Hitchin Functionals, h-Principles and Spectral Invariants

Abstract

This thesis investigates Hitchin functionals and $h$-principles for stable forms on oriented manifolds, with a special focus on $\mathrm{G}_2$ and $\widetilde{\mathrm{G}}_2$ 3- and 4-forms. Additionally, it introduces two new spectral invariants of torsion-free $\mathrm{G}_2$-structures. Part I begins by investigating an open problem posed by Bryant, $\textit{viz.}$ whether the Hitchin functional $\mathcal{H}_3$ on closed $\mathrm{G}_2$ 3-forms is unbounded above. Chapter 3 uses a scaling argument to obtain sufficient conditions for the functional $\mathcal{H}_3$ to be unbounded above and applies this result to prove the unboundedness above of $\mathcal{H}_3$ on two explicit examples of closed 7-manifolds with closed $\mathrm{G}_2$ 3-forms. Chapter 3 then proceeds to interpret this unboundedness geometrically, demonstrating an unexpected link between the functional $\mathcal{H}_3$ and fibrations, proving that the 'large volume limit' of $\mathcal{H}_3$ in each case corresponds to the adiabatic limit of a suitable fibration. The proof utilises a new, general collapsing result for singular fibrations between orbifolds, without assumptions on curvature, which is proved in Chapter 4. Chapter 5 broadens the focus of Part I to include the Hitchin functionals $\mathcal{H}_4$, $\widetilde{\mathcal{H}}_3$ and $\widetilde{\mathcal{H}}_4$ on closed $\mathrm{G}_2$ 4-forms, $\widetilde{\mathrm{G}}_2$ 3-forms and $\widetilde{\mathrm{G}}_2$ 4-forms respectively. In its main result, Chapter 5 proves that $\mathcal{H}_4,\widetilde{\mathcal{H}}_3,\widetilde{\mathcal{H}}_4$ are always unbounded above and below (whenever defined), and also that $\mathcal{H}_3$ is always unbounded below (whenever defined). As scholia, the critical points of the functionals $\mathcal{H}_4$, $\widetilde{\mathcal{H}}_3$ and $\widetilde{\mathcal{H}}_4$ are shown to be saddle points, and initial conditions of the Laplacian coflow which cannot lead to convergent solutions are shown to be dense. Part I ends with a short discussion of open questions, in Chapter 6. Part II investigates relative $h$-principles for closed, stable forms. After establishing some prerequisite algebraic results, Chapter 7 begins by proving that if a class of closed, stable forms satisfies the relative $h$-principle, then its corresponding Hitchin functional is automatically unbounded above. By utilising the technique of convex integration, Chapter 7 then obtains sufficient conditions for a class of closed, stable forms to satisfy the relative $h$-principle, a result which subsumes all previously established $h$-principles for closed stable forms. Until now, 12 of the 16 possible classes of closed stable forms have remained open questions with regard to the relative $h$-principle. In the main result of Part II, Chapters 7 and 8 prove the relative $h$-principle in 5 of these open cases. The remaining 7 cases are addressed in the final chapter of Part II, where it is conjectured that the relative $h$-principle holds in each case. Chapter 9 applies the $h$-principles established in this thesis to prove various results on the topological properties of closed $\widetilde{\mathrm{G}}_2$, $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms. Firstly, it characterises which oriented 7-manifolds admit closed $\widetilde{\mathrm{G}}_2$ forms, in the process introducing a new technique for proving the vanishing of natural cohomology classes on non-closed manifolds. Next, it introduces $\widetilde{\mathrm{G}}_2$-cobordisms of closed $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms and proves that homotopic forms are $\widetilde{\mathrm{G}}_2$-cobordant. Additionally, Chapter 9 classifies $\mathrm{SL}(3;\mathbb{C})$ 3-forms up to homotopy and provides a partial classification result on homotopy classes of $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms. Part II ends with a short discussion of open questions, in Chapter 10. Part III introduces and examines two new spectral invariants of torsion-free $\mathrm{G}_2$-structures. Although the notion of an invariant is a central theme in geometry and topology, currently, there is only one known invariant of torsion-free $\mathrm{G}_2$-structures: the $\overline{\nu}$-invariant of Crowley-Goette-Nordström. Part III defines two new invariants of torsion-free $\mathrm{G}_2$-structures, termed $\mu_3$- and $\mu_4$-invariants, by regularising the classical notion of Morse index for the Hitchin functionals $\mathcal{H}_3$ and $\mathcal{H}_4$ at their critical points. In general, there is no known way to compute $\overline{\nu}$ for $\mathrm{G}_2$-manifolds constructed via Joyce's `generalised Kummer construction'. Chapter 11 obtains closed formulae for $\mu_3$ and $\mu_4$ on the orbifolds used in Joyce's construction, leading to a conjectural discussion in Chapter 12 of how to compute $\mu_3$ and $\mu_4$ on Joyce's manifolds.