗=J∗u∈Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$$u_*={\mathcal {J}}^*u\in {{\,\mathrm{Res}\,}}^1_{0*}$$\end{document}. By (2.13) and (2.24) we have σ+=σ-\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _+=\sigma _-$$\end{document} where the 1-forms σ±\documentclass[12pt]{minimal}
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\begin{document}$$\sigma =\sigma _+=\sigma _-$$\end{document} is a real-valued harmonic 1-form on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}, and du≠0\documentclass[12pt]{minimal}
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\begin{document}$$du\ne 0$$\end{document} implies that σ≠0\documentclass[12pt]{minimal}
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\begin{document}$$\sigma \ne 0$$\end{document}.
Let F be defined in (5.4), then by Theorem 5 we haveQ4F=-16Δg|σ|g2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Q_4F=-\textstyle {1\over 6}\Delta _g|\sigma |_g^2. \end{aligned}$$\end{document}Now, by Proposition A.1 we see that |σ|g2\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _g|\sigma |_g^2\ne 0$$\end{document}. Therefore, Q4F≠0\documentclass[12pt]{minimal}
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\begin{document}$$Q_4F\ne 0$$\end{document} which implies that F≠0\documentclass[12pt]{minimal}
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\begin{document}$$F\ne 0$$\end{document}. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Preliminary steps
We first prove several preliminary statements. We will use the hyperboloid model of §3.1.
Hyperbolic Laplacian
We first write the Laplacian Δg\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _g$$\end{document} of the hyperbolic metric on H3\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {H}}^3$$\end{document} using the hyperboloid model. Consider the open coneC+:={(x~0,x~′)∈R1,3:x~0>|x~′|}.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {C}}_+:=\{({\tilde{x}}_0,{\tilde{x}}')\in {\mathbb {R}}^{1,3}:{\tilde{x}}_0>|{\tilde{x}}'|\}. \end{aligned}$$\end{document}Each point x~∈C+\documentclass[12pt]{minimal}
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\begin{document}$${\tilde{x}}\in {\mathcal {C}}_+$$\end{document} can be written in polar coordinates asx~=rx,r>0,x∈H3.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\tilde{x}}=rx,\quad r>0,\quad x\in {\mathbb {H}}^3. \end{aligned}$$\end{document}Define the d’Alembert operator on C+\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}_+$$\end{document} as □=∂x~02-∂x~12-∂x~22-∂x~32\documentclass[12pt]{minimal}
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\begin{document}$$\Box =\partial _{{\tilde{x}}_0}^2-\partial _{{\tilde{x}}_1}^2- \partial _{{\tilde{x}}_2}^2-\partial _{{\tilde{x}}_3}^2$$\end{document}. In polar coordinates it can be written as□=r-2((r∂r)2+2r∂r-Δg)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Box =r^{-2}\big ((r\partial _r)^2+2r\partial _r-\Delta _g\big ) \end{aligned}$$\end{document}where the hyperbolic Laplacian Δg\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _g$$\end{document} acts in the x variable.
Using (5.8), we derive the following useful identity: for any ψ∈C∞((0,∞))\documentclass[12pt]{minimal}
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\begin{document}$$\psi \in C^\infty ((0,\infty ))$$\end{document} and y∈H3\documentclass[12pt]{minimal}
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\begin{document}$$y\in {\mathbb {H}}^3$$\end{document}-Δgψ(⟨x,y⟩1,3)=ψ~(⟨x,y⟩1,3)whereψ~(ρ):=(1-ρ2)ψ′′(ρ)-3ρψ′(ρ)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} -\Delta _g\psi (\langle x,y\rangle _{1,3})&={\widetilde{\psi }}(\langle x,y\rangle _{1,3})\quad \text {where}\quad {\widetilde{\psi }}(\rho )\nonumber \\&:=(1-\rho ^2)\psi ''(\rho )-3\rho \psi '(\rho ) \end{aligned}$$\end{document}and the operator Δg\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _g$$\end{document} acts in the x variable (note that ψ~(ρ)\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\psi }(\rho )$$\end{document} is given by the radial part of -Δg\documentclass[12pt]{minimal}
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\begin{document}$$-\Delta _g$$\end{document} applied to ψ(ρ)\documentclass[12pt]{minimal}
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\begin{document}$$\psi (\rho )$$\end{document} by (3.4)). Indeed, it suffices to apply (5.8) to the function f(x~):=ψ(⟨x~,y⟩1,3)\documentclass[12pt]{minimal}
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\begin{document}$$f({\tilde{x}}):=\psi (\langle {\tilde{x}},y\rangle _{1,3})$$\end{document}, x~∈C+\documentclass[12pt]{minimal}
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\begin{document}$${\tilde{x}}\in {\mathcal {C}}_+$$\end{document}, and use that □f(x~)=ψ′′(⟨x~,y⟩1,3)\documentclass[12pt]{minimal}
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\begin{document}$$\Box f({\tilde{x}})=\psi ''(\langle {\tilde{x}},y\rangle _{1,3})$$\end{document}. Taking in particular ψ(ρ)=ρ-s\documentclass[12pt]{minimal}
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\begin{document}$$\psi (\rho )=\rho ^{-s}$$\end{document} where s∈C\documentclass[12pt]{minimal}
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\begin{document}$$s\in {\mathbb {C}}$$\end{document}, we get(-Δg-s(2-s))⟨x,y⟩1,3-s=s(s+1)⟨x,y⟩1,3-s-2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \big (-\Delta _g-s(2-s)\big )\langle x,y\rangle _{1,3}^{-s}=s(s+1)\langle x,y\rangle _{1,3}^{-s-2}. \end{aligned}$$\end{document}Similarly, if ν-,ν+∈S2⊂R3\documentclass[12pt]{minimal}
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\begin{document}$$\nu _-,\nu _+\in {\mathbb {S}}^2\subset {\mathbb {R}}^3$$\end{document}, then by applying (5.8) to the functionfν-,ν+(x~)=(⟨x~,(1,ν-)⟩1,3⟨x~,(1,ν+)⟩1,3)-1,x~∈C+\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} f_{\nu _-,\nu _+}({\tilde{x}})=\big (\langle {\tilde{x}},(1,\nu _-)\rangle _{1,3}\,\langle {\tilde{x}},(1,\nu _+)\rangle _{1,3}\big )^{-1},\quad {\tilde{x}}\in {\mathcal {C}}_+ \end{aligned}$$\end{document}and using that □fν-,ν+=2(1-ν-·ν+)fν-,ν+2\documentclass[12pt]{minimal}
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\begin{document}$$\Box f_{\nu _-,\nu _+}=2(1-\nu _-\cdot \nu _+)f_{\nu _-,\nu _+}^2$$\end{document}, where we recall ‘·\documentclass[12pt]{minimal}
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\begin{document}$$\cdot $$\end{document}’ denotes the Euclidean inner product, we get-Δg(P(x,ν-)P(x,ν+))=2(1-ν-·ν+)(P(x,ν-)P(x,ν+))2\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} -\Delta _g\big ( P(x,\nu _-)P(x,\nu _+)\big )=2(1-\nu _-\cdot \nu _+)\big (P(x,\nu _-)P(x,\nu _+)\big )^2\nonumber \\ \end{aligned}$$\end{document}where the Poisson kernel P(x,ν)\documentclass[12pt]{minimal}
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\begin{document}$$P(x,\nu )$$\end{document} is defined in (3.18) and the Laplacian Δg\documentclass[12pt]{minimal}
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Properties of the operators Qs\documentclass[12pt]{minimal}
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\begin{document}$$Q_s$$\end{document}
Let Qs:Cc∞(H3)→C∞(H3)\documentclass[12pt]{minimal}
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\begin{document}$$Q_s:C^\infty _{\mathrm {c}}({\mathbb {H}}^3)\rightarrow C^\infty ({\mathbb {H}}^3)$$\end{document} be the operator defined in (5.1). Using (3.4) we can rewrite it asQsf(x)=∫H3⟨x,y⟩1,3-sf(y)dvolg(y).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Q_s f(x)=\int _{{\mathbb {H}}^3} \langle x,y\rangle _{1,3}^{-s}f(y)\,d{{\,\mathrm{vol}\,}}_g(y). \end{aligned}$$\end{document}Note that the operator Qs\documentclass[12pt]{minimal}
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\begin{document}$$Q_s$$\end{document} is equivariant under the action of the group SO+(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{SO}\,}}_+(1,3)$$\end{document}:Qs(γ∗f)=γ∗(Qsf)for allγ∈SO+(1,3).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Q_s (\gamma ^*f)=\gamma ^* (Q_sf)\quad \text {for all}\quad \gamma \in {{\,\mathrm{SO}\,}}_+(1,3). \end{aligned}$$\end{document}For s>2\documentclass[12pt]{minimal}
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\begin{document}$$y\mapsto \langle x,y\rangle _{1,3}^{-s}$$\end{document} lies in L1(H3;dvolg)\documentclass[12pt]{minimal}
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\begin{document}$$L^1({\mathbb {H}}^3;d{{\,\mathrm{vol}\,}}_g)$$\end{document} and its L1\documentclass[12pt]{minimal}
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\begin{document}$$L^1$$\end{document} norm is independent of x; indeed, using the SO+(1,3)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{SO}\,}}_+(1,3)$$\end{document}-invariance we may reduce to the case x=(1,0,0,0)\documentclass[12pt]{minimal}
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\begin{document}$$x=(1,0,0,0)$$\end{document}, which can be handled by an explicit computation. Therefore, Qs:L∞(H3)→L∞(H3)\documentclass[12pt]{minimal}
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\begin{document}$$Q_s:L^\infty ({\mathbb {H}}^3)\rightarrow L^\infty ({\mathbb {H}}^3)$$\end{document}.
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\begin{document}$$Q_s$$\end{document} descends to the quotient Σ=Γ\H3\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma =\Gamma \backslash {\mathbb {H}}^3$$\end{document} as an operatorQs:L∞(Σ)→L∞(Σ),s>2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Q_s:L^\infty (\Sigma )\rightarrow L^\infty (\Sigma ),\quad s>2. \end{aligned}$$\end{document}Next, using (5.10), we get the following identity relating the operators Qs\documentclass[12pt]{minimal}
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\begin{document}$$Q_s$$\end{document} with the hyperbolic Laplacian Δg\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _g$$\end{document} on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}:(-Δg-s(2-s))Qs=Qs(-Δg-s(2-s))=s(s+1)Qs+2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (-\Delta _g-s(2-s))Q_s=Q_s(-\Delta _g-s(2-s))=s(s+1)Q_{s+2}. \end{aligned}$$\end{document}Putting together (5.14) and (5.15) and using elliptic regularity, we see that for any s>2\documentclass[12pt]{minimal}
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\begin{document}$$s>2$$\end{document}, Qs\documentclass[12pt]{minimal}
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\begin{document}$$Q_s$$\end{document} in fact extends to a smoothing operator D′(Σ)→C∞(Σ)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'(\Sigma )\rightarrow C^\infty (\Sigma )$$\end{document}, proving (5.2).
We now show that for f∈D′(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$f\in {\mathcal {D}}'(\Sigma )$$\end{document} one can obtain Qsf\documentclass[12pt]{minimal}
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\begin{document}$$Q_sf$$\end{document} as a limit of cutoff integrals:
Lemma 5.2
Fix a cutoff function χ(ρ)∈Cc∞(R)\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon >0$$\end{document} and s>2\documentclass[12pt]{minimal}
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\begin{document}$$s>2$$\end{document}, define the operatorQs,χ,ε:D′(H3)→C∞(H3),Qs,χ,εf(x)=∫H3χ(ε⟨x,y⟩1,3)⟨x,y⟩1,3-sf(y)dvolg(y).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Q_{s,\chi ,\varepsilon }:{\mathcal {D}}'({\mathbb {H}}^3)\rightarrow & {} C^\infty ({\mathbb {H}}^3),\\&Q_{s,\chi ,\varepsilon }f(x)=\int _{{\mathbb {H}}^3}\chi (\varepsilon \langle x,y\rangle _{1,3}) \langle x,y\rangle _{1,3}^{-s}f(y)\,d{{\,\mathrm{vol}\,}}_g(y). \end{aligned}$$\end{document}Note that Qs,χ,ε\documentclass[12pt]{minimal}
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\begin{document}$$Q_{s,\chi ,\varepsilon }$$\end{document} satisfies the equivariance relation (5.13) and thus descends to an operator D′(Σ)→C∞(Σ)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {D}}'(\Sigma )\rightarrow C^\infty (\Sigma )$$\end{document}. Then we have for all f∈D′(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$f\in {\mathcal {D}}'(\Sigma )$$\end{document}Qs,χ,εf→QsfinC∞(Σ)asε→+0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} Q_{s,\chi ,\varepsilon }f\rightarrow Q_s f\quad \text {in}\quad C^\infty (\Sigma )\quad \text {as}\quad \varepsilon \rightarrow +0. \end{aligned}$$\end{document}
Proof
It suffices to show that for all n≥0\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 0$$\end{document},‖Δgn(Qs-Qs,χ,ε)Δgn‖L∞(Σ)→L∞(Σ)→0asε→+0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert \Delta _g^n (Q_s-Q_{s,\chi ,\varepsilon })\Delta _g^n\Vert _{L^\infty (\Sigma )\rightarrow L^\infty (\Sigma )} \rightarrow 0\quad \text {as}\quad \varepsilon \rightarrow +0. \end{aligned}$$\end{document}By (5.9) with ψ(ρ):=ρ-s(1-χ(ερ))\documentclass[12pt]{minimal}
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\begin{document}$$\psi (\rho ):=\rho ^{-s}(1-\chi (\varepsilon \rho ))$$\end{document} we have (with each instance of Δg\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _g^{2n}$$\end{document} below acting in either x or y)Δg2n(⟨x,y⟩1,3-s(1-χ(ε⟨x,y⟩1,3)))=⟨x,y⟩1,3-sψs,χ,ε(n)(⟨x,y⟩1,3),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Delta _g^{2n}\big (\langle x,y\rangle _{1,3}^{-s}(1-\chi (\varepsilon \langle x,y\rangle _{1,3}))\big ) =\langle x,y\rangle _{1,3}^{-s}\psi _{s,\chi ,\varepsilon }^{(n)}(\langle x,y\rangle _{1,3}), \end{aligned}$$\end{document}where, putting Ts:=ρs((1-ρ2)∂ρ2-3ρ∂ρ)ρ-s\documentclass[12pt]{minimal}
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\begin{document}$$T_s:=\rho ^s\big ((1-\rho ^2)\partial _\rho ^2-3\rho \partial _\rho )\rho ^{-s}$$\end{document},ψs,χ,ε(n)(ρ):=Ts2n(1-χ(ε∙))(ρ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \psi _{s,\chi ,\varepsilon }^{(n)}(\rho ):=T_s^{2n}(1-\chi (\varepsilon \bullet ))(\rho ). \end{aligned}$$\end{document}For any f∈L∞(H3)\documentclass[12pt]{minimal}
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\begin{document}$$f\in L^\infty ({\mathbb {H}}^3)$$\end{document} we have (integrating by parts in y and using the fact that Δg\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _g$$\end{document} is formally self-adjoint)Δgn(Qs-Qs,χ,ε)Δgnf(x)=∫H3⟨x,y⟩1,3-sψs,χ,ε(n)(⟨x,y⟩1,3)f(y)dvolg(y).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Delta _g^n (Q_s-Q_{s,\chi ,\varepsilon })\Delta _g^n f(x) =\int _{{\mathbb {H}}^3} \langle x,y\rangle _{1,3}^{-s}\psi _{s,\chi ,\varepsilon }^{(n)}(\langle x,y\rangle _{1,3})f(y)\,d{{\,\mathrm{vol}\,}}_g(y). \end{aligned}$$\end{document}Estimating the Lx∞Ly1\documentclass[12pt]{minimal}
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\begin{document}$$L^\infty _xL^1_y$$\end{document} norm of the integral kernel of the latter operator we get for any δ∈(0,s-2)\documentclass[12pt]{minimal}
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\begin{document}$$\delta \in (0,s-2)$$\end{document} (we will use that δ>0\documentclass[12pt]{minimal}
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\begin{document}$$\delta > 0$$\end{document} at the end of the proof) and for some Cs,δ>0\documentclass[12pt]{minimal}
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\begin{document}$$s,\delta $$\end{document}‖Δgn(Qs-Qs,χ,ε)Δgn‖L∞(Σ)→L∞(Σ)≤Cs,δsupρ≥1|ρ-δψs,χ,ε(n)(ρ)|.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert \Delta _g^n (Q_s-Q_{s,\chi ,\varepsilon })\Delta _g^n\Vert _{L^\infty (\Sigma )\rightarrow L^\infty (\Sigma )} \le C_{s,\delta }\sup _{\rho \ge 1}|\rho ^{-\delta }\psi _{s,\chi ,\varepsilon }^{(n)}(\rho )|. \end{aligned}$$\end{document}For k∈N0\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert \psi \Vert _{\delta ,k}:=\max _{0\le j\le k}\sup _{\rho \ge 1}|\rho ^{-\delta }(\rho \partial _\rho )^j \psi (\rho )|. \end{aligned}$$\end{document}We have ‖Tsψ‖δ,k≤Cs,δ,k‖ψ‖δ,k+2\documentclass[12pt]{minimal}
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\begin{document}$$\Vert T_s\psi \Vert _{\delta ,k}\le C_{s,\delta ,k}\Vert \psi \Vert _{\delta ,k+2}$$\end{document}. Thereforesupρ≥1|ρ-δψs,χ,ε(n)(ρ)|≤Cs,δ,n‖1-χ(ερ)‖δ,4n=O(εδ),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sup _{\rho \ge 1}|\rho ^{-\delta }\psi _{s,\chi ,\varepsilon }^{(n)}(\rho )| \le C_{s,\delta ,n}\Vert 1-\chi (\varepsilon \rho )\Vert _{\delta ,4n} ={\mathcal {O}}(\varepsilon ^\delta ), \end{aligned}$$\end{document}which finishes the proof. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
In this section we prove an estimate on the norm of Aκ\documentclass[12pt]{minimal}
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\begin{document}$$A_\kappa $$\end{document} between Sobolev spaces, Lemma 5.5, which is used in the regularization argument in §5.2.3 below. Before we state this estimate, we establish a few basic properties of Aκ\documentclass[12pt]{minimal}
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\begin{document}$$A_\kappa $$\end{document}:
It is enough to show that, with ΔS2\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _{{\mathbb {S}}^2}$$\end{document} acting in the ν\documentclass[12pt]{minimal}
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\begin{document}$$\nu $$\end{document} variable,ΔS2(κ(|ν-ν′|2))=κ~(|ν-ν′|2).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Delta _{{\mathbb {S}}^2}(\kappa (|\nu -\nu '|^2))={\tilde{\kappa }}(|\nu -\nu '|^2). \end{aligned}$$\end{document}Similarly to the proof of Lemma 5.3, by SO(3)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{SO}\,}}(3)$$\end{document}-invariance we may reduce to the case ν′=(0,0,-1)\documentclass[12pt]{minimal}
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\begin{document}$$\nu $$\end{document}, in which the Laplace operator is ΔS2=(sinθ)-1∂θsinθ∂θ+(sinθ)-2∂φ2\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _{{\mathbb {S}}^2}=(\sin \theta )^{-1}\partial _\theta \sin \theta \partial _\theta +(\sin \theta )^{-2}\partial _\varphi ^2$$\end{document} and |ν-ν′|2=2+2cosθ\documentclass[12pt]{minimal}
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\begin{document}$$|\nu -\nu '|^2=2+2\cos \theta $$\end{document}. Then we computeΔS2(κ(|ν-ν′|2))=1sinθ∂θsinθ∂θκ(2+2cosθ)=4sin2θκ′′(2+2cosθ)-4cosθκ′(2+2cosθ)=κ~(2+2cosθ),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \Delta _{{\mathbb {S}}^2}(\kappa (|\nu -\nu '|^2))&={1\over \sin \theta }\partial _\theta \sin \theta \partial _\theta \kappa (2+2\cos \theta )\\&=4\sin ^2\theta \kappa ''(2+2\cos \theta )-4\cos \theta \kappa '(2+2\cos \theta )\\&={\tilde{\kappa }}(2+2\cos \theta ), \end{aligned} \end{aligned}$$\end{document}which finishes the proof. □\documentclass[12pt]{minimal}
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We can now give
Lemma 5.5
Assume that s1,s2∈R\documentclass[12pt]{minimal}
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\begin{document}$$s_1,s_2\in {\mathbb {R}}$$\end{document} and s2-s1=2ℓ\documentclass[12pt]{minimal}
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\begin{document}$$s_2-s_1=2\ell $$\end{document} for some ℓ∈N0\documentclass[12pt]{minimal}
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\begin{document}$$\ell \in {\mathbb {N}}_0$$\end{document}. Then there exists a constant C depending only on s1,s2\documentclass[12pt]{minimal}
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\begin{document}$$s_1,s_2$$\end{document} such that for all κ∈C∞([0,4])\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \in C^\infty ([0,4])$$\end{document}‖Aκ‖Hs1(S2)→Hs2(S2)≤C∑j=02ℓ‖rmax(j-ℓ,0)∂rjκ(r)‖L1([0,4]).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert A_\kappa \Vert _{H^{s_1}({\mathbb {S}}^2)\rightarrow H^{s_2}({\mathbb {S}}^2)}\le C\sum _{j=0}^{2\ell }\Vert r^{\max (j-\ell ,0)}\partial _r^j\kappa (r)\Vert _{L^1([0,4])}. \end{aligned}$$\end{document}
Proof
Define the differential operator arising from (5.21) (corresponding to 1-ΔS2\documentclass[12pt]{minimal}
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\begin{document}$$1 - \Delta _{{\mathbb {S}}^2}$$\end{document})W:=(r-4)r∂r2+(2r-4)∂r+1.\documentclass[12pt]{minimal}
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\begin{document}$$s_1,s_2$$\end{document}, whose precise value may change from line to line. We have‖Aκ‖Hs1(S2)→Hs2(S2)≤C‖(1-ΔS2)s2/2Aκ(1-ΔS2)-s1/2‖L2(S2)→L2(S2)=C‖(1-ΔS2)ℓAκ‖L2(S2)→L2(S2)=C‖AWℓκ‖L2(S2)→L2(S2)≤C‖Wℓκ‖L1([0,4]).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \Vert A_\kappa \Vert _{H^{s_1}({\mathbb {S}}^2)\rightarrow H^{s_2}({\mathbb {S}}^2)}&\le C\Vert (1-\Delta _{{\mathbb {S}}^2})^{s_2/2}A_\kappa (1-\Delta _{{\mathbb {S}}^2})^{-s_1/2}\Vert _{L^2({\mathbb {S}}^2)\rightarrow L^2({\mathbb {S}}^2)}\\&= C\Vert (1-\Delta _{{\mathbb {S}}^2})^\ell A_\kappa \Vert _{L^2({\mathbb {S}}^2)\rightarrow L^2({\mathbb {S}}^2)}\\&= C\Vert A_{W^\ell \kappa }\Vert _{L^2({\mathbb {S}}^2)\rightarrow L^2({\mathbb {S}}^2)}\\&\le C\Vert W^\ell \kappa \Vert _{L^1([0,4])}. \end{aligned} \end{aligned}$$\end{document}Here in the second equality we used that Aκ\documentclass[12pt]{minimal}
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\begin{document}$$A_\kappa $$\end{document} commutes with ΔS2\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _{{\mathbb {S}}^2}$$\end{document} by Lemma 5.4. In the third inequality we used Lemma 5.4 again. In the last inequality we used Lemma 5.3.
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\begin{document}$$k\ge \max (j-\ell ,0)$$\end{document}. Therefore, ‖Wℓκ‖L1([0,4])\documentclass[12pt]{minimal}
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\begin{document}$$\Vert W^\ell \kappa \Vert _{L^1([0,4])}$$\end{document} is bounded by the right-hand side of (5.22), which finishes the proof. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Proof of Theorem 5
Here we give the proof of Theorem 5, proceeding in several steps. In §5.2.1 we write both sides of (5.5) as integrals featuring some distributions g±\documentclass[12pt]{minimal}
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\begin{document}$$g_\pm $$\end{document} on S2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {S}}^2$$\end{document}. In §5.2.2 we introduce a change of variables which shows that the two integrals are formally equal. In §5.2.3 we prove that regularized versions of the two integrals are equal and show convergence of the regularization to finish the proof.
Denote by πΓ\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Gamma $$\end{document} the covering maps H3→Σ\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {H}}^3\rightarrow \Sigma $$\end{document} and SH3→M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$S{\mathbb {H}}^3\rightarrow M=S\Sigma $$\end{document} (which one is meant will be clear from the context). Since we can choose the representation of Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} as the quotient Γ\H3\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma \backslash {\mathbb {H}}^3$$\end{document} arbitrarily, for any given x∈Σ\documentclass[12pt]{minimal}
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\begin{document}$$x\in \Sigma $$\end{document} we may arrange that πΓ(e0)=x\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Gamma (e_0)=x$$\end{document} wheree0:=(1,0,0,0)∈H3.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} e_0:=(1,0,0,0)\in {\mathbb {H}}^3. \end{aligned}$$\end{document}Therefore, in order to prove Theorem 5 it suffices to consider the case x=πΓ(e0)\documentclass[12pt]{minimal}
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\begin{document}$$x=\pi _\Gamma (e_0)$$\end{document}, i.e. to show thatπΓ∗Q4F(e0)=-16πΓ∗Δg(σ-·σ+)(e0).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _\Gamma ^* Q_4 F(e_0)=-\textstyle {1\over 6}\pi _\Gamma ^*\Delta _g(\sigma _-\cdot \sigma _+)(e_0). \end{aligned}$$\end{document}
Reduction to the conformal boundary
We first express both sides of (5.24) in terms of some distributions g±\documentclass[12pt]{minimal}
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\begin{document}$$g_\pm $$\end{document} on the conformal boundary S2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {S}}^2$$\end{document}.
Let u∈Res01\documentclass[12pt]{minimal}
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\begin{document}$$u\in {{\,\mathrm{Res}\,}}^1_0$$\end{document}, u∗∈Res0∗1\documentclass[12pt]{minimal}
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\begin{document}$$u_*\in {{\,\mathrm{Res}\,}}^1_{0*}$$\end{document}. By Lemma 4.7 we havedu=f-ω-,du∗=f+ω+,α∧du∧du∗=-18f-f+dvolα,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} du=f_-\omega _-,\quad du_*=f_+\omega _+,\quad \alpha \wedge du\wedge du_*=-\textstyle {1\over 8}f_-f_+ d{{\,\mathrm{vol}\,}}_\alpha , \end{aligned}$$\end{document}where by (4.35), the lifts of f-∈DEu∗′(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$f_-\in {\mathcal {D}}'_{E_u^*}(M;{\mathbb {C}})$$\end{document}, f+∈DEs∗′(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$f_+\in {\mathcal {D}}'_{E_s^*}(M;{\mathbb {C}})$$\end{document} to the covering space SH3\documentclass[12pt]{minimal}
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\begin{document}$$S{\mathbb {H}}^3$$\end{document} have the form (recalling the definitions (3.14) of Φ±\documentclass[12pt]{minimal}
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\begin{document}$$\Phi _\pm $$\end{document}, B±\documentclass[12pt]{minimal}
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\begin{document}$$B_\pm $$\end{document})πΓ∗f±=Φ±-2(g±∘B±)for someg±∈D′(S2;C).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _\Gamma ^*f_\pm =\Phi _\pm ^{-2} (g_\pm \circ B_\pm )\quad \text {for some}\quad g_\pm \in {\mathcal {D}}'({\mathbb {S}}^2;{\mathbb {C}}). \end{aligned}$$\end{document}Arguing similarly to (2.21), we see that the distribution F∈D′(Σ;C)\documentclass[12pt]{minimal}
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\begin{document}$$F\in {\mathcal {D}}'(\Sigma ;{\mathbb {C}})$$\end{document} defined in (5.4) can be written as the pushforwardF(x)=14∫SxΣf-(x,v)f+(x,v)dS(v),x∈Σ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F(x)={1\over 4}\int _{S_x\Sigma } f_-(x,v)f_+(x,v)\,dS(v),\quad x\in \Sigma \end{aligned}$$\end{document}where dS is the canonical volume form on the spherical fiber SxΣ\documentclass[12pt]{minimal}
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\begin{document}$$S_x\Sigma $$\end{document}. Therefore, the lift of F to H3\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {H}}^3$$\end{document} has the formπΓ∗F(x)=14∫SxH3(Φ-(x,v)Φ+(x,v))-2g-(B-(x,v))g+(B+(x,v))dS(v).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _\Gamma ^*F(x)={1\over 4}\int _{S_x{\mathbb {H}}^3} \big (\Phi _-(x,v)\Phi _+(x,v)\big )^{-2}g_-(B_-(x,v))g_+(B_+(x,v))\,dS(v).\nonumber \\ \end{aligned}$$\end{document} We next express the harmonic 1-forms σ±\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _\pm $$\end{document} defined in (5.3) in terms of the distributions g±\documentclass[12pt]{minimal}
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\begin{document}$$g_\pm $$\end{document}:
Lemma 5.6
Using the hyperbolic metric, identify the pullbacks πΓ∗σ±\documentclass[12pt]{minimal}
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\begin{document}$$\pi _\Gamma ^*\sigma _\pm $$\end{document} with vector fields on H3\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {H}}^3$$\end{document}. Then for any x∈H3\documentclass[12pt]{minimal}
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\begin{document}$$x\in {\mathbb {H}}^3$$\end{document}πΓ∗σ±(x)=14∫S2g±(ν)v±(x,ν)dS(ν),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _\Gamma ^*\sigma _\pm (x)={1\over 4}\int _{{\mathbb {S}}^2} g_\pm (\nu )v_\pm (x,\nu )\,dS(\nu ), \end{aligned}$$\end{document}where v±(x,ν)∈SxH3⊂TxH3\documentclass[12pt]{minimal}
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\begin{document}$$v_\pm (x,\nu )\in S_x{\mathbb {H}}^3\subset T_x{\mathbb {H}}^3$$\end{document} is defined in (3.20).
Proof
By (3.72) and since du=f-ω-\documentclass[12pt]{minimal}
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\begin{document}$$du=f_-\omega _-$$\end{document}, du∗=f+ω+\documentclass[12pt]{minimal}
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\begin{document}$$du_*=f_+\omega _+$$\end{document} we haveσ±=πΣ∗(f±α∧ω±).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sigma _\pm =\pi _{\Sigma *}^{}(f_\pm \alpha \wedge \omega _\pm ). \end{aligned}$$\end{document}Recall the horizontal/vertical decomposition (2.15). For any (x,v)∈M=SΣ\documentclass[12pt]{minimal}
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\begin{document}$$(x,v)\in M=S\Sigma $$\end{document}, ξ=(ξH,ξV)∈T(x,v)M\documentclass[12pt]{minimal}
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\begin{document}$$\xi = (\xi _H, \xi _V) \in T_{(x,v)}M$$\end{document}, and a positively oriented g-orthonormal basis v,v1,v2∈TxΣ\documentclass[12pt]{minimal}
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\begin{document}$$v,v_1,v_2\in T_x\Sigma $$\end{document} we compute by (2.16) and (4.28)(α∧ω±)(x,v)(ξ,(0,v1),(0,v2))=14⟨ξH,v⟩g.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (\alpha \wedge \omega _\pm )(x,v)(\xi ,(0,v_1),(0,v_2))=\textstyle {1\over 4}\langle \xi _H,v\rangle _g. \end{aligned}$$\end{document}Using the metric g, we identify σ±\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _\pm $$\end{document} with a vector field on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}. Thenσ±(x)=14∫SxΣf±(x,v)vdS(v),x∈Σ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sigma _\pm (x)={1\over 4}\int _{S_x\Sigma }f_\pm (x,v)v\,dS(v),\quad x\in \Sigma . \end{aligned}$$\end{document}It follows that for each x∈H3\documentclass[12pt]{minimal}
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\begin{document}$$x\in {\mathbb {H}}^3$$\end{document}πΓ∗σ±(x)=14∫SxH3Φ±(x,v)-2g±(B±(x,v))vdS(v)=14∫S2g±(ν)v±(x,ν)dS(ν).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \pi _\Gamma ^*\sigma _\pm (x)&={1\over 4}\int _{S_x{\mathbb {H}}^3} \Phi _\pm (x,v)^{-2}g_\pm (B_\pm (x,v)) v\,dS(v)\\&={1\over 4}\int _{{\mathbb {S}}^2} g_\pm (\nu )v_\pm (x,\nu )\,dS(\nu ). \end{aligned} \end{aligned}$$\end{document}Here in the first equality we used (5.25). In the second equality we made the change of variables ν=B±(x,v)\documentclass[12pt]{minimal}
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\begin{document}$$\nu =B_\pm (x,v)$$\end{document} and used (3.21). □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
We note that by the preceding lemma v±(x,ν)\documentclass[12pt]{minimal}
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\begin{document}$$v_\pm (x, \nu )$$\end{document} define vector-valued Poisson kernels in the sense of [43, 51]. From Lemma 5.6 we get the following formula for the right-hand side of (5.24) in terms of the distributions g±\documentclass[12pt]{minimal}
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\begin{document}$$g_\pm $$\end{document}:
Lemma 5.7
We have (here e0\documentclass[12pt]{minimal}
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\begin{document}$$e_0$$\end{document} is defined in (5.23))-πΓ∗Δg(σ-·σ+)(e0)=18∫S2×S2(1-ν-·ν+)2g-(ν-)g+(ν+)dS(ν-)dS(ν+).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} -\pi _\Gamma ^*\Delta _g(\sigma _-\cdot \sigma _+)(e_0) ={1\over 8}\int _{{\mathbb {S}}^2\times {\mathbb {S}}^2} (1-\nu _-\cdot \nu _+)^2g_-(\nu _-)g_+(\nu _+)\,dS(\nu _-)dS(\nu _+).\nonumber \\ \end{aligned}$$\end{document}
Proof
By (3.20) we have for each ν-,ν+∈S2\documentclass[12pt]{minimal}
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\begin{document}$$\nu _-,\nu _+\in {\mathbb {S}}^2$$\end{document} and x∈H3\documentclass[12pt]{minimal}
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\begin{document}$$x\in {\mathbb {H}}^3$$\end{document}⟨v-(x,ν-),v+(x,ν+)⟩g=-⟨v-(x,ν-),v+(x,ν+)⟩1,3=P(x,ν-)P(x,ν+)(1-ν-·ν+)-1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle v_-(x,\nu _-),v_+(x,\nu _+)\rangle _g= & {} -\langle v_-(x,\nu _-),v_+(x,\nu _+)\rangle _{1,3}\\= & {} P(x,\nu _-)P(x,\nu _+)(1-\nu _-\cdot \nu _+)-1. \end{aligned}$$\end{document}With the hyperbolic Laplacian Δg\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _g$$\end{document} acting in the x variable, we then compute by (5.11)-Δg⟨v-(x,ν-),v+(x,ν+)⟩g=2(1-ν-·ν+)2(P(x,ν-)P(x,ν+))2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} -\Delta _g\langle v_-(x,\nu _-),v_+(x,\nu _+)\rangle _g=2(1-\nu _-\cdot \nu _+)^2\big (P(x,\nu _-)P(x,\nu _+)\big )^2. \end{aligned}$$\end{document}Now (5.27) follows from Lemma 5.6 by integration and using that P(e0,ν±)=1\documentclass[12pt]{minimal}
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\begin{document}$$P(e_0,\nu _\pm )=1$$\end{document} by (3.18). □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Change of variables
By (5.26) and (5.12) we can formally write the left-hand side of (5.24) as follows:πΓ∗Q4F(e0)=14∫SH3y0-4(Φ-(y,v)Φ+(y,v))-2×g-(B-(y,v))g+(B+(y,v))dS(v)dvolg(y),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _\Gamma ^*Q_4 F(e_0)= & {} {1\over 4}\int _{S{\mathbb {H}}^3}y_0^{-4} \big (\Phi _-(y,v)\Phi _+(y,v)\big )^{-2}\nonumber \\\times & {} g_-(B_-(y,v))g_+(B_+(y,v)) dS(v)d{{\,\mathrm{vol}\,}}_g(y), \end{aligned}$$\end{document}where we recall y=(y0,y1,y2,y3)∈H3\documentclass[12pt]{minimal}
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\begin{document}$$y = (y_0, y_1, y_2, y_3) \in {\mathbb {H}}^3$$\end{document}. Note that one has to take care when defining the integral above, as g±\documentclass[12pt]{minimal}
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\begin{document}$$g_\pm $$\end{document} are distributions and SH3\documentclass[12pt]{minimal}
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\begin{document}$$S{\mathbb {H}}^3$$\end{document} is noncompact, see §5.2.3 below.
On the other hand, the right-hand side of (5.24) can be expressed using (5.27) as an integral over (ν-,ν+)∈S2×S2\documentclass[12pt]{minimal}
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\begin{document}$$(\nu _-,\nu _+)\in {\mathbb {S}}^2\times {\mathbb {S}}^2$$\end{document}. To prove (5.24) and relate the two integrals we will use the change of variables Ξ:(y,v)↦(ν-,ν+,t)\documentclass[12pt]{minimal}
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\begin{document}$$\Xi :(y,v)\mapsto (\nu _-,\nu _+,t)$$\end{document}, where t∈R\documentclass[12pt]{minimal}
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\begin{document}$$t\in {\mathbb {R}}$$\end{document}, introduced in (3.16). The basic properties of Ξ\documentclass[12pt]{minimal}
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\begin{document}$$\Xi $$\end{document} are collected below in
2. The Jacobian of Ξ\documentclass[12pt]{minimal}
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\begin{document}$$\Xi $$\end{document} at (y, v) with respect to the densities dvolg(y)dS(v)\documentclass[12pt]{minimal}
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\begin{document}$$dS(\nu _-)dS(\nu _+)dt$$\end{document} is equal to 4(Φ-(y,v)Φ+(y,v))-2\documentclass[12pt]{minimal}
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\begin{document}$$4\big (\Phi _-(y,v)\Phi _+(y,v)\big )^{-2}$$\end{document}.
Remark
The identity in part 2 of the above is well-known, see [50, Theorem 8.1.1 on p. 131].
Proof
1. The identity (5.29) follows immediately from (3.17), noting that |ν--ν+|2=2(1-ν-·ν+)\documentclass[12pt]{minimal}
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\begin{document}$$|\nu _--\nu _+|^2=2(1-\nu _-\cdot \nu _+)$$\end{document}. To see (5.30), we compute by (5.29) and (3.16)Φ±(y,v)=2e±t|ν--ν+|,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Phi _\pm (y,v)={2e^{\pm t}\over |\nu _--\nu _+|}, \end{aligned}$$\end{document}which by (3.15) givesy0=Φ-(y,v)+Φ+(y,v)2=2cosht|ν--ν+|.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} y_0={\Phi _-(y,v)+\Phi _+(y,v)\over 2} ={2\cosh t\over |\nu _--\nu _+|}. \end{aligned}$$\end{document}2. Take (y,v)∈SH3\documentclass[12pt]{minimal}
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\begin{document}$$(y,v)\in S{\mathbb {H}}^3$$\end{document}. Let w∈TyH3\documentclass[12pt]{minimal}
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\begin{document}$$\langle v,w\rangle _{1,3}=0$$\end{document}. Then|dB±(y,v)(w,±w)|S2=2|dB±(y,v)(0,w)|S2=2|w|gΦ±(y,v).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} |dB_\pm (y,v)(w,\pm w)|_{{\mathbb {S}}^2}=2|dB_\pm (y,v)(0,w)|_{{\mathbb {S}}^2}={2|w|_g\over \Phi _\pm (y,v)}. \end{aligned}$$\end{document}Here in the first equality we write (w,±w)=(w,∓w)±2(0,w)\documentclass[12pt]{minimal}
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\begin{document}$$(w,\pm w)=(w,\mp w)\pm 2(0,w)$$\end{document} and use that by (3.23), dB±(y,v)(w,∓w)=0\documentclass[12pt]{minimal}
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\begin{document}$$dB_\pm (y,v)(w,\mp w)=0$$\end{document}. In the second equality we use (3.21). Denoting by X the generator of the geodesic flow and defining t by (3.16), we also have by (3.22) and (3.23)dB±(y,v)(X(y,v))=0,dt(X(y,v))=1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} dB_\pm (y,v)(X(y,v))=0,\quad dt(X(y,v))=1. \end{aligned}$$\end{document}Fix a g-orthonormal basis v,v1,v2\documentclass[12pt]{minimal}
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\begin{document}$$v,v_1,v_2$$\end{document} of TyH3\documentclass[12pt]{minimal}
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\begin{document}$$T_y{\mathbb {H}}^3$$\end{document} and consider the following basis of T(y,v)SH3\documentclass[12pt]{minimal}
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\begin{document}$$T_{(y,v)}S{\mathbb {H}}^3$$\end{document}:ξ0=X(y,v),ξ1±=(v1,±v1),ξ2±=(v2,±v2).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \xi _0=X(y,v),\quad \xi _1^\pm =(v_1,\pm v_1),\quad \xi _2^\pm =(v_2,\pm v_2). \end{aligned}$$\end{document}Since ξj-∧ξj+=2(vj,0)∧(0,vj)\documentclass[12pt]{minimal}
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\begin{document}$$\xi _j^-\wedge \xi _j^+=2 (v_j,0)\wedge (0,v_j)$$\end{document}, the value of the density dvolg(y)dS(v)\documentclass[12pt]{minimal}
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\begin{document}$$\xi _0,\xi _1^-,\xi _2^-,\xi _1^+,\xi _2^+$$\end{document} is equal to 4. On the other hand, writing (η-(ξ),η+(ξ),τ(ξ))=dΞ(y,v)(ξ)\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Using Lemma 5.8 and (5.28), we can formally write the left-hand side of (5.24) asπΓ∗Q4F(e0)=164∫(S2×S2)-×R(1-ν-·ν+)2cosh4tg-(ν-)g+(ν+)dS(ν-)dS(ν+)dt.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _\Gamma ^*Q_4F(e_0) ={1\over 64}\int \limits _{({\mathbb {S}}^2\times {\mathbb {S}}^2)_-\times {\mathbb {R}}} {(1-\nu _-\cdot \nu _+)^2\over \cosh ^4t}g_-(\nu _-)g_+(\nu _+) \,dS(\nu _-)dS(\nu _+)dt.\nonumber \\ \end{aligned}$$\end{document} Using the change of variables s=tanht\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _{{\mathbb {R}}}{dt\over \cosh ^4t}=\int _{-1}^1 (1-s^2)\,ds={4\over 3}. \end{aligned}$$\end{document}Comparing (5.32) with (5.27), we formally obtain the identity (5.24). However, our argument is incomplete since the integrals in (5.28) and (5.32) are over the noncompact manifolds SH3\documentclass[12pt]{minimal}
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\begin{document}$$g_\pm $$\end{document} are distributions. Thus one cannot immediately apply the change of variables formula to get (5.32) from (5.28), or Fubini’s Theorem to get (5.24) from (5.32). To deal with these issues, we will employ a regularization procedure.
Regularization and end of the proof
Fix a cutoff functionχ∈Cc∞(R;[0,1]),suppχ⊂[-2,2],χ|[-1,1]=1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} I_\varepsilon :=\int _{{\mathbb {H}}^3} \chi (\varepsilon y_0) y_0^{-4} \pi _\Gamma ^*F(y)\,d{{\,\mathrm{vol}\,}}_g(y). \end{aligned}$$\end{document}(As before, we embed H3\documentclass[12pt]{minimal}
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\begin{document}$$I_\varepsilon $$\end{document} converges to the left-hand side of (5.24):Iε→πΓ∗Q4F(e0)asε→+0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} I_\varepsilon \rightarrow \pi _\Gamma ^*Q_4F(e_0)\quad \text {as}\quad \varepsilon \rightarrow +0. \end{aligned}$$\end{document}By (5.34) and (5.27), the proof of (5.24) (and thus of Theorem 5) is finished once we show thatIε→148∫S2×S2(1-ν-·ν+)2g-(ν-)g+(ν+)dS(ν-)dS(ν+)asε→+0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} I_\varepsilon \rightarrow {1\over 48}\int \limits _{{\mathbb {S}}^2\times {\mathbb {S}}^2} (1-\nu _-\cdot \nu _+)^2g_-(\nu _-)g_+(\nu _+)\, dS(\nu _-)dS(\nu _+)\quad \text {as}\quad \varepsilon \rightarrow +0.\nonumber \\ \end{aligned}$$\end{document}By (5.26) we have the following regularized version of (5.28):Iε=14∫SH3χ(εy0)y0-4(Φ-(y,v)Φ+(y,v))-2×g-(B-(y,v))g+(B+(y,v))dS(v)dvolg(y).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} I_\varepsilon= & {} {1\over 4}\int _{S{\mathbb {H}}^3}\chi (\varepsilon y_0)y_0^{-4} \big (\Phi _-(y,v)\Phi _+(y,v)\big )^{-2}\\\times & {} g_-(B_-(y,v))g_+(B_+(y,v)) \,dS(v)d{{\,\mathrm{vol}\,}}_g(y). \end{aligned}$$\end{document}Making the change of variables (ν-,ν+,t)=Ξ(y,v)\documentclass[12pt]{minimal}
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\begin{document}$$(\nu _-,\nu _+,t)=\Xi (y,v)$$\end{document} and using Lemma 5.8, we then get the following regularized version of (5.32) (we keep in mind that g±\documentclass[12pt]{minimal}
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\begin{document}$$g_\pm $$\end{document} are merely distributions so that all of the integrals around these lines are understood in the distributional sense):Iε=164∫S2×S2×Rχ(2εcosht|ν--ν+|)(1-ν-·ν+)2cosh4t×g-(ν-)g+(ν+)dS(ν-)dS(ν+)dt.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} I_\varepsilon= & {} {1\over 64}\int \limits _{{\mathbb {S}}^2\times {\mathbb {S}}^2\times {\mathbb {R}}} \chi \Big ({2\varepsilon \cosh t\over |\nu _--\nu _+|}\Big ) {(1-\nu _-\cdot \nu _+)^2\over \cosh ^4t}\\\times & {} g_-(\nu _-)g_+(\nu _+) \,dS(\nu _-)dS(\nu _+)dt. \end{aligned}$$\end{document}For r≥0\documentclass[12pt]{minimal}
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\begin{document}$$r\ge 0$$\end{document}, define the functionψε(r):=34∫Rχ(2εcoshtr)cosh-4tdt.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \psi _\varepsilon (r):={3\over 4}\int _{{\mathbb {R}}}\chi \Big ({2\varepsilon \cosh t\over \sqrt{r}}\Big ) \cosh ^{-4}t\,dt. \end{aligned}$$\end{document}Note that ψε∈C∞([0,∞))\documentclass[12pt]{minimal}
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\begin{document}$$r\ll \varepsilon ^2$$\end{document}. We now haveIε=148∫S2×S2ψε(|ν--ν+|2)(1-ν-·ν+)2g-(ν-)g+(ν+)dS(ν-)dS(ν+).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} I_\varepsilon ={1\over 48}\int \limits _{{\mathbb {S}}^2\times {\mathbb {S}}^2} \psi _\varepsilon (|\nu _--\nu _+|^2)(1-\nu _-\cdot \nu _+)^2g_-(\nu _-)g_+(\nu _+)\, dS(\nu _-)dS(\nu _+).\nonumber \\ \end{aligned}$$\end{document}Recalling that |ν--ν+|2=2(1-ν-·ν+)\documentclass[12pt]{minimal}
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\begin{document}$$|\nu _--\nu _+|^2=2(1-\nu _-\cdot \nu _+)$$\end{document}, we see from (5.37) that it suffices to prove the following version of (5.35):∫S2×S2(1-ψε(|ν--ν+|2))|ν--ν+|4g-(ν-)×g+(ν+)dS(ν-)dS(ν+)→0asε→+0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int \limits _{{\mathbb {S}}^2\times {\mathbb {S}}^2}\big (1- & {} \psi _\varepsilon (|\nu _--\nu _+|^2)\big ) |\nu _--\nu _+|^4g_-(\nu _-)\nonumber \\\times & {} g_+(\nu _+)\,dS(\nu _-)dS(\nu _+)\rightarrow 0\quad \text {as}\quad \varepsilon \rightarrow +0. \end{aligned}$$\end{document}If g±\documentclass[12pt]{minimal}
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\begin{document}$$g_\pm $$\end{document} were smooth functions on S2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {S}}^2$$\end{document}, then (5.38) would follow from the Dominated Convergence Theorem since by (5.33) we have ψε(r)→1\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon \rightarrow +0$$\end{document} for all r>0\documentclass[12pt]{minimal}
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\begin{document}$$r>0$$\end{document}. However, g±\documentclass[12pt]{minimal}
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\begin{document}$$g_\pm $$\end{document} are merely distributions, so one has to be more careful. We start by establishing the Sobolev regularity of g±\documentclass[12pt]{minimal}
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\begin{document}$$g_\pm $$\end{document} by following the standard proof of the Fredholm property in anisotropic Sobolev spaces. (We use the proof in [20]; one could alternatively carefully examine the proof in [29].) See the papers of Adam–Baladi [1, §3.3], Guillarmou–Poyferré–Bonthonneau [30, Appendix A], and Dyatlov [19] for a general discussion of Sobolev regularity thresholds for the Pollicott–Ruelle resolvent.
Lemma 5.9
We have g±∈H-2-δ(S2)\documentclass[12pt]{minimal}
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\begin{document}$$g_\pm \in H^{-2-\delta }({\mathbb {S}}^2)$$\end{document} for all δ>0\documentclass[12pt]{minimal}
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\begin{document}$$\delta >0$$\end{document}.
Proof
We show the regularity of g-\documentclass[12pt]{minimal}
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\begin{document}$$g_-$$\end{document}, with g+\documentclass[12pt]{minimal}
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\begin{document}$$g_+$$\end{document} handled similarly. Recall that g-\documentclass[12pt]{minimal}
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\begin{document}$$g_-$$\end{document} is related to the distribution f-∈DEu∗′(M;C)\documentclass[12pt]{minimal}
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\begin{document}$$f_-\in {\mathcal {D}}'_{E_u^*}(M;{\mathbb {C}})$$\end{document} by (5.25). Since Φ-\documentclass[12pt]{minimal}
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\begin{document}$$\Phi _-$$\end{document} is smooth and B-\documentclass[12pt]{minimal}
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\begin{document}$$B_-$$\end{document} is a submersion, it suffices to show that f-∈H-2-δ(M)\documentclass[12pt]{minimal}
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\begin{document}$$f_-\in H^{-2-\delta }(M)$$\end{document}.
By Lemma 4.7, we have (X-2)f-=0\documentclass[12pt]{minimal}
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\begin{document}$$(X-2)f_-=0$$\end{document}, that is f-\documentclass[12pt]{minimal}
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\begin{document}$$f_-$$\end{document} is a Pollicott–Ruelle resonant state for the operator P=-iX\documentclass[12pt]{minimal}
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\begin{document}$$P=-iX$$\end{document} corresponding to the resonance λ0=-2i\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0=-2i$$\end{document}, see §2.3.2. Given that Pollicott–Ruelle resonant states are eigenfunctions of P on anisotropic Sobolev spaces (see (4.10)), it suffices to show that one can choose the order function m in the definition of the weight G(ρ,ξ)=m(ρ,ξ)log(1+|ξ|)\documentclass[12pt]{minimal}
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\begin{document}$$G(\rho ,\xi )=m(\rho ,\xi )\log (1+|\xi |)$$\end{document} such that the Fredholm property (4.11) holds on the anisotropic Sobolev space HG,0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}_{G,0}$$\end{document} for Imλ≥-2\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Im}\,}}\lambda \ge -2$$\end{document} and HG,0⊂H-2-δ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}_{G,0}\subset H^{-2-\delta }$$\end{document}; the latter is equivalent to requiring that m≥-2-δ\documentclass[12pt]{minimal}
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\begin{document}$$m\ge -2-\delta $$\end{document} everywhere.
In [20, §§3.3–3.4] the Fredholm property (4.11) is shown using propagation of singularities and microlocal radial estimates. Following the proof of [20, Proposition 3.4], we see that one only needs to check that the low regularity radial estimate [20, Proposition 2.7] applies to the operator P-λ\documentclass[12pt]{minimal}
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\begin{document}$$P-\lambda $$\end{document} (where Imλ≥-2\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{Im}\,}}\lambda \ge -2$$\end{document}) at the radial sink Eu∗\documentclass[12pt]{minimal}
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\begin{document}$$E_u^*$$\end{document} (see (2.4)) in the space H-2-δ\documentclass[12pt]{minimal}
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\begin{document}$$H^{-2-\delta }$$\end{document}. (The high regularity radial estimate [20, Proposition 2.6] would apply once m is sufficiently large on Es∗\documentclass[12pt]{minimal}
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\begin{document}$$E_s^*$$\end{document}, which can be arranged.) The threshold regularity for this estimate is computed in [22, Theorem E.54]. In our setting, since the operator P is symmetric on L2(M;dvolα)\documentclass[12pt]{minimal}
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\begin{document}$$L^2(M;d{{\,\mathrm{vol}\,}}_\alpha )$$\end{document} and it has order k=1\documentclass[12pt]{minimal}
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\begin{document}$$k=1$$\end{document}, it is enough that2+(-2-δ)Hp|ξ||ξ|<0onEu∗\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} 2+(-2-\delta ){H_p|\xi |\over |\xi |}<0\quad \text {on}\quad E_u^* \end{aligned}$$\end{document}where p(ρ,ξ)=⟨X(ρ),ξ⟩\documentclass[12pt]{minimal}
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\begin{document}$$p(\rho ,\xi )=\langle X(\rho ),\xi \rangle $$\end{document} is the principal symbol of P and its Hamiltonian flow is given by etHp(ρ,ξ)=(φt(ρ),dφt-T(ρ)ξ)\documentclass[12pt]{minimal}
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\begin{document}$$e^{tH_p}(\rho ,\xi )=(\varphi _t(\rho ),d\varphi _t^{-T}(\rho )\xi )$$\end{document}, see [20, §3.1]. Choosing the norm |ξ|\documentclass[12pt]{minimal}
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\begin{document}$$|\xi |$$\end{document} induced by the Sasaki metric and using (3.7), we see thatHp|ξ||ξ|=1onEu∗,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {H_p|\xi |\over |\xi |}=1\quad \text {on}\quad E_u^*, \end{aligned}$$\end{document}which means that the threshold regularity condition for the radial estimate is satisfied and the proof is finished. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Coming back to the proof of (5.38), we rewrite it as⟨Aκεg-,g+¯⟩L2(S2)→0asε→+0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle A_{\kappa _\varepsilon } g_-,\overline{g_+}\rangle _{L^2({\mathbb {S}}^2)}\rightarrow 0\quad \text {as}\quad \varepsilon \rightarrow +0, \end{aligned}$$\end{document}where the operator Aκε\documentclass[12pt]{minimal}
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\begin{document}$$A_{\kappa _\varepsilon }$$\end{document} is given by (5.20):Aκεf(ν+)=∫S2κε(|ν--ν+|2)f(ν-)dS(ν-)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} A_{\kappa _\varepsilon }f(\nu _+)=\int _{{\mathbb {S}}^2}\kappa _\varepsilon (|\nu _--\nu _+|^2)f(\nu _-)\,dS(\nu _-) \end{aligned}$$\end{document}and the function κε∈C([0,4])\documentclass[12pt]{minimal}
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\begin{document}$$\kappa _\varepsilon \in C([0,4])$$\end{document} is given by (using (5.33) and (5.36) in the second equality below)κε(r):=43r2(1-ψε(r))=r2∫R(1-χ(2εcoshtr))cosh-4tdt.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \kappa _\varepsilon (r):={4\over 3}r^2(1-\psi _\varepsilon (r))=r^2\int _{{\mathbb {R}}}\bigg (1-\chi \Big ({2\varepsilon \cosh t\over \sqrt{r}}\Big )\bigg )\cosh ^{-4} t\,dt. \end{aligned}$$\end{document}Using Lemma 5.9, we have in particular g±∈H-5/2(S2)\documentclass[12pt]{minimal}
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\begin{document}$$g_\pm \in H^{-5/2}({\mathbb {S}}^2)$$\end{document}. Thus to finish the proof of (5.39), and thus of Theorem 5, it remains to prove the norm bound‖Aκε‖H-5/2(S2)→H5/2(S2)→0asε→+0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert A_{\kappa _\varepsilon }\Vert _{H^{-5/2}({\mathbb {S}}^2)\rightarrow H^{5/2}({\mathbb {S}}^2)}\rightarrow 0\quad \text {as}\quad \varepsilon \rightarrow +0. \end{aligned}$$\end{document}To show (5.40), we will bound the norms of Aκε\documentclass[12pt]{minimal}
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\begin{document}$$A_{\kappa _\varepsilon }$$\end{document} between Sobolev spaces using Lemma 5.5. To do this we estimate the derivatives of κε\documentclass[12pt]{minimal}
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\begin{document}$$\kappa _\varepsilon $$\end{document}:
Lemma 5.10
Let j,k∈N0\documentclass[12pt]{minimal}
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\begin{document}$$j,k\in {\mathbb {N}}_0$$\end{document}. Then there exists C depending only on j, k such that for all ε∈(0,1]\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon \in (0,1]$$\end{document}‖rk∂rjκε(r)‖L1([0,4])≤Cε4,k≥j;Cε4log(1/ε),k=j-1;Cε2(3+k-j),k≤j-2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert r^k\partial ^j_r \kappa _\varepsilon (r)\Vert _{L^1([0,4])}\le {\left\{ \begin{array}{ll} C\varepsilon ^4,&{} k\ge j;\\ C\varepsilon ^4\log (1/\varepsilon ),&{} k=j-1;\\ C\varepsilon ^{2(3+k-j)},&{} k\le j-2. \end{array}\right. } \end{aligned}$$\end{document}
Proof
Throughout the proof we denote by C a constant depending only on j, k whose precise value might change from line to line.
1. For any G(s)∈C∞([0,∞))\documentclass[12pt]{minimal}
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\begin{document}$$G(s)\in C^\infty ([0,\infty ))$$\end{document} which is constant near s=∞\documentclass[12pt]{minimal}
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\begin{document}$$s=\infty $$\end{document} defineΦG(τ):=∫RG(2coshtτ)cosh-4tdt,τ>0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Phi _G(\tau ):=\int _{{\mathbb {R}}}G\Big ({2\cosh t\over \sqrt{\tau }}\Big )\cosh ^{-4}t\,dt,\quad \tau > 0. \end{aligned}$$\end{document}We have the identityτ∂τΦG=-12Φs∂sG.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \tau \partial _\tau \Phi _G=-\textstyle {1\over 2}\Phi _{s\partial _s G}. \end{aligned}$$\end{document}Moreover, we have the estimateG|[-1,1]=0⟹|ΦG(τ)|≤C‖G‖L∞1+τ2,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} G|_{[-1,1]}=0\quad \Longrightarrow \quad |\Phi _G(\tau )|\le {C\Vert G\Vert _{L^\infty }\over 1+\tau ^2}, \end{aligned}$$\end{document}which can be proved by bounding |ΦG(τ)|\documentclass[12pt]{minimal}
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\begin{document}$$|\Phi _G(\tau )|$$\end{document} by ‖G‖L∞\documentclass[12pt]{minimal}
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\begin{document}$$\cosh t\ge \sqrt{\tau }/2$$\end{document} and using that ∫cosh-4tdt=tanht-13tanh3t+C\documentclass[12pt]{minimal}
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\begin{document}$$\lambda ={4\over \tau }\rightarrow 0$$\end{document}.
Combining Lemma 5.5 and Lemma 5.10, we get‖Aκε‖H-5/2→H3/2≤Cε2,‖Aκε‖H-5/2→H7/2≤C.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert A_{\kappa _\varepsilon }\Vert _{H^{-5/2}\rightarrow H^{3/2}}\le C\varepsilon ^2,\quad \Vert A_{\kappa _\varepsilon }\Vert _{H^{-5/2}\rightarrow H^{7/2}}\le C. \end{aligned}$$\end{document}By interpolation in Sobolev spaces (taking f∈H-5/2(S2)\documentclass[12pt]{minimal}
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\begin{document}$$\Vert v\Vert _{H^1({\mathbb {S}}^2)}^2$$\end{document} is bounded by ⟨(1-ΔS2)v,v⟩L2(S2)≤C‖v‖L2(S2)‖v‖H2(S2)\documentclass[12pt]{minimal}
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\begin{document}$$\langle (1-\Delta _{{\mathbb {S}}^2})v,v\rangle _{L^2({\mathbb {S}}^2)}\le C\Vert v\Vert _{L^2({\mathbb {S}}^2)}\Vert v\Vert _{H^2({\mathbb {S}}^2)}$$\end{document} for v:=(1-ΔS2)3/4Aκεf\documentclass[12pt]{minimal}
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\begin{document}$$v:=(1-\Delta _{{\mathbb {S}}^2})^{3/4}A_{\kappa _\varepsilon }f$$\end{document}) we then have‖Aκε‖H-5/2→H5/2≤Cε.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert A_{\kappa _\varepsilon }\Vert _{H^{-5/2}\rightarrow H^{5/2}}\le C\varepsilon . \end{aligned}$$\end{document}This gives (5.40) and finishes the proof of Theorem 5.
Acknowledgements
We would like to thank Bernd Sturmfels and Nathan Ilten-Gee for useful discussions related to Lemma 4.8, and Tomasz Mrowka for discussions related to Proposition A.1. We are also grateful to the anonymous referees for many suggestions on improving the article. MC and BD have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 725967). MC is further supported by an Ambizione Grant (Project Number 201806) from the Swiss National Science foundation. BD has received further funding from the Deutsche Forschungsgemeinschaft (German Research Foundation, DFG) through the Priority Programme (SPP) 2026 “Geometry at Infinity”. SD was supported by the National Science Foundation (NSF) CAREER Grant DMS-1749858 and a Sloan Research Fellowship. GPP was supported by the Leverhulme trust and EPSRC Grant EP/R001898/1. Part of this project was carried out while the four authors were participating in the Mathematical Sciences Research Institute Program in Microlocal Analysis in Fall 2019, supported by the NSF Grant DMS-1440140.
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The purpose of this appendix is to give an elementary proof of the fact that there are no harmonic 1-forms of constant nonzero length on closed hyperbolic 3-manifolds:
Proposition A.1
Let (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document} be a compact hyperbolic 3-manifold (see §3.1). Assume that ω∈C∞(Σ;T∗Σ)\documentclass[12pt]{minimal}
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\begin{document}$$\omega =0$$\end{document}.
Remark
Proposition A.1 follows directly from the more general work of [55]. The presentation in the appendix borrows from ideas in [39].
To prove Proposition A.1 we argue by contradiction. Assume that ω≠0\documentclass[12pt]{minimal}
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\begin{document}$$\star $$\end{document} is the Hodge star)dω=0,δω=0,|ω|g=1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d\omega =0,\quad \delta \omega =0,\quad |\omega |_g=1. \end{aligned}$$\end{document}Using the metric g, define the dual vector field to ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document},W∈C∞(Σ;TΣ),|W|g=ω(W)=1.\documentclass[12pt]{minimal}
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1. The Levi-Civita covariant derivative ∇W\documentclass[12pt]{minimal}
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\begin{document}$$T\Sigma $$\end{document}. This endomorphism is symmetric with respect to the metric g; indeed we compute for any two vector fields Y,Z∈C∞(Σ;TΣ)\documentclass[12pt]{minimal}
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\begin{document}$$Y,Z\in C^\infty (\Sigma ;T\Sigma )$$\end{document}0=dω(Y,Z)=Yg(W,Z)-Zg(W,Y)-g(W,[Y,Z])=g(∇YW,Z)-g(∇ZW,Y).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} 0=d\omega (Y,Z)&=Yg(W,Z)-Zg(W,Y)-g(W,[Y,Z]) \\&=g(\nabla _YW,Z)-g(\nabla _ZW,Y). \end{aligned} \end{aligned}$$\end{document}Taking Z:=W\documentclass[12pt]{minimal}
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\begin{document}$$Z:=W$$\end{document} and using that g(∇YW,W)=12Yg(W,W)=0\documentclass[12pt]{minimal}
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\begin{document}$$g(\nabla _YW,W)={1\over 2}Yg(W,W)=0$$\end{document} we see that the vector field W is geodesible, that is∇WW=0.\documentclass[12pt]{minimal}
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\begin{document}$$\delta \omega =0$$\end{document}, the vector field W is also divergence free; that is,tr(∇W)=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{tr}\,}}(\nabla W)=0. \end{aligned}$$\end{document}2. We next claim thattr((∇W)2)=2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\,\mathrm{tr}\,}}((\nabla W)^2)=2. \end{aligned}$$\end{document}To see this, take locally defined vector fields Y1,Y2\documentclass[12pt]{minimal}
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\begin{document}$$\nabla _W Y_j=0$$\end{document}. These can be obtained using parallel transport along the flow lines of W (which are geodesics since ∇WW=0\documentclass[12pt]{minimal}
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\begin{document}$$\nabla _WW=0$$\end{document}). We compute1=g(∇W∇YjW-∇Yj∇WW+∇∇YjWW-∇∇WYjW,Yj)=Wg(∇YjW,Yj)-g(∇YjW,∇WYj)+g(∇∇YjWW,Yj)-g(∇∇WYjW,Yj)=Wg(∇YjW,Yj)+g((∇W)2Yj,Yj).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} 1&=g(\nabla _{W}\nabla _{Y_j}W-\nabla _{Y_j}\nabla _WW+\nabla _{\nabla _{Y_j}W}W-\nabla _{\nabla _WY_j}W,Y_j) \\&= Wg(\nabla _{Y_j}W,Y_j)-g(\nabla _{Y_j}W,\nabla _WY_j)\\&\quad +g(\nabla _{\nabla _{Y_j}W}W,Y_j)-g(\nabla _{\nabla _WY_j}W,Y_j) \\&=Wg(\nabla _{Y_j}W,Y_j)+g((\nabla W)^2Y_j,Y_j). \end{aligned} \end{aligned}$$\end{document}Here in the first line we used that Σ\documentclass[12pt]{minimal}
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\begin{document}$$-1$$\end{document}, in the second line we used (A.2), and in the last line we used that ∇WYj=0\documentclass[12pt]{minimal}
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\begin{document}$$j=1,2$$\end{document} and using again (A.2) we get2=Wtr(∇W)+tr((∇W)2)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} 2=W{{\,\mathrm{tr}\,}}(\nabla W)+{{\,\mathrm{tr}\,}}((\nabla W)^2) \end{aligned}$$\end{document}and (A.4) now follows from (A.3).
3. From (A.2), (A.3), and (A.4) we see that ∇W\documentclass[12pt]{minimal}
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We are now ready to finish the proof of Proposition A.1. We can approximate the 1-form ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} by a closed 1-form with rational periods (integrals over closed curves on Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}); indeed, for an appropriate choice of linear isomorphism H1(Σ;C)≃Cb1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$H^1(\Sigma ;{\mathbb {C}})\simeq {\mathbb {C}}^{b_1(\Sigma )}$$\end{document} the forms with rational periods correspond to points in Qb1(Σ)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Q}}^{b_1(\Sigma )}$$\end{document}. In particular, we can find a number q∈N\documentclass[12pt]{minimal}
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\begin{document}$$q\in {\mathbb {N}}$$\end{document} and a closed 1-form ω~\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{\omega }}$$\end{document} with integer periods such thatsupΣ|ω-q-1ω~|g≤12.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sup _\Sigma |\omega -q^{-1}{\widetilde{\omega }}|_g\le \textstyle {1\over 2}. \end{aligned}$$\end{document}Since ω~\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{\omega }}$$\end{document} has integer periods, we can write ω~=df\documentclass[12pt]{minimal}
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\begin{document}$$\omega (W)=1$$\end{document}, (A.5) implies that Wf=ω~(W)>0\documentclass[12pt]{minimal}
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\begin{document}$$Wf={\widetilde{\omega }}(W)>0$$\end{document} which in turn gives df≠0\documentclass[12pt]{minimal}
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\begin{document}$$df\ne 0$$\end{document} everywhere, that is f is a fibration. Next, for each x∈Σ\documentclass[12pt]{minimal}
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\begin{document}$$x\in \Sigma $$\end{document} define the one-dimensional spacesE~±(x):=(RW(x)⊕E±(x))∩kerdf(x),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\widetilde{E}}_\pm (x):=(\mathbb RW(x)\oplus E_\pm (x))\cap \ker df(x), \end{aligned}$$\end{document}then the tangent bundle of each fiber f-1(c)\documentclass[12pt]{minimal}
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\begin{document}$$f^{-1}(c)$$\end{document} decomposes into a direct sum E~+⊕E~-\documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{E}}_+\oplus {\widetilde{E}}_-$$\end{document}. Since Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} is orientable, so is f-1(c)\documentclass[12pt]{minimal}
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\begin{document}$$f^{-1}(c)$$\end{document}, which implies that f-1(c)\documentclass[12pt]{minimal}
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\begin{document}$$f^{-1}(c)$$\end{document} is topologically a torus. Then Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} is a torus bundle over a circle, which gives a contradiction because such bundles do not admit hyperbolic metrics: by the homotopy long exact sequence of a fibration the fundamental group of Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} contains a subgroup isomorphic to Z⊕Z\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}\oplus {\mathbb {Z}}$$\end{document}, which is impossible for compact negatively curved manifolds by Preissman’s Theorem [45, Theorem 12.19].
By the Gray Stability Theorem (see [31, Theorem 2.2.2]), any perturbation of a contact form is a conformal perturbation up to pullback by a diffeomorphism.
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