University of Cambridge
Department of Pure Mathematics and Mathematical Statistics
Deformations of Cayley
submanifolds
Matthias Ohst
Trinity College
This dissertation is submitted for the degree of
Doctor of Philosophy
September 2015

Declaration
This dissertation is the result of my own work and includes nothing which is
the outcome of work done in collaboration except as declared in the Preface
and specified in the text.
It is not substantially the same as any that I have submitted, or, is being
concurrently submitted for a degree or diploma or other qualification at the
University of Cambridge or any other University or similar institution except
as declared in the Preface and specified in the text. I further state that no
substantial part of my dissertation has already been submitted, or, is being
concurrently submitted for any such degree, diploma or other qualification
at the University of Cambridge or any other University of similar institution
except as declared in the Preface and specified in the text.
Cambridge, 29 September 2015 Matthias Ohst

Abstract
Cayley submanifolds of R8 were introduced by Harvey and Lawson as an in-
stance of calibrated submanifolds, extending the volume-minimising properties
of complex submanifolds in Kähler manifolds. More generally, Cayley submani-
folds are 4-dimensional submanifolds which may be defined in an 8-manifold M
equipped with a certain differential 4-form Φ invariant at each point under the
spin representation of Spin(7). If this 4-form Φ is closed, then the holonomy
of M is contained in Spin(7) and Cayley submanifolds are calibrated minimal
submanifolds.
McLean studied the deformations of closed Cayley submanifolds. The de-
formation problem is elliptic but in general obstructed. We show that for a
generic choice of Spin(7)-structure, there are no obstructions, and hence the
moduli space is a finite-dimensional smooth manifold.
Then we study the deformations of compact, connected Cayley submanifolds
with non-empty boundary contained in a given submanifold W of M , where we
require that the Cayley submanifolds meet the submanifold W orthogonally.
We show that for a generic choice of Spin(7)-structure, the Cayley submanifolds
are rigid. Moreover, we show that also for a generic choice of the submanifoldW ,
the Cayley submanifolds are rigid. We further discuss some examples for this
deformation theory.
In addition, we study the deformations of asymptotically cylindrical Cayley
submanifolds inside asymptotically cylindrical Spin(7)-manifolds. We prove an
index formula for the operator of Dirac type that arises as the linearisation of
the deformation map and show that for a generic choice of Spin(7)-structure,
there are no obstructions, and hence the moduli space is a finite-dimensional
smooth manifold whose dimension is equal to the index of the operator of
Dirac type. We further construct examples of asymptotically cylindrical Cayley
submanifolds inside the asymptotically cylindrical Riemannian 8-manifolds
with holonomy Spin(7) constructed by Kovalev.

Acknowledgements
First and foremost, I wish to thank my supervisor, Alexei Kovalev, for all his
support and guidance. I am also grateful to Trinity College and the Engineering
and Physical Sciences Research Council for financial support. Finally, I would
like to thank my family and friends.
vii

Contents
1 Introduction 1
1.1 Historical background and motivation . . . . . . . . . . . . . . . 1
1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Preliminaries 13
2.1 Geometry and analysis . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Tubular Neighbourhood Theorem . . . . . . . . . . . . . 13
2.1.2 Adapted tubular neighbourhood . . . . . . . . . . . . . . 14
2.1.3 Implicit Function Theorem for Banach spaces . . . . . . 16
2.1.4 Hölder spaces . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Calibrated geometry . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Introduction to calibrated geometry . . . . . . . . . . . . 18
2.2.2 Cayley subspaces of R8 . . . . . . . . . . . . . . . . . . . 20
2.2.3 Some facts about the cross products . . . . . . . . . . . 22
2.2.4 Spin(7)-structures on 8-manifolds . . . . . . . . . . . . . 26
2.2.5 Special Lagrangian subspaces of Cn . . . . . . . . . . . . 30
2.2.6 Calabi–Yau manifolds and special Lagrangian submanifolds 31
2.3 Smoothness of deformation maps . . . . . . . . . . . . . . . . . 34
3 Closed Cayley submanifolds 37
3.1 Deformations of closed Cayley submanifolds . . . . . . . . . . . 37
3.2 Index formula for the operator of Dirac type . . . . . . . . . . . 44
3.3 Genericity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Varying the Spin(7)-structure . . . . . . . . . . . . . . . . . . . 49
3.5 Remark about torsion-free Spin(7)-structures . . . . . . . . . . . 54
ix
4 Compact Cayley submanifolds with boundary 55
4.1 Elliptic boundary problems . . . . . . . . . . . . . . . . . . . . . 55
4.2 Deformations of compact Cayley submanifolds with boundary . 57
4.3 Remarks about the dimension of the scaffold . . . . . . . . . . . 67
4.4 Varying the Spin(7)-structure . . . . . . . . . . . . . . . . . . . 69
4.5 Varying the scaffold . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6 Remark about torsion-free Spin(7)-structures . . . . . . . . . . . 86
4.7 Non-ellipticity of boundary conditions for first-order boundary
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Examples of compact Cayley submanifolds with boundary 95
5.1 Volume minimising property . . . . . . . . . . . . . . . . . . . . 95
5.2 Holonomy contained in SU(2)× SU(2) . . . . . . . . . . . . . . 101
5.3 Bryant–Salamon construction . . . . . . . . . . . . . . . . . . . 103
5.4 Cayley deformations of special Lagrangian submanifolds . . . . 106
5.5 Cayley deformations of coassociative submanifolds . . . . . . . . 109
5.6 Cayley deformations of associative submanifolds . . . . . . . . . 111
6 Asymptotically cylindrical Cayley submanifolds 113
6.1 Asymptotically cylindrical manifolds . . . . . . . . . . . . . . . 114
6.1.1 Analysis on asymptotically cylindrical manifolds . . . . . 114
6.1.2 Atiyah–Patodi–Singer Index Theorem . . . . . . . . . . . 118
6.1.3 Relative Euler class and generalised Gauss–Bonnet–Chern
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.1.4 Volume minimising property . . . . . . . . . . . . . . . . 122
6.2 Deformations of asymptotically cylindrical Cayley submanifolds 123
6.3 Index formula for the operator of Dirac type . . . . . . . . . . . 125
6.3.1 Main index formula . . . . . . . . . . . . . . . . . . . . . 125
6.3.2 Relation between Bev and D˜ and alternative index formula 127
6.3.3 Additional assumptions . . . . . . . . . . . . . . . . . . . 130
6.4 Varying the Spin(7)-structure . . . . . . . . . . . . . . . . . . . 134
x
7 Examples of asymptotically cylindrical Cayley submanifolds 143
7.1 Examples in manifolds with holonomy Spin(7) . . . . . . . . . . 143
7.1.1 Weighted projective spaces and connected sum . . . . . . 143
7.1.2 Cayley submanifold in a Spin(7)-manifold constructed
from CP41,1,1,1,4 . . . . . . . . . . . . . . . . . . . . . . . . 144
7.1.3 Cayley submanifolds in Spin(7)-manifolds constructed
from a hypersurface in CP51,1,1,1,4,4 . . . . . . . . . . . . . 157
7.2 Relation to other calibrations . . . . . . . . . . . . . . . . . . . 165
7.2.1 Special Lagrangian calibration . . . . . . . . . . . . . . . 166
7.2.2 Coassociative calibration . . . . . . . . . . . . . . . . . . 168
7.2.3 Complex surfaces . . . . . . . . . . . . . . . . . . . . . . 170
7.2.4 Associative calibration . . . . . . . . . . . . . . . . . . . 172
Concluding remarks 175
Bibliography 177
xi

1 Introduction
This thesis deals with a class of minimal submanifolds called calibrated submani-
folds, which increased in popularity over the recent years following advances in
special holonomy and relevant areas of mathematical physics. In this chapter,
we first give some motivation and an overview of previous work by other authors
in Section 1.1. Then we present the main results of this thesis in Section 1.2.
We finish this chapter with an outline of the thesis in Section 1.3.
1.1 Historical background and motivation
The notion of calibrated geometry was introduced by Harvey and Lawson in the
foundational paper [HL82]. A calibration is a closed k-form on a Riemannian
manifold (M,g) such that φx|V ≤ volV for all x ∈M and every oriented k-dimen-
sional subspace V of TxM . Then an oriented k-dimensional submanifoldX ofM
is calibrated if φ|X = volX . Harvey and Lawson proved that calibrated sub-
manifolds are minimal submanifolds; in fact, closed calibrated submanifolds are
minimisers of the volume functional rather than just critical points. But while
to be minimal is a second-order partial differential equation on the embedding
X ↪→M , to be calibrated is a first-order partial differential equation.
The primary example for a calibration is the form 1
k!ω
k on a complex Kähler
manifold, where ω is the Kähler form. This follows from the Wirtinger In-
equality. The corresponding calibrated submanifolds are precisely the complex
submanifolds. In fact, the volume-minimising property of complex submanifolds
in Kähler manifolds, which was known before, was a motivation for Harvey
and Lawson’s work.
1
Further examples of calibrations can be found in manifolds with special
holonomy as follows. Let G := Hol(g) ⊆ O(n) be the holonomy group of a
Riemannian manifold (M,g). Then G acts on Λk(Rn)∗, the k-forms on Rn.
Suppose that there is a non-zero, G-invariant k-form φ0 on Rn. By rescaling φ0,
if necessary, we can assume that φ0|V ≤ volV for every oriented k-dimensional
subspace V of Rn and that equality holds for at least one subspace. Parallel
transport then yields a parallel k-form φ on M which satisfies φx|V ≤ volV
for all x ∈M and every oriented k-dimensional subspace V of TxM . Further-
more, φ is closed since ∇φ = 0 for the Levi-Civita connection ∇. So φ is a
calibration.
The above example of a Kähler manifold corresponds to G = U(n2 ). In [HL82],
Harvey and Lawson defined four calibrations that exist in Riemannian manifolds
with holonomy contained in SU(n2 ), G2, or Spin(7). Note that any such manifold
is Ricci-flat.
In this thesis, we focus on the Cayley calibration, which exists in Riemannian
manifolds with holonomy contained in Spin(7). Using the algebra of the
octonions, Harvey and Lawson defined a 4-form Φ0 on R8, which is given in
coordinates (x1, . . . , x8) by
Φ0 := dx1234 + (dx12 − dx34) ∧ (dx56 − dx78)
+ (dx13 + dx24) ∧ (dx57 + dx68)
+ (dx14 − dx23) ∧ (dx58 − dx67) + dx5678 ,
where we write dxi...j for dxi ∧ · · · ∧ dxj. This form Φ0 is a calibration, and
the subgroup of GL(8,R) preserving Φ0 is isomorphic to Spin(7), viewed as a
subgroup of SO(8). The calibrated submanifolds are called Cayley.
For an 8-manifold M , the existence of a (topological) ‘Spin(7)-structure’ Φ
(i.e., a 4-form onM which can be identified pointwise with Φ0) can be character-
ised via algebraic topology by a certain equation involving characteristic classes
of M [LM89]. The extra condition dΦ = 0, which implies that Φ is parallel and
the holonomy of M is contained in Spin(7) [Fer86], is harder to solve. The first
examples of Riemannian 8-manifolds with holonomy Spin(7) were constructed
2
by Bryant [Bry87]. Complete examples were obtained by Bryant and Salamon
[BS89]. The first examples of closed Riemannian 8-manifolds with holonomy
Spin(7) were constructed by Joyce [Joy96]. He also provided examples of closed
Cayley submanifolds inside these manifolds [Joy00].
The other calibrations introduced in [HL82] are the special Lagrangian
calibration on Calabi–Yau manifolds and the associative and coassociative
calibrations on manifolds with holonomy contained in G2. There are known
easy relations between these calibrations. In fact, Harvey and Lawson noted
that the geometry of Cayley submanifolds includes the geometries of other
calibrations. In particular, if the holonomy reduces to a proper subgroup
of Spin(7), then Cayley submanifolds can be constructed out of submanifolds
that are calibrated with respect to another calibration. However, their behaviour
is essentially governed by the other calibration, and results about the other
calibrations need not necessarily generalise to the Cayley calibration.
McLean [McL98] studied the deformations of closed calibrated submani-
folds for the calibrations introduced in [HL82]. The moduli spaces of special
Lagrangian and of coassociative submanifolds are finite-dimensional smooth
manifolds whose dimensions are topological invariants of the submanifolds. On
the other hand, the deformation problems for associative and Cayley submani-
folds, which are elliptic, are in general obstructed. The corresponding Zariski
tangent spaces can be given in terms of harmonic spinors of certain twisted
Dirac operators. For associative submanifolds, Gayet [Gay14] proved that for
a generic choice of (closed) G2-structure, there are no obstructions, and hence
associative submanifolds are rigid.
Later, McLean’s results were extended to larger classes of submanifolds
that are not necessarily closed. For the special Lagrangian, coassociative,
and associative calibrations, one such class are compact submanifolds with
boundary. See Butscher [But03], Kovalev and Lotay [KL09], and Gayet and
Witt [GW11], respectively. In all three cases, there is a constraint on the
boundary, namely to lie in an appropriately chosen fixed submanifold, which we
will occasionally call a scaffold (by analogy with [But03] although it need not be
a scaffold in the sense of [But03]). For the coassociative calibration, there is a
3
further constraint, namely that the coassociative submanifolds meet the scaffold
orthogonally. The moduli spaces of special Lagrangian and of coassociative
submanifolds with these constraints are finite-dimensional smooth manifolds,
and the dimension of the moduli space of special Lagrangian submanifolds
is a topological invariant of the submanifold. The deformation problem for
associative submanifolds with boundary on the scaffold is an elliptic boundary
problem and in general obstructed. Moreover, Gayet [Gay14] proved that
for a generic choice of (closed) G2-structure and also for a generic choice
of the scaffold, there are no obstructions, and hence the moduli space is a
finite-dimensional smooth manifold.
For the coassociative and special Lagrangian calibrations, another class of
submanifolds to which McLean’s results were extended are asymptotically
cylindrical submanifolds. A complete Riemannian manifold (M,g) is called
asymptotically cylindrical if there are a compact subset K ⊆ M , a closed
Riemannian manifold (N,h), and a diffeomorphism Ψ : (0,∞)×N →M \K
such that
|∇k∞(Ψ ∗(g)− g∞)| = O(e−δt) for all k ∈ N
for some δ > 0, where t denotes the projection onto the (0,∞)-factor and
∇∞ is the Levi-Civita connection of (0,∞)×N with respect to the product
metric g∞ = dt2 + h. For the deformations of asymptotically cylindrical
coassociative and special Lagrangian submanifolds, see Joyce and Salur [JS05]
and Salur and Todd [ST10], respectively. The corresponding moduli spaces of
deformations are finite-dimensional smooth manifolds whose dimensions are
topological invariants of the submanifolds.
Asymptotically cylindrical manifolds with holonomy SU(3) and G2 were con-
structed by Kovalev [Kov03] and Kovalev and Nordström [KN10], respectively.
In particular, Kovalev and Nordström also constructed examples of asymptot-
ically cylindrical coassociative submanifolds inside asymptotically cylindrical
7-manifolds with holonomy G2. The first examples of asymptotically cylindrical
Riemannian 8-manifolds with holonomy Spin(7) were constructed by Kovalev
[Kov13].
4
Note that the two classes of submanifolds require different concepts of
boundary conditions: local boundary conditions (e.g., Dirichlet or Neumann
boundary conditions) for compact submanifolds with boundary and global
boundary conditions (e.g., defined by spectral projections) for asymptotically
cylindrical submanifolds.
1.2 Main results
Our first main theorem is the following genericity statement in Section 3.4 about
the deformations of closed Cayley submanifolds. The proof largely amounts to
finding out how McLean’s deformation map depends on the Spin(7)-structure.
In the following theorem, we say that a statement holds for generic x ∈ X,
where X is a topological space. By this we mean that the set of all x ∈ X
for which the statement is true is a residual set, that is, it contains a set
which is the intersection of countably many open dense subsets. For the
topologies and norms used in the following theorem, see Section 3.4. Note
that every Spin(7)-structure Φ induces a metric g(Φ) (since Spin(7) ⊆ SO(8)),
and the Spin(7)-structure Φ is not uniquely determined by the metric g(Φ); in
fact, the Spin(7)-structures inducing the same metric as Φ are parametrised
by an O(8)/Spin(7)-bundle [BS89, page 835], and SO(8)/Spin(7) ∼= RP7 is
7-dimensional.
Theorem 1.1. Let M be an 8-manifold with a Spin(7)-structure Φ, and let X
be a closed Cayley submanifold of M .
Then for every generic Spin(7)-structure Ψ that is close to Φ and inducing
the same metric as Φ, the moduli space of all Cayley submanifolds of (M,Ψ)
that are close to X is either empty or a smooth manifold of dimension indD,
where D is the operator of Dirac type that arises as the linearisation of the
deformation map and is defined in (3.1).
Note that X itself need not be in the moduli space since X need not be Cayley
with respect to the Spin(7)-structure Ψ . Also, if indD < 0, then the moduli
space is necessarily empty for a generic Spin(7)-structure Ψ . Furthermore,
5
the statement of the above theorem remains true for the larger class of all
Spin(7)-structures Ψ that are close to Φ (not necessarily inducing the same
metric as Φ).
We proceed to study the deformations of compact Cayley submanifolds with
boundary. We require that the deformations meet the scaffold orthogonally,
but unlike the previous results for the special Lagrangian, coassociative, and
associative calibrations, we allow a range of dimensions for the scaffold. Ideally,
one would like to get an elliptic first-order boundary problem as for compact
associative submanifolds with boundary. But it turns out that an appropriate
first-order boundary problem will not be elliptic (see Section 4.7). Instead, we
will consider a second-order boundary problem. A similar approach was used
in [KL09] for the deformation theory of compact coassociative submanifolds
with boundary. On the other hand, we prove genericity results similar to those
in [Gay14] for compact associative submanifolds with boundary.
We first prove the following proposition in Section 4.2, which is a key technical
result needed for the proofs of the next two theorems.
Proposition 1.2. Let M be an 8-manifold with a Spin(7)-structure, let X
be a compact, connected Cayley submanifold of M with non-empty boundary,
and let W be a submanifold of M with ∂X ⊆ W such that X and W meet
orthogonally.
Then the moduli space of all local deformations of X as a Cayley submanifold
of M with boundary on W and meeting W orthogonally is included in the
solution space of the boundary problem (4.18), which is a second-order elliptic
boundary problem with index 0.
In particular, the Zariski tangent space is finite-dimensional. The usefulness
of the boundary problem (4.18) lies in the fact that the solution space is a
finite set under appropriate genericity assumptions. In particular, we show the
following two theorems (they have the same hypotheses as in Proposition 1.2)
in Sections 4.4 and 4.5, respectively, where we vary the Spin(7)-structure
(Theorem 1.3) and the submanifold W (Theorem 1.4).
6
Theorem 1.3. Let M be an 8-manifold with a Spin(7)-structure Φ, let X be a
compact, connected Cayley submanifold of M with non-empty boundary, and let
W be a submanifold of M with ∂X ⊆ W such that X and W meet orthogonally.
Then for every generic Spin(7)-structure Ψ that is close to Φ and inducing
the same metric as Φ, the moduli space of all Cayley submanifolds of (M,Ψ)
that are close to X with boundary on W and meeting W orthogonally is a finite
set (possibly empty).
The statement of the above theorem remains true for the larger class of all
Spin(7)-structures Ψ that are close to Φ (not necessarily inducing the same
metric as Φ). In addition, the statement is also true for the class of all torsion-
free Spin(7)-structures Ψ that are close to Φ (not necessarily inducing the
same metric as Φ), that is, Spin(7)-structures Ψ with dΨ = 0 (assuming Φ is
torsion-free); see Section 4.6.
Theorem 1.4. Let M be an 8-manifold with a Spin(7)-structure, let X be a
compact, connected Cayley submanifold of M with non-empty boundary, and let
W be a submanifold of M with ∂X ⊆ W such that X and W meet orthogonally.
Then for every generic local deformation W ′ of W with ∂X ⊆ W ′, the moduli
space of all Cayley submanifolds of M that are close to X with boundary on W ′
and meeting W ′ orthogonally is a finite set (possibly empty).
The statement of the above theorem remains true for the larger class of all
local deformationsW ′ ofW (not necessarily satisfying ∂X ⊆ W ′). Furthermore,
if we allow to vary both Φ and W , then we also get a genericity statement.
We further discuss some examples for this deformation theory in Chapter 5,
where we also show versions of the volume minimising property and relate our
deformation theory to the deformation theories of compact special Lagrangian,
coassociative, and associative submanifolds with boundary.
Then we study the deformations of asymptotically cylindrical Cayley sub-
manifolds. We first prove, in Section 6.3.1, the following index formula for the
operator of Dirac type that arises as the linearisation of McLean’s deformation
map using the Atiyah–Patodi–Singer Index Theorem. Note that a Spin(7)-
structure on an asymptotically cylindrical 8-manifold induces a G2-structure on
7
the cross-section at infinity (for the appropriate relation, see Section 2.2.2), and
the cross-section at infinity of an asymptotically cylindrical Cayley submanifold
is associative with respect to this G2-structure.
Theorem 1.5. Let M be an asymptotically cylindrical 8-manifold with an
asymptotically cylindrical Spin(7)-structure asymptotic to (0,∞)×N , where
N is a 7-manifold with torsion-free G2-structure, let X be an asymptotically
cylindrical Cayley submanifold of M asymptotic to (0,∞)× Y , where Y is a
closed associative submanifold of N , and let D be the operator of Dirac type that
arises as the linearisation of the deformation map, which is defined in (3.1).
If λ > 0 is small enough, then
indλD =
1
2χ(X)−
1
2σ(X)−
∫
X
e(νMX)− dim ker D˜2 +
η(D˜)− η(Bev)
2 ,
where χ(X) is the Euler characteristic of X, σ(X) is the signature of X,
e(νMX) is the Euler density of the normal bundle, D˜ : Γ(νNY )→ Γ(νNY ) is
the (twisted) Dirac operator that arises as the linearisation of the deformation
map for associative deformations of Y , and Bev : Ωev(Y )→ Ωev(Y ) is the odd
signature operator on Y .
Here indλD denotes the index of D with respect to exponentially weighted
Hölder spaces with weight λ (see Section 6.1.1). For the definitions of the
quantities in the above index formula, see the full statement in Section 6.3.1.
We also prove an alternative index formula that involves the spectral flow
between the two operators Bev and D˜ (Proposition 6.22). Furthermore, we
show the following genericity statement in Section 6.4.
Theorem 1.6. Let M be an asymptotically cylindrical 8-manifold with an
asymptotically cylindrical Spin(7)-structure asymptotic to (0,∞)×N , where N
is a 7-manifold with a G2-structure, and let X be an asymptotically cylindrical
Cayley submanifold of M asymptotic to (0,∞) × Y , where Y is a closed
associative submanifold of N .
If λ > 0 is small enough, then for every generic asymptotically cylindrical
Spin(7)-structure Ψ with rate λ that is close to Φ, with the same asymptotic limit
8
as Φ, and inducing the same metric as Φ, the moduli space of all asymptotically
cylindrical Cayley submanifolds of (M,Ψ) with rate λ that are close to X and
with the same asymptotic limit as X is either empty or a smooth manifold of
dimension indλD, where D is the operator of Dirac type that arises as the
linearisation of the deformation map and is defined in (3.1).
The statement of the above theorem remains true for the larger class of all
asymptotically cylindrical Spin(7)-structures Ψ with rate λ that are close to Φ
and with the same asymptotic limit as Φ (not necessarily inducing the same
metric as Φ). We also show two variations of the above theorem, where the
asymptotic limits of the deformations of X and the Spin(7)-structures Ψ are
not necessarily fixed.
We further construct, in Chapter 7, examples of asymptotically cylindrical
Cayley submanifolds inside the asymptotically cylindrical Riemannian 8-mani-
folds with holonomy Spin(7) constructed by Kovalev in [Kov13] and apply
Theorem 1.5 to calculate the indices for these Cayley submanifolds. Moreover,
we relate the deformation theory of asymptotically cylindrical Cayley submani-
folds to the deformation theories of asymptotically cylindrical coassociative
and special Lagrangian submanifolds.
1.3 Outline of the thesis
Chapter 2 contains mostly background material. We give short reviews of
the Tubular Neighbourhood Theorem (including a version adapted to the
deformations of compact submanifolds with boundary on a given submanifold),
the Implicit Function Theorem, and Hölder spaces in Section 2.1. Then we give
an introduction to the Cayley calibration and the other calibrations defined
by Harvey and Lawson in Section 2.2. We finish Chapter 2 by reviewing a
smoothness result about deformation maps in Section 2.3.
In Chapter 3, we present the deformation theory of closed Cayley submani-
folds. We first review, in Section 3.1, McLean’s foundational result and its
extension to Spin(7)-structures that are not necessarily torsion-free. Then in
9
Section 3.2, we prove an index formula for the operator of Dirac type that
arises as the linearisation of McLean’s deformation map. In Section 3.3, we
review some results regarding genericity. We further prove Theorem 1.1 in
Section 3.4. We finish Chapter 3 with a remark (Section 3.5) about why we
use the wider space of all Spin(7)-structures in Theorem 1.1 rather than just
torsion-free Spin(7)-structures.
In Chapter 4, we present the deformation theory of compact, connected Cayley
submanifolds with non-empty boundary. We first review elliptic boundary
problems in Section 4.1. Then we prove Proposition 1.2 in Section 4.2. We
proceed by discussing the extremal dimensions 3 and 7 for the scaffold W in
Section 4.3. We prove Theorems 1.3 and 1.4 in Sections 4.4 and 4.5, respectively.
We further show in Section 4.6 that Theorem 1.3 remains true if we restrict to
the smaller class of all torsion-free Spin(7)-structures. We finish Chapter 4 by
proving in Section 4.7 that a particular first-order boundary problem for the
operator of Dirac type that arises as the linearisation of the deformation map
is not elliptic. That is why we consider a second-order boundary problem for
the deformations of compact Cayley submanifolds with boundary.
In Chapter 5, we present and discuss some examples for the deformation
theory of compact Cayley submanifolds with boundary. We first show, in
Section 5.1, versions of the volume minimising property for this class of sub-
manifolds. In Section 5.2, we discuss examples with a smooth k-dimensional
moduli space (for 0 ≤ k ≤ 4) inside a manifold with holonomy SU(2)× SU(2).
Then in Section 5.3, we construct a rigid compact Cayley submanifold with
boundary inside the complete Riemannian 8-manifold with holonomy Spin(7)
constructed by Bryant and Salamon in [BS89]. We further relate the deforma-
tion theory of compact Cayley submanifolds with boundary to the deformation
theories of compact special Lagrangian, coassociative, and associative submani-
folds with boundary in Sections 5.4–5.6.
In Chapter 6, we present the deformation theory of asymptotically cylindrical
Cayley submanifolds. We first review, in Section 6.1, the Fredholm theory of
elliptic operators on asymptotically cylindrical manifolds, the Atiyah–Patodi–
Singer Index Theorem, the relative Euler class, and the generalised Gauss–
10
Bonnet–Chern Theorem, and prove an extension of the volume-minimising prop-
erty to asymptotically cylindrical calibrated submanifolds. Then in Section 6.2,
we extend the results of Section 3.1 to the asymptotically cylindrical setting.
We proceed by proving Theorem 1.5 in Section 6.3.1. In Section 6.3.2, we in-
vestigate the relation between the two operators Bev and D˜, whose η-invariants
appear in the index formula of Theorem 1.5, resulting in an alternative index
formula that involves the spectral flow (Proposition 6.22). In specific examples
of asymptotically cylindrical Cayley submanifolds (which can usually be found
if the holonomy of the cross-section at infinity of the Spin(7)-manifold is a
proper subgroup of G2), the cross-section at infinity of the Cayley submanifold
has certain properties that help to simplify the general index formula. In Sec-
tion 6.3.3, we prove simplified index formulae (which only involve topological
quantities) under such special assumptions. We finish Chapter 6 by proving
Theorem 1.6 and two variations, where the asymptotic limits are not necessarily
fixed, in Section 6.4.
In Chapter 7, we present and discuss some examples for the deformation
theory of asymptotically cylindrical Cayley submanifolds. In particular, in
Section 7.1, we provide examples of asymptotically cylindrical Cayley sub-
manifolds inside the asymptotically cylindrical Riemannian 8-manifolds with
holonomy Spin(7) constructed by Kovalev in [Kov13] and calculate the indices
of these Cayley submanifolds. We further relate the deformation theory of
asymptotically cylindrical Cayley submanifolds to the deformation theories of
asymptotically cylindrical coassociative and special Lagrangian submanifolds
in Section 7.2.
11

2 Preliminaries
This chapter contains mostly background material. We give short reviews
of the Tubular Neighbourhood Theorem (including a version adapted to the
deformations of compact submanifolds with boundary on a given submanifold),
the Implicit Function Theorem, and Hölder spaces in Section 2.1. Then we give
an introduction to the Cayley calibration and the other calibrations defined
by Harvey and Lawson in Section 2.2. We finish this chapter by reviewing a
smoothness result about deformation maps in Section 2.3.
2.1 Geometry and analysis
In this section, we give short reviews of the Tubular Neighbourhood Theorem
(including a version adapted to the deformations of compact submanifolds with
boundary on a given submanifold), the Implicit Function Theorem, and Hölder
spaces.
2.1.1 Tubular Neighbourhood Theorem
We want to study the deformations of certain types of submanifolds. For
this we need to parametrise nearby submanifolds. We will use the Tubular
Neighbourhood Theorem for this. The following theorem is a consequence of
the proof of [Lan95, Theorem IV.5.1].
Theorem 2.1 (Tubular Neighbourhood Theorem). Let (M, g) be a Riemannian
manifold, let X be a closed submanifold of M , and let νMX be the normal
bundle of X.
13
Then there are an open subset U ⊆ νMX containing the 0-section and an
open subset V ⊆M containing X such that the exponential map exp |U : U → V
is a diffeomorphism.
Therefore, if s : X → U is a section of νMX (and hence it is a submanifold
of νMX), then exp ◦s : X → M is a submanifold of M . Since X is compact,
there is an ε > 0 such that
{(x, v) ∈ νMX : |v| < ε} ⊆ U .
So any section s ∈ Γ(νMX) with ∥s∥C0 < ε defines a submanifold of M .
Since X is compact, there is an ε′ > 0 with ε′ ≤ ε such that for each
x ∈ X the exponential map expx : {v ∈ TxM : |v| < ε′} → M exists and
is a diffeomorphism onto its image. Let h : {(x, v) ∈ TM : |v| < ε′} → U ,
h := (exp |U)−1 ◦ exp, and let π : νMX → X denote the projection. Then
there is a δ > 0 with δ ≤ ε′ such that if s ∈ Γ(TM |X) with ∥s∥C1 < δ, then
π ◦ h ◦ s : X → X is a diffeomorphism.
So if f : X → M is C1-close to the inclusion iX : X → M , then there are
a diffeomorphism k : X → X and a section s ∈ Γ(νMX) with small C1-norm
such that f(k(x)) = expx(s(x)) for all x ∈ X. So C1-close submanifolds are
parametrised by sections of the normal bundle with small C1-norm.
A local deformation is a submanifold of the form exps(X) for some s ∈ Γ(νMX)
with small ∥s∥C0 , where exps : X →M , x ↦→ expx(s(x)).
2.1.2 Adapted tubular neighbourhood
Let (M, g) be a Riemannian manifold, let X be a compact submanifold of M
with boundary, and let W be a submanifold of M with ∂X ⊆ W such that
X and W meet orthogonally (see Definition 4.4). In Chapter 4, we want to
parametrise submanifolds near X with boundary on W by normal vector fields
using the exponential map. In general, we cannot use the exponential map
of the metric g since it does not preserve W (i.e., W is not totally geodesic
with respect to g). Here we construct a metric gˆ on M such that W is totally
14
geodesic with respect to gˆ. Such a construction was previously used in [But03]
and later in [KL09] and [Gay14].
Lemma 2.2 (cf. [But03, Proposition 6]). Let (M, g) be a Riemannian manifold,
let W be a submanifold of M , and let Y be closed submanifold of W .
Then there are an open neighbourhood U of Y in M and a metric gˆ on M
such that W ∩ U is totally geodesic with respect to gˆ and such that gˆx = gx for
all x ∈ W .
Proof. By the Tubular Neighbourhood Theorem (Theorem 2.1), there are an
open neighbourhood U ′ of Y in M , an open subset U˜ ⊆ νMW |W∩U ′ containing
the 0-section, and an open subset V ⊆ M containing W ∩ U ′ such that the
exponential map exp |U˜ : U˜ → V is a diffeomorphism. Let π : νMW → W
denote the projection, and let gν denote the metric on the fibres of νMW .
Then g˜ := π∗(g|W ) + gν defines a metric on U˜ such that the 0-section is totally
geodesic with respect to g˜. Let χ : M → [0, 1] be a smooth function such that
χ ≡ 0 outside U ′ and χ ≡ 1 in some neighbourhood U of Y in M contained
in U ′. Define
gˆ := χ · ((exp |U˜)−1)∗(g˜) + (1− χ) · g .
Then gˆ satisfies all the conditions.
As a consequence of the Tubular Neighbourhood Theorem (Theorem 2.1)
and Lemma 2.2, we obtain the following version of the Tubular Neighbourhood
Theorem, which is adapted to local deformations of X with boundary on W .
Proposition 2.3. Let (M, gˆ) be an n-dimensional Riemannian manifold, let
X be a compact submanifold of M with boundary, let W be a submanifold of M
with ∂X ⊆ W such that X and W meet orthogonally, and let νˆMX be the
normal bundle of X. Suppose that there is an open neighbourhood U of Y in M
such that W ∩ U is totally geodesic.
Then there are an open subset Uˆ ⊆ νˆMX containing the 0-section and an
n-dimensional submanifold Vˆ of M containing X and with boundary such
that the exponential map eˆxp|Uˆ : Uˆ → Vˆ is a diffeomorphism and such that
eˆxpx(v) ∈ W for all (x, v) ∈ Uˆ with x ∈ ∂X and v ∈ νˆW∂X.
15
So C1-close submanifolds with boundary on W are parametrised by appro-
priate sections of the normal bundle with small C1-norm using the exponential
map eˆxp of the metric gˆ.
2.1.3 Implicit Function Theorem for Banach spaces
Here we give a short review of the Implicit Function Theorem for Banach
spaces. This will be needed in order to show that certain moduli spaces are
smooth manifolds.
Theorem 2.4 (Implicit Function Theorem, [Lan93, Theorem XIV.2.1]). Let
X, Y , and Z be Banach spaces, let U ⊆ X and V ⊆ Y be open subsets, let
f : U × V → Z be a smooth map, and let (x0, y0) ∈ U × V . Suppose that
(df)(x0,y0)|Ty0Y : Ty0Y → Tf(x0,y0)Z
is an isomorphism.
Then there are open neighbourhoods U0 ⊆ X of x0 and V0 ⊆ Y of y0 and a
smooth map g : U0 → V0 such that
{(x, y) ∈ U0 × V0 : f(x, y) = f(x0, y0)} = {(x, g(x)) : x ∈ U0} .
Suppose that E and F are Banach spaces, W ⊆ E is an open subset,
f : W → F is a smooth map, and e0 ∈ W . If (df)e0 : Te0E → Tf(e0)F is
surjective and ker(df)e0 is finite-dimensional, then the hypotheses in the above
theorem apply for X = ker(df)e0 , Y a closed complement of ker(df)e0 in Te0E
(which exists since ker(df)e0 is finite-dimensional), and Z = F . Here we regard
E ∼= Te0E ∼= ker(df)e0 ⊕ Y . So f−1({f(e0)}) is a smooth submanifold of E in
a neighbourhood of e0, and the tangent space at e0 is ker(df)e0 .
Lemma 2.5. Let W be a Banach manifold, let Z be a Banach space, let
f : W → Z be a smooth map, and let x ∈ W . Suppose that ker(df)x is
finite-dimensional and that im(df)x is closed and has a closed complement.
16
If there is a submanifold M of W with x ∈ M and dimM = dimker(df)x
such that M ⊆ f−1({f(x)}), then there is a neighbourhood U ⊆ W of x such
that
f−1({f(x)}) ∩ U =M ∩ U .
Proof. Let V := im(df)x, let πV : Z → V be a continuous projection (which
exists since V is closed and has a closed complement), and let k := dim ker(df)x.
Define
g : W → V , x ↦→ πV (f(x)) .
Then (dg)x : TxW → V is surjective, and the kernel has dimension k. Hence
there is a neighbourhood U ′ ⊆ W of x such that g−1({g(x)})∩U ′ is a manifold
of dimension k. Now
M ∩ U ′ ⊆ f−1({f(x)}) ∩ U ′ ⊆ g−1({g(x)}) ∩ U ′ .
Furthermore, M ∩ U ′ and g−1({g(x)}) ∩ U ′ are manifolds of dimension k. So
there is a neighbourhood U ⊆ U ′ of x such that M ∩U = g−1({g(x)})∩U .
2.1.4 Hölder spaces
Let (M, g) be a Riemannian manifold, let E be a vector bundle overM equipped
with a metric connection ∇, let k ≥ 0 be an integer, and let 0 < α < 1. For a
Ck-section s of E, define
∥s∥Ck :=
k∑
i=0
∥∇is∥∞ . (2.1)
Furthermore, let
Γ := {γ : [0, 1]→M minimal geodesic on M}
and
∥s∥[α] := sup
γ∈Γ
|(Pγ)−1(s(γ(1)))− s(γ(0))|
|γ|α , (2.2)
17
where Pγ denotes the parallel transport along γ, and define
∥s∥Ck,α := ∥s∥Ck + ∥∇ks∥[α] . (2.3)
The Hölder space Ck,α(E) consists of all Ck-sections with finite ∥ · ∥Ck,α-norm.
Note that if M is compact (with or without boundary), then
C∞(E) =
∞⋂
k=0
Ck,α(E) , (2.4)
where C∞(E) consists of all sections that are smooth up to the boundary.
The C∞-topology on C∞(E) is the initial topology induced by the inclusions
C∞(E) ↪→ Ck,α(E) (sometimes also called the Whitney C∞-topology). Note
that C∞(E) is a Fréchet space with the C∞-topology, and hence a complete
metric space. So a residual set is dense by the Baire Category Theorem.
Lemma 2.6. Let M be a compact manifold, let E be a vector bundle over M ,
let k ≥ 0 be an integer, and let 0 < β < α < 1. Then the closed unit ball
of Ck,α(E) is compact in Ck,β(E).
Proof. The closed unit ball of Ck,α(E) is relatively compact in Ck,β(E) since
the embedding operator Ck,α(E) ↪→ Ck,β(E) is compact [RS82, Proposition 1
in Section 2.3.2.6]. It is closed since the Hölder norm ∥ · ∥Ck,α is lower semi-
continuous on Ck,β(E).
2.2 Calibrated geometry
In this section, we give an introduction to the Cayley calibration and the other
calibrations defined by Harvey and Lawson in [HL82].
2.2.1 Introduction to calibrated geometry
Here we give a short introduction to calibrated geometry (see, for example,
[HL82], [Joy00]).
18
Definition 2.7. Let (M, g) be a Riemannian manifold. A k-form φ on M is
called a calibration if φ is closed and φx|V ≤ volV for all x ∈ M and every
oriented k-dimensional subspace V of TxM . Here volV is the volume form
(induced by the metric g and the orientation on V ) and φx|V ≤ volV means
that φx|V = λ volV with λ ≤ 1.
Now let (M, g) be a Riemannian manifold with a calibration φ ∈ Ωk(M). An
oriented k-dimensional submanifold X of M is called a calibrated submanifold
or φ-submanifold if φx|TxX = volTxX for all x ∈ X.
Suppose that X is a submanifold of M with φx|TxX = ±volTxX for all x ∈ X.
Then φ|X is a volume form of X which determines an orientation of X, and if
we consider X as an oriented submanifold with this induced orientation, then
it is a calibrated submanifold of M . We will therefore often only talk about
submanifolds of M and choose the orientation induced by the calibration φ.
Proposition 2.8 ([HL82, Theorem II.4.2]). Let (M, g) be a Riemannian mani-
fold, let φ be a calibration on M , and let X be a closed calibrated submanifold
of M . Then X is volume-minimising in its homology class.
If Y is another closed submanifold of M in the same homology class as X
such that Y is volume-minimising in its homology class, then Y is also calibrated
with respect to φ.
The following proof is taken from [Joy07, Proposition 4.1.4].
Proof. Let n := dimX, and let [X] ∈ Hn(M ;R) and [φ] ∈ Hn(M ;R) be
the homology and cohomology classes of X and φ, respectively (here we use
singular homology and de Rham cohomology, which we identify with singular
cohomology using de Rham’s Theorem). Then
⟨[φ], [X]⟩ =
∫
X
φ|X =
∫
X
volX = vol(X) .
If Y is another closed n-dimensional submanifold of M with [Y ] = [X] in
Hn(M ;R), then
⟨[φ], [X]⟩ = ⟨[φ], [Y ]⟩ =
∫
Y
φ|Y
(∗)
≤
∫
Y
volY = vol(Y ) .
19
Thus vol(Y ) ≥ vol(X), and X is volume-minimising in its homology class.
If vol(Y ) = vol(X), then the inequality (∗) becomes an equality, and hence
φ|Y = volY , that is, Y is calibrated with respect to φ.
2.2.2 Cayley subspaces of R8
Here we review some facts from [HL82, Chapter IV] about the octonions and
the Cayley calibration. On H⊕H define a product by
(a, b) · (c, d) := (ac− d¯b, da+ bc¯) (2.5)
for all a, b, c, d ∈ H. Then the normed algebra H ⊕ H is isomorphic to the
octonions O. Let 1, i, j, k ∈ H be the standard basis for H (i.e., i2 = j2 = k2 =
ijk = −1). Let e := (0, 1) ∈ H⊕H ∼= O. Then (a, b) = a+ be for a, b ∈ H. So
we will use the basis e1 = 1, e2 = i, e3 = j, e4 = k, e5 = e, e6 = ie, e7 = je,
e8 = ke on R8 ∼= O. Furthermore, let ImO denote the span of e2, . . . , e8; and
for x ∈ O, let Im x := x− ⟨x, e1⟩e1 ∈ ImO.
The octonions carry several cross products, defined as follows:
x× y := 12(y¯x− x¯y) = Im(y¯x) ,
x× y× z := 12
(
x(y¯z)− z(y¯x)
)
,
w×x× y× z := 14
(
w¯(x× y× z)− x¯(y× z×w) + y¯(z×w×x)− z¯(w×x× y)
)
for all w, x, y, z ∈ O. These are alternating. We have
|x× y| = |x ∧ y| and |x× y × z| = |x ∧ y ∧ z| (2.6)
for all x, y, z ∈ O, where ‘∧’ is the exterior product on R8. If w, x, y, z ∈ O are
mutually orthogonal, then
x× y = y¯x ,
x× y × z = x(y¯z) ,
w × x× y × z = w¯(x(y¯z)) .
20
Define a 3-form φ0 ∈ Λ3(R7)∗ by
φ0(x, y, z) := ⟨x, y × z⟩ (2.7)
for all x, y, z ∈ ImO ∼= R7 and a 4-form Φ0 ∈ Λ4(R8)∗ by
Φ0(w, x, y, z) := ⟨w, x× y × z⟩ (2.8)
for all w, x, y, z ∈ O ∼= R8.
Let ψ0 ∈ Λ4(R7)∗ be the Euclidean Hodge dual of φ0. If we regard R8 ∼=
R⊕ R7 and use t as the coordinate on R, then
Φ0 = dt ∧ φ0 + ψ0 . (2.9)
In particular, Φ0 is self-dual. In coordinates (x1, . . . , x8) of R8, we have
φ0 = dx123 + dx145 − dx167 + dx246 + dx257 + dx347 − dx356 ,
ψ0 = − dx1247 + dx1256 + dx1346 + dx1357 − dx2345 + dx2367 + dx4567 ,
Φ0 = dx1234 + dx1256 − dx1278 + dx1357 + dx1368 + dx1458 − dx1467
− dx2358 + dx2367 + dx2457 + dx2468 − dx3456 + dx3478 + dx5678 ,
where we write dxi...j for dxi ∧ · · · ∧ dxj. Note that here we have used the
same convention as in [HL82] and [McL98]; but this is, for example, a different
convention than in [Joy00], which is equivalent to this one by an orientation-
reversing isomorphism.
The subgroup of GL(7,R) preserving φ0 is isomorphic to the Lie group G2,
and the subgroup of GL(8,R) preserving Φ0 is isomorphic to Spin(7). Since
G2 is the stabiliser of φ0 and G2 ⊆ SO(7), it is also the stabiliser of ψ0. If
we consider the action of G2 on R⊕ R7 where it acts trivially on R, then G2
preserves Φ0, and hence G2 ⊆ Spin(7).
21
Let ρ7 : Spin(7) → SO(7) be the standard double cover, let SO(7) act on
O ∼= R ⊕ R7 via the trivial action on R and the usual action on R7, and let
ρ8 : Spin(7)→ SO(8) be the real spin representation of Spin(7). Then
ρ8(g)(x)× ρ8(g)(y) = ρ7(g)(x× y) ,
ρ8(g)(x)× ρ8(g)(y)× ρ8(g)(z) = ρ8(g)(x× y × z) ,
ρ8(g)(w)× ρ8(g)(x)× ρ8(g)(y)× ρ8(g)(z) = ρ7(g)(w × x× y × z)
for all g ∈ Spin(7), w, x, y, z ∈ O.
The forms φ0 and ψ0 are calibrations on R7, and the form Φ0 is a calibration
on R8. A 3-dimensional subspace V ⊆ R7 is called associative if φ0|V = volV for
some orientation of V , a 4-dimensional subspace V ⊆ R7 is called coassociative
if ψ0|V = volV for some orientation of V , and a 4-dimensional subspace V ⊆ R8
is called Cayley if Φ0|V = volV for some orientation of V .
Let V ⊆ R7 be a linear subspace. Then V is associative if and only if
R ⊕ V ⊆ R ⊕ R7 ∼= R8 is Cayley. V ⊆ R7 is coassociative if and only if
{0} ⊕ V ⊆ R⊕ R7 ∼= R8 is Cayley. Both facts follow from (2.9).
Let K := (Sp(1) × Sp(1) × Sp(1))/{±(1, 1, 1)}. The group K acts on
H⊕H ∼= R8 via [p, q, r] · (u, v) = (puq¯, rvq¯) for p, q, r ∈ Sp(1), u, v ∈ H. This
action preserves Φ0, and hence K ⊆ Spin(7). The Grassmannian of Cayley
subspaces is isomorphic to Spin(7)/K. It has dimension 21− 3 · 3 = 12.
A linear subspace V ⊆ R8 is Cayley if and only if Im(w × x × y × z) = 0
for all w, x, y, z ∈ V . Since Im(w × x× y × z) is alternating in w, x, y, z ∈ O,
we can think of it as a 4-form with values in ImO ∼= R7. Denote this form by
τ0 ∈ Λ4(R8)⊗ R7.
2.2.3 Some facts about the cross products
Here we collect some facts about the cross products on the octonions. Lemma 2.9
will be used in Section 2.2.4 to define τ , Lemma 2.10 will be used in Section 2.2.4
to derive (2.20), and Lemma 2.13 will be used in Chapter 3 for the proof of
Theorem 1.1. Lemmas 2.9–2.12 will be used in the proof of Lemma 2.13.
22
Lemma 2.9. For all a, b, c, d ∈ O, we have
Im(a× b× c×d) = −a× (b× c×d)+ ⟨a, b⟩(c×d)+ ⟨a, c⟩(d× b)+ ⟨a, d⟩(b× c) .
Proof. Both sides of the above equation are multilinear in a, b, c, d. Hence it
suffices to prove the equation for a basis of O. In particular, for each pair we
can assume that they are either orthogonal or the same. Furthermore, note
that both sides of the equation are invariant under cyclic permutation of b, c, d.
If a, b, c, d are mutually orthogonal, then
Im(a× b× c× d) = Im(a¯(b(c¯d))) = (b(c¯d))× a = −a× (b× c× d)
as required.
If b = c, then Im(a× b× c× d) = 0, b× c× d = 0, b× c = 0, and
⟨a, b⟩(c× d) + ⟨a, c⟩(d× b) = ⟨a, b⟩(b× d+ d× b) = 0 .
So the equation holds.
If a = b and a, c, d are mutually orthogonal, then Im(a × b × c × d) = 0.
Furthermore, a ⊥ b× c× d, and so
−a× (b× c× d) = a¯(a(c¯d)) = (a¯a)(c¯d) = |a|2(d× c) = −⟨a, b⟩(c× d) .
Lemma 2.10. For all a, b, c, d ∈ O, we have
⟨a× b, c× d⟩ = −⟨a, b× c× d⟩+ ⟨a, c⟩⟨b, d⟩ − ⟨a, d⟩⟨b, c⟩ .
Proof. Both sides of the above equation are multilinear in a, b, c, d. Hence it
suffices to prove the equation for a basis of O. In particular, for each pair we can
assume that they are either orthogonal or the same. Furthermore, note that both
sides of the equation are alternating in c, d. Also, ⟨a, b× c× d⟩ = Φ0(a, b, c, d)
is alternating in a, b, c, d. Hence both sides of the equation are invariant
under exchanging (a, b) and (c, d). So we can assume that b, c, d are mutually
orthogonal or that a = c, b = d, and a and b are orthogonal.
23
If b, c, d are mutually orthogonal, then
⟨a× b, c× d⟩ = ⟨Im(b¯a), Im(d¯c)⟩ = ⟨b¯a, d¯c⟩ − (Re(b¯a))(Re(d¯c))
= ⟨a, b(d¯c)⟩ − ⟨a, b⟩⟨c, d⟩ = ⟨a, b× d× c⟩ = −⟨a, b× c× d⟩ .
If a = c, b = d, and a and b are orthogonal, then
⟨a× b, c× d⟩ = |a× b|2 = |a|2|b|2 = ⟨a, c⟩⟨b, d⟩ .
Lemma 2.11. For all a, b, c, d ∈ O, we have
⟨Im(a× b× c× d), a× b⟩ = 0 .
Proof. Using Lemmas 2.9 and 2.10, we get
⟨Im(a× b× c× d), a× b⟩ = −⟨a× (b× c× d), a× b⟩+ ⟨a, b⟩⟨c× d, a× b⟩
+ ⟨a, c⟩⟨d× b, a× b⟩+ ⟨a, d⟩⟨b× c, a× b⟩
= ⟨a, (b× c× d)× a× b⟩ − ⟨a, a⟩⟨b× c× d, b⟩
+ ⟨a, b⟩⟨b× c× d, a⟩ − ⟨a, b⟩⟨c, d× a× b⟩
+ ⟨a, b⟩⟨c, a⟩⟨d, b⟩ − ⟨a, b⟩⟨c, b⟩⟨d, a⟩
− ⟨a, c⟩⟨d, b× a× b⟩+ ⟨a, c⟩⟨d, a⟩⟨b, b⟩
− ⟨a, c⟩⟨d, b⟩⟨b, a⟩ − ⟨a, d⟩⟨b, c× a× b⟩
+ ⟨a, d⟩⟨b, a⟩⟨c, b⟩ − ⟨a, d⟩⟨b, b⟩⟨c, a⟩
= 0
since ⟨w, x× y × z⟩ = Φ0(w, x, y, z) is alternating in w, x, y, z ∈ O.
Lemma 2.12. If a, b, c, d, e, f ∈ O are mutually orthogonal, then
⟨a× b× c, d× e× f⟩ = 0 .
24
Proof. Let (e1, . . . , e8) ∈ O8 be a positive orthonormal frame of O with
a = |a|e1, b = |b|e2, c = |c|e3, d = |d|e4, e = |e|e5, and f = |f |e6. Then
e1 × e2 × e3 ⊥ e1, e2, e3 and e4 × e5 × e6 ⊥ e4, e5, e6, and therefore
⟨e1 × e2 × e3, e4 × e5 × e6⟩
=
8∑
i=1
⟨e1 × e2 × e3, ei⟩⟨e4 × e5 × e6, ei⟩
= Φ0(e7, e1, e2, e3)Φ0(e7, e4, e5, e6) + Φ0(e8, e1, e2, e3)Φ0(e8, e4, e5, e6)
= Φ0(e7, e1, e2, e3)Φ0(e7, e4, e5, e6)− (∗Φ0)(e7, e4, e5, e6)(∗Φ0)(e7, e1, e2, e3)
= 0
since ∗Φ0 = Φ0, where ∗ is the Hodge star operator on O.
Lemma 2.13. For all a, b, c, d, v, w ∈ O, we have
⟨Im(a× b× c× d), v × w⟩ = (w♭ ∧ (v ⌟ Φ0)− v♭ ∧ (w ⌟ Φ0))(a, b, c, d) .
Proof. Both sides of the above equation are multilinear in a, b, c, d, v, w. Hence
it suffices to prove the equation for a basis of O. In particular, for each pair we
can assume that they are either orthogonal or the same. Furthermore, note
that both sides of the equation are alternating in a, b, c, d and alternating in
v, w. So we can assume that a, b, c, d, v, w are mutually orthogonal, that a = v,
and a, b, c, d, w are mutually orthogonal, or that a = v, b = w, and a, b, c, d are
mutually orthogonal.
If a, b, c, d, v, w are mutually orthogonal, then
(w♭ ∧ (v ⌟ Φ0)− v♭ ∧ (w ⌟ Φ0))(a, b, c, d) = 0
and
⟨Im(a×b×c×d), v×w⟩ = −⟨a×(b×c×d), v×w⟩ = −⟨b×c×d, a×v×w⟩ = 0
by Lemmas 2.9, 2.10, and 2.12.
25
If a = v, and a, b, c, d, w are mutually orthogonal, then
⟨Im(a× b× c× d), a× w⟩ = −⟨a× (b× c× d), a× w⟩
= −|a|2⟨w, b× c× d⟩
= (w♭ ∧ (a ⌟ Φ0)− a♭ ∧ (w ⌟ Φ0))(a, b, c, d)
by Lemmas 2.9 and 2.10.
If a = v, b = w, and a, b, c, d are mutually orthogonal, then
(b♭ ∧ (a ⌟ Φ0)− a♭ ∧ (b ⌟ Φ0))(a, b, c, d) = Φ0(a, a, c, d) + Φ0(b, b, c, d) = 0
and ⟨Im(a× b× c× d), a× b⟩ = 0 by Lemma 2.11.
2.2.4 Spin(7)-structures on 8-manifolds
Here we recall some basic facts about Spin(7)-structures on 8-manifolds and
Cayley submanifolds (see, for example, [HL82], [Joy00]).
Let (x1, . . . , x8) be coordinates on R8, and write dxi...j for dxi ∧ · · · ∧ dxj.
The 4-form Φ0 on R8 defined in Section 2.2.2 satisfies
Φ0 = dx1234 + (dx12 − dx34) ∧ (dx56 − dx78)
+ (dx13 + dx24) ∧ (dx57 + dx68)
+ (dx14 − dx23) ∧ (dx58 − dx67) + dx5678 ,
(2.10)
and the subgroup of GL(8,R) preserving Φ0 is isomorphic to Spin(7), viewed
as a subgroup of SO(8). Note that Φ0 is self-dual.
LetM be an 8-manifold. Suppose that there is a 4-form Φ onM such that for
each x ∈M there is a linear isomorphism ix : TxM → R8 with (ix)∗(Φ0) = Φx
(in a neighbourhood of each point, this isomorphism can be chosen to depend
smoothly on x). Then Φ induces a Spin(7)-structure on M . Conversely, if M
has a Spin(7)-structure, then there is such a 4-form Φ. Via such an identification
ix : TxM → R8 of Φx with Φ0, the metric g0 of R8 induces a metric (ix)∗(g0)
on TxM . Since Spin(7) ⊆ SO(8), this metric is independent of the chosen
26
identification, and we get a well-defined Riemannian metric g = g(Φ) and
orientation on M . By abuse of notation, we will refer to the 4-form Φ as a
Spin(7)-structure. The Spin(7)-structure is called torsion-free if ∇Φ = 0, where
∇ is the Levi-Civita connection of (M, g). This is equivalent to dΦ = 0 [Fer86,
Theorem 5.3].
If M is an 8-manifold, then there exists a Spin(7)-structure on M if and only
if M is orientable and spin and
p1(M)2 − 4p2(M) + 8e(M) = 0 (2.11)
for some orientation of M [LM89, Theorem 10.7 in Chapter IV], where pi(M)
is the i-th Pontryagin class and e(M) is the Euler class of M .
Now let M be an 8-manifold with a Spin(7)-structure Φ. Then we have
pointwise orthogonal splittings [Fer86, Lemmas 3.1 and 3.3]
Λ2M = Λ27M ⊕ Λ221M and
Λ4M = Λ41M ⊕ Λ47M ⊕ Λ427M ⊕ Λ435M .
(2.12)
Here ΛpM := ΛpT ∗M , and ΛkℓM corresponds to an irreducible representation
of Spin(7) of dimension ℓ. Furthermore, M possesses a 2-fold cross product
TM × TM → Λ27M ,
v × w := 2π7(v♭ ∧ w♭) = 12(v♭ ∧ w♭ − ∗(v♭ ∧ w♭ ∧ Φ)) (2.13)
and a 3-fold cross product TM × TM × TM → TM ,
u× v × w := (u ⌟ (v ⌟ (w ⌟ Φ)))♯ . (2.14)
They satisfy |v × w| = |v ∧ w|, |u× v × w| = |u ∧ v ∧ w|, and
h(a× b, c× d) = −Φ(a, b, c, d) + g(a, c)g(b, d)− g(a, d)g(b, c) (2.15)
27
by (2.6) and Lemma 2.10, where h is the induced metric on Λ27M . There is also
a vector-valued 4-form τ ∈ Ω4(M,Λ27M) (also called the 4-fold cross product),
τ(a, b, c, d) := −a×(b×c×d)+g(a, b)(c×d)+g(a, c)(d×b)+g(a, d)(b×c) , (2.16)
which satisfies
h(τ, v × w) = w♭ ∧ (v ⌟ Φ)− v♭ ∧ (w ⌟ Φ) (2.17)
by Lemma 2.13. Note that
w♭ ∧ (v ⌟ Φ)− v♭ ∧ (w ⌟ Φ) ∈ Λ47M , (2.18)
which follows from [Bry87, page 548].
Let x ∈ M , and let (e1, . . . , e8) be a basis of TxM . We call (e1, . . . , e8) a
Spin(7)-frame if
Φ = e1234 + e1256 − e1278 + e1357 + e1368 + e1458 − e1467
− e2358 + e2367 + e2457 + e2468 − e3456 + e3478 + e5678 ,
(2.19)
where (e1, . . . , e8) is the dual coframe. Note that if (e1, . . . , e8) is a Spin(7)-
frame, then it is an orthonormal frame since Spin(7) ⊆ SO(8). Furthermore,
ei × ej = ±ek × eℓ if and only if Φ(ei, ej, ek, eℓ) = ∓1 for pairwise distinct
i, j, k, ℓ ∈ {1, . . . , 8} by (2.15). So (2.19) shows that
e1 × e5 = e2 × e6 = e3 × e7 = e4 × e8 ,
e1 × e6 = −e2 × e5 = e3 × e8 = −e4 × e7 ,
e1 × e7 = −e2 × e8 = −e3 × e5 = e4 × e6 ,
e1 × e8 = e2 × e7 = −e3 × e6 = −e4 × e5 .
(2.20)
If e1, e2, e3 ∈ TxM are orthogonal unit vectors and e5 ∈ TxM is a unit
vector that is orthogonal to e1, e2, e3, e1 × e2 × e3, then there are (uniquely de-
termined) e4, e6, e7, e8 ∈ TxM such that (e1, . . . , e8) is a Spin(7)-frame, namely
e4 = −e1 × e2 × e3, e6 = −e1×e2×e5, e7 = −e1×e3×e5, and e8 = e2×e3×e5.
28
We have Φx|V ≤ volV for all x ∈ M and every oriented 4-dimensional
subspace V of TxM , where volV is the volume form (induced by the metric g
and the orientation on V ) and φx|V ≤ volV means that φx|V = λ volV with
λ ≤ 1. An orientable 4-dimensional submanifold X of M is called Cayley if
Φ|X = volX for some orientation of X. This is equivalent to τ |X = 0. If the
Spin(7)-structure Φ is torsion-free, then Φ is a calibration on M , and Cayley
submanifolds are minimal submanifolds [HL82, Theorem 4.2 in Chapter II].
Now suppose that X is a Cayley submanifold ofM . Then Λ2−X is isomorphic
to a subbundle of Λ27M |X [McL98, Section 6] via the embedding
Λ2−X → Λ27M |X , α ↦→ 2π7(α) = 12(α− ∗(α ∧ Φ)) , (2.21)
where we extend α ∈ Λ2−X to Λ2M |X by v ⌟ α = 0 for all v ∈ νMX. Let E
denote the orthogonal complement of Λ2−X in Λ27M |X . So
Λ27M |X ∼= Λ2−X ⊕ E , (2.22)
and E has rank 4. Furthermore,
E = {α ∈ Λ27M |X : α|TX = 0} . (2.23)
The cross products restrict to
TX × TX → Λ2−X , TX × νMX → E , νMX × νMX → Λ2−X (2.24)
and
TX × TX × TX → TX , TX × TX × νMX → νMX ,
TX × νMX × νMX → TX , νMX × νMX × νMX → νMX .
(2.25)
Lemma 2.14 ([Joy00, Proposition 10.8.6]). Let M be an 8-manifold with a
Spin(7)-structure Φ, and let ρ : M →M be an involution with ρ∗(Φ) = Φ. Suppose
that the fixed-point set of ρ contains neither an isolated point nor an 8-dimensional
component. Then the fixed-point set of ρ is a Cayley submanifold of M .
29
2.2.5 Special Lagrangian subspaces of Cn
Here we review some facts from [HL82, Chapters III and IV] about the special
Lagrangian calibration and its relation to the Cayley calibration.
On Cn ∼= R2n, take coordinates z1 = x1 + xn+1i, . . . , zn = xn + x2ni, and
define
ω0 := dx1 ∧ dxn+1 + · · ·+ dxn ∧ dx2n ,
Ω0 := dz1 ∧ · · · ∧ dzn .
(2.26)
The stabiliser of (g0, ω0, Ω0) is isomorphic to SU(n), where g0 is the Euclidean
metric on R2n. The forms ω0 and Ω0 satisfy
1
n! ω
n
0 = (−1)n(n−1)/2
(
i
2
)n
Ω0 ∧Ω0 . (2.27)
The form ReΩ0 is a calibration on Cn. An n-dimensional subspace V ⊆ Cn is
called special Lagrangian if (ReΩ0)|V = volV for some orientation of V . This is
equivalent to ω0|V = 0, (ImΩ0)|V = 0. So every special Lagrangian subspace is
Lagrangian. The Grassmannian of special Lagrangian subspaces is isomorphic
to SU(n)/SO(n). It has dimension (n2 − 1)− n(n− 1)/2 = (n2 + n− 2)/2.
More generally, the form Re(eiθΩ0) for θ ∈ R is a calibration on Cn, and
an n-dimensional subspace V ⊆ Cn is called special Lagrangian with phase
angle θ if (Re(eiθΩ0))|V = volV for some orientation of V . This is equivalent to
ω0|V = 0, (Im(eiθΩ0))|V = 0. A linear subspace V ⊆ Cn is Lagrangian if and
only if it is special Lagrangian with phase angle θ for some θ ∈ R.
Now let n = 4, and consider O ∼= C4 via the complex structure J : O→ O,
x ↦→ xe. Then
Φ0 = −12 ω0 ∧ ω0 +ReΩ0 , (2.28)
where Φ0 is the Cayley calibration on R8 as defined in Section 2.2.2. This
shows that if V ⊆ R8 is a special Lagrangian subspace, then it is Cayley as
it is calibrated with respect to ReΩ0 and ω0|V = 0. If V ⊆ R8 is a complex
2-dimensional subspace, then it is also Cayley (with the opposite orientation)
as it is calibrated with respect to 12 ω0 ∧ ω0 and (ReΩ0)|V = 0. Note that there
30
are Cayley subspaces that are neither special Lagrangian nor complex (e.g., the
subspace V spanned by e1, e2, e3 + e5, e4 + e6, where (−12 ω0 ∧ ω0)|V = 12volV
and (ReΩ0)|V = 12volV ).
2.2.6 Calabi–Yau manifolds and special Lagrangian
submanifolds
Let (M,ω) be a 2n-dimensional symplectic manifold, let J : TM → TM be a
compatible almost complex structure (i.e., g : TM×TM → R, (v, w) ↦→ ω(v, Jw)
is a Riemannian metric on M), and let Ω be a non-vanishing complex-valued
(n, 0)-form. We will assume that Ω is normalised, that is,
1
n! ω
n = (−1)n(n−1)/2
(
i
2
)n
Ω ∧ Ω¯ . (2.29)
If the almost complex structure J is integrable (in particular, M is Kähler) and
∇Ω = 0, where ∇ is the Levi-Civita connection of (M, g), then Hol(g) ⊆ SU(n).
A symplectic manifold with a compatible, integrable complex structure and
parallel, normalised complex-valued (n, 0)-form is called a Calabi–Yau mani-
fold.
Conversely, if Hol(g) ⊆ SU(n), then there are an integrable complex struc-
ture J with associated fundamental (1, 1)-form ω and a non-vanishing complex-
valued (n, 0)-form Ω such that ∇ω = 0 (and hence dω = 0), ∇Ω = 0, and
such that (2.29) holds. Furthermore, given (g, J), the form Ω is unique up to
multiplication of a constant eiθ, θ ∈ R. The form Ω is then called a holomorphic
volume form.
Let M be a Calabi–Yau manifold with Kähler form ω and holomorphic
volume form Ω. An n-dimensional submanifold X of M is called a special
Lagrangian submanifold if it is calibrated with respect to the calibration ReΩ.
This is equivalent to ω|X = 0, (ImΩ)|X = 0. A submanifold X of M is called a
special Lagrangian submanifold with phase angle θ if it is calibrated with respect
to the calibration Re(eiθΩ). This is equivalent to ω|X = 0, (Im(eiθΩ))|X = 0.
31
Proposition 2.15 (cf. [HL82, Proposition 2.17 in Chapter III]). Let M be a
Calabi–Yau manifold, and let X be a connected Lagrangian submanifold of M .
Then X is minimal (i.e., the mean curvature vanishes) if and only if X is
special Lagrangian with phase angle θ for some θ ∈ R.
Remark 2.16. Note, in particular, that this proposition implies that every
minimal Lagrangian submanifold of a Calabi–Yau manifold is orientable.
Harvey and Lawson stated this proposition for minimal submanifolds of Cn.
The following proof is based on their proof.
Proof. First assume that X is oriented. Let Ω be the holomorphic volume
form of M , and let ∇ be the Levi-Civita connection of M . Then ∇Ω = 0.
By [HL82, Proposition 1.14 in Chapter III], Ω|X = λ volX for some λ : X → C
with |λ(x)| = 1 for all x ∈ X (note that the normalisation (2.29) is important
for this). Define ϑ : X → R/2πZ by λ = e−iϑ. Then X is a special Lagrangian
submanifold with phase angle θ if and only if ϑ(x) = θ for all x ∈ X.
Let (e1, . . . , en) be a positive local orthonormal frame ofX (where n := dimX),
and extend e1, . . . , en to a neighbourhood U in M such that (e1, . . . , en) is an
orthonormal system spanning a Lagrangian subspace at each point in U . Then
(e1, . . . , en, Je1, . . . , Jen) is an orthonormal frame of M , where J : TM → TM
is the complex structure (which is orthogonal).
Let v ∈ Γ(TX). Then
0 = (∇vΩ)(e1, . . . , en)
= v.(Ω(e1, . . . , en))−
n∑
i=1
Ω(e1, . . . ,∇vei, . . . , en) . (2.30)
Now Ω is an (n, 0)-form, and Ω|X = e−iϑvolX . Hence
Ω(e1, . . . , ei−1,Jei, ei+1, . . . , en)= iΩ(e1, . . . , en)= ie−iϑ and
Ω(e1, . . . , ei−1,Jej, ei+1, . . . , en)= iΩ(e1, . . . , ei−1, ej, ei+1, . . . , en)=0 for j ̸= i.
32
Since (e1, . . . , en, Je1, . . . , Jen) is an orthonormal frame, we therefore get
Ω(e1, . . . ,∇vei, . . . , en) =
n∑
j=1
Ω(e1, . . . , ei−1, ej, ei+1, . . . , en)g(∇vei, ej)
+
n∑
j=1
Ω(e1, . . . , ei−1, Jej, ei+1, . . . , en)g(∇vei, Jej)
= e−iϑg(∇vei, ei) + ie−iϑg(∇vei, Jei) .
So by (2.30), we have
v.(e−iϑ) = e−iϑ
n∑
i=1
(
g(∇vei, ei) + ig(∇vei, Jei)
)
.
Since v.(e−iϑ) = −ie−iϑ(v.ϑ) and 2g(∇vei, ei) = v.(g(ei, ei)) = 0, we get
v.ϑ = −
n∑
i=1
g(∇vei, Jei) . (2.31)
We have g([v, ei], Jei) = 0 since v and ei are tangent vectors and Jei is a normal
vector to X (here [ · , · ] is the Lie bracket on M). Further, J is orthogonal
and parallel with respect to ∇. Hence
g(∇vei, Jei) = g(∇eiv, Jei) + g([v, ei], Jei)
= −g(v,∇eiJei)
= −g(v, J∇eiei)
= g(Jv,∇eiei) .
So by (2.31), we get
v.ϑ = −
n∑
i=1
g(∇eiei, Jv) = g(nH, Jv) , (2.32)
where H is the mean curvature vector. So ϑ is constant if and only if H ≡ 0,
that is, X is special Lagrangian with phase angle θ for some θ ∈ R if and only
if X is minimal.
33
For general X, we get a well-defined function ϑ : X → R/πZ. Since X is
locally orientable, the above proof shows that ϑ is constant. Choose θ ∈ R/2πZ
with ϑ = θ in R/πZ. Then eiθΩ|X is a nowhere vanishing (real) top-dimensional
form, showing that X is orientable.
Now let M be a complex 4-dimensional Calabi–Yau manifold with Kähler
form ω and holomorphic volume form Ω. Let Φ := −12 ω ∧ ω + ReΩ. Equa-
tion (2.28) shows that this defines a Spin(7)-structure on M . Since ω and
ReΩ are closed (note that ∇Ω = 0 implies dΩ = 0 as ∇ is torsion-free),
this Spin(7)-structure is torsion-free. Hence every Calabi–Yau manifold is a
Spin(7)-manifold. Every special Lagrangian submanifold of this Calabi–Yau
manifold is a Cayley submanifold with respect to this Spin(7)-structure. Also
every complex 2-dimensional submanifold is a Cayley submanifold (with the
reversed orientation). But there are Cayley submanifolds that are neither
special Lagrangian nor complex.
Note that eiθΩ is also a holomorphic volume form, andΦθ := −12 ω ∧ ω +Re(eiθΩ)
defines a Spin(7)-structure. Special Lagrangian submanifolds with respect to
ReΩ are not calibrated with respect to Φθ unless eiθ = 1. So we actually have
a family of different Spin(7)-structures. Complex 2-dimensional submanifolds
of M are Cayley submanifolds for all of these structures but the only minimal
Lagrangian submanifolds that are Cayley submanifolds with respect to Φθ are
special Lagrangian submanifolds with phase angle θ.
2.3 Smoothness of deformation maps
In this section, we present a result about the smoothness of deformation maps
as maps between Banach spaces. The following proposition is a corollary of
[Bai01, Theorem 2.2.15], which we extend to vector-valued forms. For an open
subset U of a vector bundle E over a manifold X, define
Γ(U) := {s ∈ Γ(E) : s(X) ⊆ U} . (2.33)
We will use a similar definition for the Hölder spaces Ck,α with k ≥ 0 an integer
and 0 < α < 1.
34
Proposition 2.17 (cf. [Bai01, Theorem 2.2.15]). Let (M, g) be a Riemannian
manifold, let X be a compact submanifold of M , let TX := ⨁i(T ∗X)⊗i and
TM := ⨁i(T ∗M)⊗i be the tensor algebras of X and M , respectively, let E be a
vector bundle over M equipped with a connection ∇, let U be an open tubular
neighbourhood of the 0-section in νMX such that the exponential map defines
a diffeomorphism from U to an open neighbourhood V of X in M , let k ≥ 1,
ℓ ≥ 0 be integers, and let 0 < α < 1. For (x, v) ∈ U , let πx,v : Eexpx(v) → Ex
denote the parallel transport along the curve [0, 1] → M , t ↦→ expx((1 − t)v)
(so for all s ∈ Γ(U), the map πs : (exps)∗E → E|X is an isomorphism of vector
bundles).
Then the map
Ψ : Ck,α(U)⊕ Ck−1+ℓ,α((TM ⊗ E)|V )→ Ck−1,α(TX ⊗ E|X) ,
(s,Θ) ↦→ (idTX⊗πs)((exps)∗Θ)
(2.34)
is of class Cℓ.
[Bai01, Theorem 2.2.15] corresponds to the case when E is trivial.
Proof. Define π : V → X by π(expx(v)) := x for (x, v) ∈ U . Since X is
compact, there are open subsets U1, . . . , Un ⊆ X with U1 ∪ · · · ∪ Un = X such
that E|Ui is trivial for i = 1, . . . , n. Let λ˜1, . . . , λ˜n : X → R be a partition
of unity subordinate to the cover (U1, . . . , Un). Extend these to functions
λ1, . . . , λn : V → R by λi(y) := λ˜i(π(y)) for y ∈ V . Then (λ1, . . . , λn) is a
partition of unity subordinate to the cover (V1, . . . , Vn), where Vi := π−1(Ui)
for i = 1, . . . , n. If the proposition is true for λ1Θ, . . . , λnΘ, then it is also true
for Θ since Ψ is linear in Θ. So w.l.o.g. we may assume that the support of Θ
is contained in some Vi, say in V1.
Now let ψ : E|U1 → Rn be a trivialising map. Let ρ : X → R be a smooth
function with supp(ρ) ⊆ U1 and ρ|π(supp(Θ)) ≡ 1. Let φ˜ : E|X → Rn be defined
by φ˜(x, e) := ρ(x)ψ(x, e) for x ∈ U1, e ∈ Ex and φ˜(x, e) := 0 for x ∈ X \ U1,
e ∈ Ex. Let φ : E|V → Rn be defined by φ(y, e) := φ˜(π(y), πx,v(e)) for y ∈ V ,
e ∈ Ey, where v ∈ (νMX)x is such that (x, v) ∈ U and expπ(y)(v) = y.
35
Now Ψ is of class Cℓ if and only if (idTX⊗φ˜) ◦ Ψ is of class Cℓ since
E|π(supp(Θ)) → π(supp(Θ))× Rn , (x, e) ↦→ (x, φ˜(x, e))
is a smooth diffeomorphism. We have
(idTX⊗φ˜)(Ψ(s)) = (idTX⊗φ˜)((idTX⊗πs)((exps)∗Θ))
= (idTX⊗φ)((exps)∗Θ)
= (exps)∗((idTM |V⊗φ)(Θ)) .
But (idTM |V⊗φ)(Θ) ∈ Γ(TM |V ⊗ Rn), and the map
Ck,α(U)→ Ck−1,α(TX ⊗ Rn) , s ↦→ (exps)∗((idTM |V⊗φ)(Θ))
is of class Cℓ by [Bai01, Theorem 2.2.15] (note that the proof of [Bai01,
Theorem 2.2.15] also works if Θ is just of class Ck−1+ℓ,α). Now the claim follows
as Ψ is linear in Θ.
Corollary 2.18. Let (M, g) be a Riemannian manifold, let X be a compact
submanifold of M , let TX := ⨁i(T ∗X)⊗i and TM :=⨁i(T ∗M)⊗i be the tensor
algebras of X and M , respectively, let E be a vector bundle over M equipped
with a connection ∇, let U be an open tubular neighbourhood of the 0-section
in νMX such that the exponential map defines a diffeomorphism from U to an
open neighbourhood V of X in M , let Θ ∈ Γ((TM ⊗ E)|V ), let k ≥ 1 be an
integer, and let 0 < α < 1. For (x, v) ∈ U , let πx,v : Eexpx(v) → Ex denote the
parallel transport along the curve [0, 1] → M , t ↦→ expx((1 − t)v) (so for all
s ∈ Γ(U), the map πs : (exps)∗E → E|X is an isomorphism of vector bundles).
Then the map
Ck,α(U)→ Ck−1,α(TX ⊗ E|X) , s ↦→ (idTX⊗πs)((exps)∗Θ) (2.35)
is smooth.
36
3 Closed Cayley submanifolds
In this chapter, we present the deformation theory of closed Cayley submani-
folds. We first review, in Section 3.1, McLean’s foundational result and its
extension to Spin(7)-structures that are not necessarily torsion-free. Then in
Section 3.2, we prove an index formula for the operator of Dirac type that arises
as the linearisation of McLean’s deformation map. In Section 3.3, we review
some results regarding genericity. We further show, in Section 3.4, that for a
generic Spin(7)-structure, closed Cayley submanifolds form a smooth moduli
space (Theorem 1.1). We finish this chapter with a remark (Section 3.5) about
why we use the wider space of all Spin(7)-structures in Theorem 1.1 rather
than just torsion-free Spin(7)-structures.
3.1 Deformations of closed Cayley submanifolds
McLean [McL98] proved that if M is an 8-manifold with a torsion-free Spin(7)-
structure and X is a closed Cayley submanifold of M , then the Zariski tangent
space to the moduli space of all local deformations of X as a Cayley submanifold
of M is isomorphic to the space of harmonic twisted spinors. Gutowski, Ivanov,
and Papadopoulos [GIP03] generalised this result to manifolds with Spin(7)-
structures that have torsion. Here we give an explicit formula for the operator
of Dirac type that arises as the linearisation of the deformation map in terms of
the Levi-Civita connection. The equivalence of the operators can also be seen
using the explicit formula for a Spin(7)-connection in [Iva04, Theorem 1.1].
37
Theorem 3.1 (cf. [McL98, Theorem 6–3], [GIP03, Section 13]). Let M be an
8-manifold with a Spin(7)-structure Φ, and let X be a Cayley submanifold of M .
Then the Zariski tangent space to the moduli space of all local deformations
of X as a Cayley submanifold of M can be identified with the kernel of the
operator D : Γ(νMX)→ Γ(E),
Ds :=
4∑
i=1
ei ×∇⊥eis+
8∑
i=5
(∇sΦ)(ei, e2, e3, e4)(ei × e1) , (3.1)
where the vector bundle E of rank 4 over X is defined as in (2.23), (ei)i=1,...,4 is
any positive local orthonormal frame of X, (ei)i=5,...,8 is any local orthonormal
frame of νMX, and ∇⊥ is the induced connection on νMX.
Note that the second part of the operator (3.1) is an operator of order 0 that
vanishes if the Spin(7)-structure Φ is torsion-free (i.e., if ∇Φ = 0).
Remark 3.2. Suppose that X has a spin structure, and let S+ and S− denote
the positive and negative spinor bundles, respectively (note that both bundles
are quaternionic line bundles). Then there is a quaternionic line bundle F
over X such that [McL98, Section 6]
νMX ⊗R C ∼= S− ⊗C F and E ⊗R C ∼= S+ ⊗C F . (3.2)
If the Spin(7)-structure is torsion-free, then D can be identified with a negative
twisted Dirac operator [McL98, Theorem 6–3], that is, the negative Dirac
operator associated to the bundle S⊗C F with the tensor product connection
(cf. [LM89, Proposition 5.10 in Chapter II]). If the Spin(7)-structure is not
torsion-free, then D may not be a negative twisted Dirac operator but the
symbol of D is still the same as the symbol of a negative twisted Dirac operator,
and hence D is always an operator of Dirac type.
We now give a proof of Theorem 3.1. This proof is mostly based on [McL98].
Besides introducing some notation, the proofs of our main theorems also build
up on this proof. In particular, we need the deformation map from [McL98]
later.
38
Proof. The Tubular Neighbourhood Theorem (Theorem 2.1) asserts that there
is an open tubular neighbourhood U ⊆ νMX of the 0-section such that the
exponential map exp |U : U → exp(U) is a diffeomorphism. For a normal vector
field s ∈ Γ(U), let exps : X →M , x ↦→ expx(s(x)). Furthermore, let
F : Γ(U)→ Ω4(X,Λ27M |X) , s ↦→ (exps)∗(τ) , (3.3)
where τ ∈ Ω4(M,Λ27M) is defined as in (2.16).
Local deformations of X are parametrised by sections s ∈ Γ(U) via the
exponential map (see Section 2.1.1). With this identification, Xs := exps(X)
is Cayley if and only if (exps)∗(τ) = τ |Xs = 0. So the moduli space of
all local deformations of X as a Cayley submanifold of M can be identified
with F−1({0}).
Let s ∈ Γ(νMX). We have
(dF )0(s) =
d
dtF (ts)
⏐⏐⏐⏐⏐
t=0
= ddt(expts)
∗(τ)
⏐⏐⏐⏐⏐
t=0
= (Lsτ)|X . (3.4)
Let (e1, . . . , e8) be a local orthonormal frame of M such that (e1, . . . , e4) is a
positive orthonormal frame of X. Then
τ =
8∑
i=2
(ei ∧ (e1 ⌟ Φ)− e1 ∧ (ei ⌟ Φ))⊗ (e1 × ei) (3.5)
by (2.17), where (e1, . . . , e8) is the dual coframe of (e1, . . . , e8). Let
τi := ei ∧ (e1 ⌟ Φ)− e1 ∧ (ei ⌟ Φ) .
So τ = ∑8i=2 τi ⊗ (e1 × ei). Then
(Lsτ)|X =
8∑
i=2
(Lsτi)|X ⊗ (e1 × ei) +
8∑
i=2
τi|X ⊗ ∇˜s(e1 × ei)
=
8∑
i=2
(Lsτi)|X ⊗ (e1 × ei)
since τi|X = 0 as X is Cayley, where ∇˜ is the induced connection on Λ27M . In
particular, the result is independent of ∇˜.
39
Now
Lsτi = (Lsei) ∧ (e1 ⌟ Φ) + ei ∧ ((Lse1) ⌟ Φ) + ei ∧ (e1 ⌟ LsΦ)
− (Lse1) ∧ (ei ⌟ Φ)− e1 ∧ ((Lsei) ⌟ Φ)− e1 ∧ (ei ⌟ LsΦ) .
Note that (Lsei)(ej) = s.(ei(ej)) − ei(Lsej) = −g(Lsej, ei) for i, j = 1, . . . , 8
and that
(ei ∧ (ej ⌟ Φ))|X =
⎧⎪⎨⎪⎩δijvolX for i = 1, . . . , 4, j = 1, . . . , 8,0 for i = 5, . . . , 8, j = 1, . . . , 8.
Indeed, ei|X = 0 for i = 5, . . . , 8, and if i ≤ 4, let (f1, f2, f3) be such that
(ei, f1, f2, f3) is a positive orthonormal frame of X. Then
(ei∧(ej⌟Φ))(ei, f1, f2, f3) = Φ(ej, f1, f2, f3) = g(ej, f1×f2×f3) = g(ej, ei) = δij
for j = 1, . . . , 8. So
((Lsei) ∧ (e1 ⌟ Φ))|X = −
8∑
j=1
g(Lsej, ei)(ej ∧ (e1 ⌟ Φ))|X
= −g(Lse1, ei)volX for i = 2, . . . , 8,
(ei ∧ ((Lse1) ⌟ Φ))|X =
8∑
j=1
g(Lse1, ej)(ei ∧ (ej ⌟ Φ))|X
=
⎧⎪⎨⎪⎩g(Lse1, ei)volX for i = 2, 3, 4,0 for i = 5, . . . , 8,
−((Lse1) ∧ (ei ⌟ Φ))|X =
8∑
j=1
g(Lsej, e1)(ej ∧ (ei ⌟ Φ))|X
=
⎧⎪⎨⎪⎩g(Lsei, e1)volX for i = 2, 3, 4,0 for i = 5, . . . , 8,
−(e1 ∧ ((Lsei) ⌟ Φ))|X = −
8∑
j=1
g(Lsei, ej)(e1 ∧ (ej ⌟ Φ))|X
= −g(Lsei, e1)volX for i = 2, . . . , 8.
40
Hence
((Lsei) ∧ (e1 ⌟ Φ) + ei ∧ ((Lse1) ⌟ Φ))|X =
⎧⎪⎨⎪⎩0 for i = 2, 3, 4,−g(Lse1, ei)volX for i = 5, . . . , 8
and
−((Lse1)∧ (ei ⌟Φ)+e1∧ ((Lsei)⌟Φ))|X =
⎧⎪⎨⎪⎩0 for i = 2, 3, 4,−g(Lsei, e1)volX for i = 5, . . . , 8.
Since ∇ is the Levi-Civita connection of (M, g),
−g(Lse1, ei)− g(Lsei, e1) = s.(g(e1, ei)) + g(∇e1s, ei) + g(∇eis, e1)
= g(∇e1s, ei) + g(∇eis, e1) .
Note that
(ei ∧ (ej ⌟ LsΦ))|X = ei ∧ (ej ⌟ (LsΦ)|X) = δij(LsΦ)|X for i, j = 1, . . . , 4
since ei ∧ (ej ⌟ volX) = δijvolX for i, j = 1, . . . , 4. So
(ei ∧ (e1 ⌟ LsΦ)− e1 ∧ (ei ⌟ LsΦ))|X =
⎧⎪⎨⎪⎩0 for i = 2, 3, 4,−e1 ∧ (ei ⌟ LsΦ)|X for i = 5, . . . , 8.
Let i ∈ {5, . . . , 8}. Now
(LsΦ)(ei, e2, e3, e4) = (∇sΦ)(ei, e2, e3, e4) + Φ(∇eis, e2, e3, e4)
+ Φ(ei,∇e2s, e3, e4) + Φ(ei, e2,∇e3s, e4)
+ Φ(ei, e2, e3,∇e4s) .
We have
Φ(∇eis, e2, e3, e4) = g(∇eis, e2 × e3 × e4) = g(∇eis, e1)
41
and
Φ(ei,∇e2s, e3, e4) = −g(∇e2s, ei × e3 × e4) ,
Φ(ei, e2,∇e3s, e4) = −g(∇e3s, ei × e4 × e2) ,
Φ(ei, e2, e3,∇e4s) = −g(∇e4s, ei × e2 × e3) .
Further,
e2 × (ei × e3 × e4) = e3 × (ei × e4 × e2) = e4 × (ei × e2 × e3)
= τ(ei, e2, e3, e4) = −ei × (e2 × e3 × e4) = e1 × ei
by (2.16) since τ is alternating.
So together we get
(Lsτ)(e1, e2, e3, e4)
=
8∑
i=5
(g(∇e1s, ei) + g(∇eis, e1)− (LsΦ)(ei, e2, e3, e4))(e1 × ei)
=
8∑
i=5
(g(∇e1s, ei) + g(∇eis, e1)− (∇sΦ)(ei, e2, e3, e4))(e1 × ei)
−
8∑
i=5
(Φ(∇eis, e2, e3, e4) + Φ(ei,∇e2s, e3, e4))(e1 × ei)
−
8∑
i=5
(Φ(ei, e2,∇e3s, e4) + Φ(ei, e2, e3,∇e4s))(e1 × ei)
=
8∑
i=5
g(∇e1s, ei)(e1 × ei) +
8∑
i=5
(∇sΦ)(ei, e2, e3, e4)(ei × e1)
+
8∑
i=5
g(∇e2s, ei × e3 × e4)(e2 × (ei × e3 × e4))
+
8∑
i=5
g(∇e3s, ei × e4 × e2)(e3 × (ei × e4 × e2))
+
8∑
i=5
g(∇e4s, ei × e2 × e3)(e4 × (ei × e2 × e3))
42
=
4∑
i=1
8∑
j=5
g(∇eis, ej)(ei × ej) +
8∑
i=5
(∇sΦ)(ei, e2, e3, e4)(ei × e1)
=
4∑
i=1
ei ×∇⊥eis+
8∑
i=5
(∇sΦ)(ei, e2, e3, e4)(ei × e1)
since (ei × ek × eℓ)i=5,...,8 is an orthonormal frame of νMX for k, ℓ ∈ {2, 3, 4}
with k ̸= ℓ. Hence
(dF )0(s) = volX ⊗ (0, Ds) , (3.6)
where we used the splitting (2.22), that is, (0, Ds) ∈ Λ2−X ⊕ E ∼= Λ27M |X .
The following proposition can be done by similar methods as in [McL98].
Proposition 3.3. Let M be a smooth 8-manifold with a smooth Spin(7)-
structure, let X be a smooth closed Cayley submanifold of M , and let 0 < α < 1.
If the operator D : Γ(νMX)→ Γ(E) defined in (3.1) is surjective, then the
moduli space of all smooth Cayley submanifolds of M that are C1,α-close to X
is a smooth manifold of dimension dim kerD = indD.
Proof. If Y is a 4-dimensional submanifold of M that is C1-close to X, then
Λ2−Y → Λ27M |Y , α ↦→ 2π7(α) is injective. So we can regard Λ2−Y as a
subbundle of Λ27M |Y . Note that τ |Y is orthogonal to Λ2−Y by (2.17) since
(v♭ ∧ (w ⌟ Φ))|Y = g(v, w)Φ|Y for v, w ∈ Γ(TY ). In particular, if
πE : Λ27M |X → E (3.7)
denotes the orthogonal projection, then πE((exps)∗(τ)) = 0 implies (exps)∗(τ) = 0
for s ∈ Γ(U), where τ ∈ Ω4(M,Λ27M) is defined as in (2.16) and U ⊆ νMX is
defined as in the proof of Theorem 3.1. So if we modify the definition of F
from the proof of Theorem 3.1 to
F : Γ(U)→ Ω4(X,E) ∼= Γ(E) , s ↦→ πE((exps)∗(τ)) , (3.8)
then the moduli space of all local deformations of X as a Cayley submanifold
of M can still be identified with F−1({0}).
43
The operator D is an operator of Dirac type, and hence elliptic. In fact, the
symbol σD of D is given by σD(x, ξ)s =
∑4
i=1 ei×ξis = ξ×s for x ∈ X, ξ ∈ TxX.
So σD(x, ξ) : (νMX)x → Ex is bijective if ξ ̸= 0. By Corollary 2.18, F extends to
a smooth map F1,α : C1,α(U)→ C0,α(E). So we can apply the Implicit Function
Theorem (Theorem 2.4) to deduce that (F1,α)−1({0}) is a smooth manifold
near 0 (in the C1,α-topology) of dimension dim kerD (see Section 2.1.3). Note
that the equation F1,α(s) = 0 is a nonlinear partial differential equation of
order 1 whose linearisation at 0 is elliptic, and hence the linearisation is elliptic
near 0 (in the C1,α-topology). So all elements of (F1,α)−1({0}) near 0 are smooth
by elliptic regularity [Mor66, Theorem 6.8.1].
3.2 Index formula for the operator of Dirac type
Here we prove an index formula for the operator of Dirac type that arises as
the linearisation of McLean’s deformation map.
Proposition 3.4. Let M be an 8-manifold with a Spin(7)-structure, let X be
a closed Cayley submanifold of M , and let D : Γ(νMX)→ Γ(E) be defined as
in (3.1). Then
indD = 12χ(X)−
1
2σ(X)− [X] · [X] , (3.9)
where χ(X) is the Euler characteristic, σ(X) is the signature, and [X] · [X] is
the self-intersection number of X.
Proof. We first recall some facts from Remark 3.2. Suppose that X has a spin
structure, and let S+ and S− denote the positive and negative spinor bundles,
respectively (note that both bundles are quaternionic line bundles). Then there
is a quaternionic line bundle F over X such that
νMX ⊗R C ∼= S− ⊗C F and E ⊗R C ∼= S+ ⊗C F . (3.10)
The symbol of D is the same as the symbol of the negative twisted Dirac
operator (the negative Dirac operator associated to the bundle S⊗C F with
44
the tensor product connection), and the index of an elliptic operator depends
only on the symbol of the operator [LM89, Corollary III.7.9]. So w.l.o.g. we
may assume that the Spin(7)-structure is torsion-free.
The Atiyah–Singer Index Theorem yields
indD = −
∫
X
(Aˆ(X) ch(F ))4 , (3.11)
where Aˆ(X) is the total Aˆ-class
Aˆ(X) = 1− 124p1(X) (3.12)
and ch(F ) is the Chern character
ch(F ) = 2 + c1(F ) +
1
2(c1(F )
2 − 2c2(F )) . (3.13)
Here p1(X) is the first Pontryagin class of X, and ci(F ) is the i-th Chern class
of F . Note that c1(F ) = 0 since F has a quaternionic structure. Hence
− (Aˆ(X) ch(F ))4 = 112p1(X) + c2(F ) . (3.14)
Now (3.10) implies that
− p1(νMX) = c2(νMX ⊗R C) = 2c2(S−) + 2c2(F ) (3.15)
since ch(S− ⊗C F ) = ch(S−) ch(F ) and c1(νMX ⊗R C) = c1(S−) = c1(F ) = 0.
We further have
c2(S−) =
1
2e(X)−
1
4p1(X) , (3.16)
where e(X) is the Euler class of X. Combining (3.11), (3.14), (3.15), and (3.16)
yields
indD = 13
∫
X
p1(X)− 12
∫
X
e(X)− 12
∫
X
p1(νMX) . (3.17)
This formula is valid even when X does not have a spin structure.
45
The definition (2.10) of the Spin(7)-form Φ0 on R8 shows that the interior
product with the Spin(7)-form Φ gives an isomorphism
Λ2−νMX ∼= Λ2−T ∗X . (3.18)
So
p1(νMX)− 2e(νMX) = p1(Λ2−νMX) = p1(Λ2−T ∗X) = p1(X)− 2e(X) (3.19)
by Lemma 3.5. Hence
indD = 12
∫
X
e(X)− 16
∫
X
p1(X)−
∫
X
e(νMX) . (3.20)
Now (3.9) follows from this formula using the generalised Gauss–Bonnet Theorem
∫
X
e(X) = χ(X) , (3.21)
the Hirzebruch Signature Theorem
1
3
∫
X
p1(X) = σ(X) , (3.22)
and the relation ∫
X
e(νMX) = [X] · [X] (3.23)
between the Euler class of the normal bundle and the self-intersection number.
Lemma 3.5. Let V be an oriented vector bundle of rank 4. Then
p1(Λ2−V ) = p1(V )− 2e(V ) . (3.24)
Proof. Let F∇ and F∇¯ denote the curvatures of V and Λ2−V , respectively. If
F∇ =
⎛⎜⎜⎜⎜⎜⎜⎝
0 a b c
−a 0 d e
−b −d 0 f
−c −e −f 0
⎞⎟⎟⎟⎟⎟⎟⎠
46
for a local, positively oriented, orthonormal frame (e1, . . . , e4), then
F∇¯ =
⎛⎜⎜⎜⎝
0 d− c b+ e
−(d− c) 0 f − a
−(b+ e) −(f − a) 0
⎞⎟⎟⎟⎠
for the orthonormal frame 1√2(e1∧ e2− e3∧ e4, e1∧ e3+ e2∧ e4, e1∧ e4− e2∧ e3).
Since
− tr(F∇¯)2 = 2((d− c)2 + (b+ e)2 + (f − a)2) ,
− tr(F∇)2 = 2(a2 + b2 + c2 + d2 + e2 + f 2) , and
Pf(F∇) = af − be+ cd ,
we get
− tr(F∇¯)2 = − tr(F∇)2 − 4Pf(F∇) ,
which proves (3.24) by the definition of the Pontryagin and Euler classes in
terms of the curvature.
3.3 Genericity
In the next section, we will prove that for a generic Spin(7)-structure, the
moduli space of closed Cayley submanifolds is a smooth finite-dimensional
manifold. Here we review some results that will be used in the proof.
Definition 3.6. Let X be a topological space. We say that a statement holds
for generic x ∈ X if the set of all x ∈ X for which the statement is true is a
residual set, that is, it contains a set which is the intersection of countably
many open dense subsets.
Note that if X is a complete metric space, then a residual set is dense by
the Baire Category Theorem.
Theorem 3.7 (cf. [Sch93, Proposition 2.24]). Let X, Y , and Z be separable
Banach spaces, let U ⊆ X and V ⊆ Y be open subsets, let f : U × V → Z be a
47
Ck-map (1 ≤ k ≤ ∞), let (x0, y0) ∈ U × V , and let ℓ ∈ Z be such that ℓ < k.
Suppose that for all (x, y) ∈ U × V with f(x, y) = f(x0, y0):
(i) (df)(x,y) : TxX × TyY → Tf(x,y)Z is surjective and
(ii) (df)(x,y)|TxX : TxX → Tf(x,y)Z is Fredholm with index ℓ.
Then the set of all y ∈ V such that the operator (df)(x,y)|TxX : TxX → Tf(x,y)Z
is surjective for all x ∈ U with f(x, y) = f(x0, y0) is the intersection of countably
many open dense subsets.
To achieve the surjectivity (i) in the above theorem, we will use the following
lemma.
Lemma 3.8. Let X be a topological vector space, let V be closed subspace of X
with finite codimension, and let W be a dense subspace of X. Then X = V +W .
Proof. The proof works by induction. It is trivially true if V = X. Otherwise,
X \ V is open and non-empty. Hence there is some w ∈ W \ V since W is
dense. Then V + ⟨w⟩ is a closed subspace of X whose codimension is smaller
than the codimension of V .
Theorem 3.7 needs Banach spaces but the space of smooth sections of a
vector bundle is not a Banach space. So we will show the genericity for Hölder
spaces first and then use the following lemma to get a genericity statement
for C∞.
Lemma 3.9. Let M be a smooth, compact manifold, let E be a smooth vector
bundle over M , let k0 ≥ 0 be an integer, let 0 < α < 1, let Mk0 ⊆ Ck0,α(E), let
Mk :=Mk0 ∩Ck,α(E) for k ≥ k0 + 1, and let M∞ :=Mk0 ∩C∞(E). Suppose
that Mk is the intersection of countably many open dense subsets of Ck,α(E)
for all k ≥ k0. Then M∞ is the intersection of countably many open dense
subsets of C∞(E).
Proof. For each k ≥ k0, there are countable many open dense subsets (Uk,n)n≥1
of Ck,α(E) such that
Mk =
∞⋂
n=1
Uk,n .
48
Let k ≥ k0 and n ≥ 1. Then Uk,n ∩ Ck′,α(E) is open in Ck′,α(E) for all k′ ≥ k.
Furthermore, Mk′ ⊆ Uk,n ∩ Ck′,α(E) for all k′ ≥ k, and hence Uk,n ∩ Ck′,α(E)
is dense in Ck′,α(E) for all k′ ≥ k since Mk′ is dense in Ck′,α(E) by the Baire
Category Theorem. So Uk,n ∩ C∞(E) is open and dense in C∞(E).
Therefore, M∞ is the intersection of countably many open dense subsets of
C∞(E) since
M∞ =
∞⋂
k=k0
Mk =
∞⋂
k=k0
∞⋂
n=1
Uk,n .
3.4 Varying the Spin(7)-structure
Here we prove that for a generic Spin(7)-structure, the moduli space of closed
Cayley submanifolds is a finite-dimensional smooth manifold. A similar result
for associative submanifolds was proved by Gayet [Gay14]. Lemma 3.12 below
will be the key lemma of the proof. In the following theorem, we use the
C∞-topology for the space of all Spin(7)-structures.
Theorem 3.10 (Theorem 1.1). Let M be a smooth 8-manifold with a smooth
Spin(7)-structure Φ, let X be a smooth, closed Cayley submanifold of M , and
let 0 < α < 1.
Then for every generic smooth Spin(7)-structure Ψ that is C1,α-close to Φ
and inducing the same metric as Φ, the moduli space of all smooth Cayley
submanifolds of (M,Ψ) that are C1,α-close to X is either empty or a smooth
manifold of dimension indD, where D is defined in (3.1).
Note that if indD < 0, then the moduli space is necessarily empty for a
generic Spin(7)-structure. Furthermore, the statement of the above theorem
remains true for the larger class of all smooth Spin(7)-structures Ψ that are
C1,α-close to Φ (not necessarily inducing the same metric as Φ).
Remark 3.11. For the proof, we will assume that M is compact. If M is
non-compact, the statement of the theorem is true for both the weak and the
strong C∞-topology on M . Since X is compact, we can work on a compact
neighbourhood K of X in M for the proof. Note that the map restricting a
49
Spin(7)-structure to K has the property that the preimage of a residual set is
again a residual set. Also, ‘Ψ that is C1,α-close to Φ’ should be interpreted as
such that the restriction of Ψ to K is C1,α-close to the restriction of Φ to K.
Proof. Recall from (2.12) that we have a pointwise orthogonal splitting
Λ4M ∼= Λ41M ⊕ Λ47M ⊕ Λ427M ⊕ Λ435M .
There are an open tubular neighbourhood V ⊆ Λ41M ⊕ Λ47M ⊕ Λ435M of the
0-section and a smooth bundle morphism Θ : V → Λ4M (we will also denote
the map Γ(V )→ Ω4(M), χ ↦→ Θ ◦ χ by Θ) such that [Joy00, Definition 10.5.8
and Proposition 10.5.9]
(i) Θ(0) = Φ,
(ii) Θ(χ) is a Spin(7)-structure on M for each χ ∈ Γ(V ),
(iii) if Ψ is a Spin(7)-structure on M that it C0-close to Φ, then Ψ = Θ(χ) for
some unique χ ∈ Γ(V ), and
(iv) (dΘ)0(χ) = χ for all χ ∈ Γ(Λ41M ⊕ Λ47M ⊕ Λ435M).
So Θ parametrises Spin(7)-structures on M that are C0-close to Φ.
Let
πE : Λ2M |X → E (3.25)
denote the orthogonal projection, where the vector bundle E of rank 4 over X
is defined as in (2.23). Furthermore, let U ⊆ νMX be defined as in the proof
of Theorem 3.1, and let
F˜ : Γ(U)⊕ Γ(V )→ Ω4(X,E) ∼= Γ(E) , (s, χ) ↦→ πE((exps)∗(τΘ(χ))) , (3.26)
where τΘ(χ) ∈ Ω4(M,Λ2M) is defined as in (2.16) with respect to the Spin(7)-
structure Θ(χ). So the moduli space of all Cayley submanifolds of (M,Θ(χ))
near X can be identified with F˜ ( · , χ)−1({0}).
Lemma 3.12. Let e ∈ Γ(Λ27M), and let χ := h(τΦ, e), where h is the metric
on Λ27M (note that χ ∈ Ω47(M) by (2.17) and (2.18)). Then
(dF˜ )(0,0)(0, χ) = −πE(e|X) . (3.27)
50
Proof. Since χ ∈ Ω47(M), there is a path (Φt)t∈(−ε,ε) of Spin(7)-structures on M
with Φ0 = Φ and ddtΦt
⏐⏐⏐
t=0
= χ such that the metric induced by Φt is the same
metric as the metric induced by Φ for t ∈ (−ε, ε) [Kar05, Proposition 5.3.1].
Write Φt = Θ(χt) with ddtχt
⏐⏐⏐
t=0
= χ.
Let (e1, . . . , e8) be a local orthonormal frame of M such that (e1, . . . , e4) is a
positive frame of X, and let (e1, . . . , e8) be the dual coframe. Then
(dF˜ )(0,0)(0, χ) =
d
dtF˜ (0, χt)
⏐⏐⏐⏐⏐
t=0
= ddtπE
(
τΘ(χt)
⏐⏐⏐
X
)⏐⏐⏐⏐⏐
t=0
= ddt
8∑
i=2
(ei ∧ (e1 ⌟ Φt)− e1 ∧ (ei ⌟ Φt))|X ⊗ πE(e1 × ei)
⏐⏐⏐⏐⏐
t=0
=
8∑
i=2
(ei ∧ (e1 ⌟ χ)− e1 ∧ (ei ⌟ χ))|X ⊗ πE(e1 × ei)
= −
8∑
i=5
χ(ei, e2, e3, e4)volX ⊗ (e1 × ei)
= −
8∑
i=5
h(τΦ(ei, e2, e3, e4), e)volX ⊗ (e1 × ei)
=
8∑
i=5
h(ei × (e2 × e3 × e4), e)volX ⊗ (e1 × ei)
=
8∑
i=5
h(e, ei × e1)volX ⊗ (e1 × ei)
= −volX ⊗ πE(e|X)
by (2.16) and (2.17).
By Proposition 2.17, F˜ extends to a map
F˜1,α : C1,α(U)⊕ C1,α(V )→ C0,α(E) (3.28)
of class C1. For fixed χ ∈ C1,α(V ), the equation
F˜1,α(s, χ) = πE((exps)∗(τΘ(χ))) = 0 (3.29)
51
is a nonlinear partial differential equation of order 1 in s. Furthermore, the
linearisation at 0 is elliptic for χ = 0. Hence there is a C1,α-neighbourhood
U˜1 ⊆ C1,α(U)⊕ C1,α(V ) of (0, 0) such that
(dF˜1,α)(s,χ)|C1,α(νMX) : C1,α(νMX)→ C0,α(E) (3.30)
is an elliptic differential operator of order 1 for all (s, χ) ∈ U˜1. In particular,
this operator is Fredholm for all (s, χ) ∈ U˜1, and hence its image is a closed
subspace with finite codimension. Furthermore, if (s, χ) ∈ U˜1 satisfies (3.29)
and χ ∈ Ck,α(V ) (for some k ≥ 1), then s ∈ Ck+1,α(νMX) by elliptic regularity
[Mor66, Theorem 6.8.1] since τΘ(χ) ∈ Ck,α(Λ4M ⊗ Λ2M).
The proof of Lemma 3.12 implies that if (s, χ) ∈ C1,α(U) ⊕ C1,α(V ) is
C1,α-close to (0, 0) and satisfies (3.29), then τΘ(χ)|Xs ∈ C1,α(Λ4Xs ⊗ Es) and
C1,α(V )→ C1,α(Λ4Xs ⊗ Es), χ ↦→ τΘ(χ)|Xs is surjective, where Es is defined as
in (2.23) with respect to the Cayley submanifold Xs := exps(X). So there is a
C1,α-neighbourhood U˜2 ⊆ C1,α(U)⊕ C1,α(V ) of (0, 0) such that if (s, χ) ∈ U˜2
satisfies (3.29) and χ ∈ Ck,α(V ) (for some k ≥ 1), then the image of the
operator
(dF˜1,α)(s,χ)|Ck,α(Λ41M⊕Λ47M⊕Λ435M) : Ck,α(Λ41M ⊕ Λ47M ⊕ Λ435M)→ C0,α(E)
is exactly Ck,α(E) since exps : X → exps(X) is a diffeomorphism, and hence the
map Ck,α(Λ4Xs ⊗ Es)→ Ck,α(E), χ ↦→ πE((exps)∗(χ)) is a linear isomorphism
(note that s ∈ Ck+1,α(νMX) by above).
For k ≥ 1, let
F˜k,α := F˜1,α|C1,α(U)⊕Ck,α(V ) : C1,α(U)⊕ Ck,α(V )→ C0,α(E) .
Then F˜k,α is of class Ck by Proposition 2.17. Let U˜ := U˜1∩ U˜2. Then the above
argumentation shows that if (s, χ) ∈ U˜ ∩ (C1,α(U)⊕ Ck,α(V )) satisfies (3.29),
52
then
(dF˜k,α)(s,χ) : C1,α(νMX)⊕ Ck,α(Λ41M ⊕ Λ47M ⊕ Λ435M)→ C0,α(E)
is surjective by Lemma 3.8 since Ck,α(E) is dense in C0,α(E), and
(dF˜k,α)(s,χ)|C1,α(νMX) = (dF˜1,α)(s,χ)|C1,α(νMX) : C1,α(νMX)→ C0,α(E)
is Fredholm. Furthermore, the Fredholm index is the same for all of these
operators as we may assume w.l.o.g. that U˜1 is connected. So the Fredholm
index is indD since D = (dF˜ )(0,0)|Γ(νMX).
Let U˜3 ⊆ C1,α(U) and U˜4 ⊆ C1,α(V ) be open neighbourhoods of 0 such that
U˜3 × U˜4 ⊆ U˜ , let U1,α be the set of all χ ∈ U˜4 such that the operator (3.30) is
surjective for all s ∈ U˜3, and define Uk,α := U1,α∩Ck,α(Λ41M⊕Λ47M⊕Λ435M) for
k ≥ 1. Let k0 := max{indD+1, 1}. Then Theorem 3.7 implies that Uk,α is the
intersection of countably many open dense subsets of Ck,α(Λ41M⊕Λ47M⊕Λ435M)
for k ≥ k0. So if we define U∞ := ⋂∞k=k0 Uk,α, then U∞ is the intersection of
countably many open dense subsets of C∞(Λ41M⊕Λ47M⊕Λ435M) by Lemma 3.9.
So for every generic χ ∈ C∞(V ) that is C1,α-close to 0, the moduli space of
all solutions s ∈ C∞(U) of (3.29) that are C1,α-close to 0 is either empty or a
smooth manifold of dimension indD.
Note that the proof also shows the following Ck,α-version, where we use the
Ck,α-topology for the space of all Spin(7)-structures.
Theorem 3.13. Let k ≥ 1, let 0 < α < 1, let M be an 8-manifold of
class Ck+1,α with a Spin(7)-structure Φ of class Ck,α, and let X be a closed
Cayley submanifold of M of class Ck+1,α. Suppose that k > indD, where D is
defined in (3.1).
Then for every generic Spin(7)-structure Ψ of class Ck,α that is C1,α-close
to Φ and inducing the same metric as Φ, the moduli space of all Cayley
submanifolds of (M,Ψ) of class Ck+1,α that are C1,α-close to X is either empty
or a Ck-manifold of dimension indD.
53
3.5 Remark about torsion-free Spin(7)-structures
If M is a closed 8-manifold with a torsion-free Spin(7)-structure Φ, then the
tangent space to the space of all torsion-free Spin(7)-structures on M at Φ can
be identified with [Joy00, Theorem 10.7.1]
{LvΦ : v ∈ Γ(TM)} ⊕ (H41(M)⊕H47(M)⊕H435(M)) , (3.31)
where H4i (M) is the space of all harmonic forms in Ω4i (M). If χ = LvΦ, then
(dF˜ )(0,0)(0, χ) lies in the image of D by Lemma 3.14 below. So it does not
contribute to the surjectivity of (dF˜ )(0,0). But the dimension of cokerD is not
in general less than b41(M) + b47(M) + b435(M) (see below). That is why we
consider the larger space of all Spin(7)-structures on M .
Lemma 3.14. Let M be an 8-manifold with a Spin(7)-structure Φ, let X be a
closed Cayley submanifold of M , and let v ∈ Γ(TM). Then
(dF˜ )(0,0)(0,LvΦ) = D(v|X)⊥ , (3.32)
where (v|X)⊥ is the pointwise orthogonal projection of v|X onto νMX.
Proof. Let φtv denote the flow generated by v, and let Φt := (φtv)∗Φ. Then
τΦt = (φtv)∗τΦ. Hence
d
dtτΦt
⏐⏐⏐⏐⏐
t=0
= ddt(φ
t
v)∗τΦ
⏐⏐⏐⏐⏐
t=0
= LvτΦ .
So the result follows from the proof of Theorem 3.1 since τΦ|X = 0.
In [Gay14, last paragraph of Section 4.4], Gayet constructs a closed 7-mani-
fold M˜ with a torsion-free G2-structure such that given n ∈ N, there is a closed
associative submanifold X˜ for which the dimension of the cokernel of the asso-
ciated Dirac operator has dimension at least n. So if we define M := S1 × M˜
and X := S1 × X˜, then the dimension of cokerD will be at least n.
54
4 Compact Cayley submanifolds
with boundary
In this chapter, we present the deformation theory of compact, connected Cayley
submanifolds with non-empty boundary. We first review elliptic boundary
problems in Section 4.1. Then in Section 4.2, we show that the moduli space
of compact Cayley submanifolds with boundary in a chosen fixed submanifold
(in the following called scaffold) and meeting the scaffold orthogonally is
included in the solution space of a second-order elliptic boundary problem with
index 0 (Proposition 1.2). We proceed in Section 4.3 by discussing the extremal
dimensions 3 and 7 for the scaffold. We further show, in Section 4.4, that for a
generic Spin(7)-structure, the moduli space is a finite set (Theorem 1.3). In
Section 4.5, we show that also for a generic deformation of the scaffold, the
moduli space is a finite set (Theorem 1.4). Then in Section 4.6, we show that
Theorem 1.3 remains true if we restrict to the smaller class of all torsion-free
Spin(7)-structures. We finish this chapter by proving in Section 4.7 that a
particular first-order boundary problem for the operator of Dirac type that
arises as the linearisation of the deformation map is not elliptic. That is why
we consider a second-order boundary problem in Section 4.2.
4.1 Elliptic boundary problems
Here we give a short review on elliptic boundary problems and the regularity
of their solutions. In this section, we assume that all differential operators have
smooth coefficients, although later we also use generalisations to differential
operators whose coefficients are just in appropriate Hölder spaces.
55
Definition 4.1. Let X be a compact manifold with boundary, let E and F
be vector bundles over X, let G1, . . . , Gk be vector bundles over ∂X, and let
P : Γ(E) → Γ(F ) and B1 : Γ(E) → Γ(G1), . . . , Bk : Γ(E) → Γ(Gk) be linear
differential operators with principal symbols σP and σB1 , . . . , σBk , respectively.
Then the boundary problem
⎧⎪⎨⎪⎩ Ps = t in X,(B1s, . . . , Bks) = (h1, . . . , hk) on ∂X (4.1)
(where t ∈ Γ(F ), h1 ∈ Γ(G1), . . . , hk ∈ Γ(Gk) are given; s ∈ Γ(E)) is called
elliptic if
(i) P is elliptic in X (i.e., σP (x, ξ) : Ex → Fx is bijective for all x ∈ X,
ξ ∈ T ∗xX \ {0}) and
(ii) for all x ∈ ∂X, ξ ∈ T ∗x∂X ⊆ T ∗xX (we extend ξ from Tx∂X to TxX by
ξ(ux) = 0, where ux is the inward-pointing unit normal vector), the map
M+x,ξ → (G1)x ⊕ · · · ⊕ (Gk)x ,
f ↦→ (σB1(x, ξ − i∂tηx)f(0), . . . , σBk(x, ξ − i∂tηx)f(0))
(4.2)
is bijective, where ηx is the interior conormal ofX (i.e., ηx(ux) = 1 and ηx|Tx∂X = 0)
andM+x,ξ is the set of all f ∈ C∞(R, Ex) with σP (x, ξ − i∂tηx)f(t) = 0 which
are bounded on R+.
Compare [Hör85, Definition 20.1.1]. This condition is sometimes also called
the Lopatinski–Shapiro condition.
Theorem 4.2 (cf. [Hör85, Theorems 20.1.2 and 20.1.8] and [RS82, Theorem 3
in Section 3.1.1.4]). Let X be a compact manifold with boundary, let E and F
be vector bundles over X, let G1, . . . , Gk be vector bundles over ∂X, and let
P : Γ(E) → Γ(F ) and B1 : Γ(E) → Γ(G1), . . . , Bk : Γ(E) → Γ(Gk) be linear
differential operators of orders p and b1, . . . , bk, respectively. Suppose that
p > b1, . . . , bk and that (4.1) is an elliptic boundary problem.
56
Then the map
Ck,α(E)→ Ck−p,α(F )⊕ Ck−b1,α(G1)⊕ · · · ⊕ Ck−bk,α(Gk) ,
s ↦→ (Ps,B1s, . . . , Bks)
(4.3)
is Fredholm for all k ≥ p, 0 < α < 1, and the index is independent of k, α.
Furthermore, the kernel consists of smooth sections and the L2-orthogonal
complement of the image consists of smooth sections.
Theorem 4.3 (cf. [Mor66, Theorem 6.8.2]). Let X be a compact manifold with
boundary, let E and F be vector bundles over X, let G1, . . . , Gk be vector bundles
over ∂X, let P : Γ(E)→ Γ(F ) and B1 : Γ(E)→ Γ(G1), . . . , Bk : Γ(E)→ Γ(Gk)
be non-linear differential operators of orders p and b1, . . . , bk, respectively.
Suppose that p > b1, . . . , bk and that the linearisation of (P,B1, . . . , Bk) at the
0-section is an elliptic boundary problem.
Then there is an ε > 0 such that if s ∈ Cp(E) with ∥s∥Cp < ε satisfies⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
Ps = 0 in X,
B1s = 0 on ∂X,
...
Bks = 0 on ∂X,
(4.4)
then s is smooth.
4.2 Deformations of compact Cayley
submanifolds with boundary
The next proposition is the first step for the proofs of Theorems 1.3 and 1.4.
Definition 4.4. Let (M, g) be a Riemannian manifold, let X be a compact
submanifold of M with non-empty boundary, let W be a submanifold of M
with ∂X ⊆ W , and let u ∈ Γ(νX∂X) be a unit normal vector field of ∂X in X.
Then X and W meet orthogonally if u ∈ Γ(νMW |∂X).
57
Proposition 4.5 (Proposition 1.2). Let M be an 8-manifold with a Spin(7)-
structure Φ, let X be a compact Cayley submanifold of M with boundary, and let
W be a submanifold of M with ∂X ⊆ W such that X and W meet orthogonally.
Then the moduli space of all local deformations of X as a Cayley submanifold
of M with boundary on W and meeting W orthogonally is included in the
solution space of the boundary problem (4.18) below, which is a second-order
elliptic boundary problem with index 0.
In particular, the ‘Zariski tangent space’ (the kernel of the linearisation
of the deformation map) is finite-dimensional and elliptic regularity applies
(i.e., all solutions are smooth). Note that the dimension of W is at least 3 since
∂X ⊆ W and ∂X is 3-dimensional. Furthermore, the dimension of W is at
most 7 since M is 8-dimensional and X and W meet orthogonally. In fact, if
W were 8-dimensional, then there would be essentially no constraint on the
boundary, which would lead to an infinite-dimensional moduli space. For a
discussion of the dimensions 3 and 7, see Section 4.3.
Proof. In Section 2.1.2, we modified the metric g to a metric gˆ with gˆx = gx
for all x ∈ ∂X such that W is totally geodesic with respect to gˆ. Then there
is an open tubular neighbourhood Uˆ ⊆ νˆMX of the 0-section such that local
deformations of X with boundary on W are parametrised by sections sˆ ∈ Γ(Uˆ)
with sˆ|∂X ∈ Γ(νW∂X) (Proposition 2.3), where νˆMX is the normal bundle
with respect to the metric gˆ. Note that νˆMX ∼= (TM |X)/TX ∼= νMX. Let
U ⊆ νMX be the image of Uˆ under this isomorphism.
For a normal vector field s ∈ Γ(U), let sˆ ∈ Γ(Uˆ) be the corresponding
vector field under the isomorphism νMX ∼= νˆMX, and define eˆxps : X → M ,
x ↦→ eˆxpx(sˆ(x)), where eˆxp : Uˆ → M is the exponential map of the metric gˆ.
Let
πE : Λ27M |X → E (4.5)
denote the orthogonal projection, where the vector bundle E of rank 4 over X
is defined as in (2.23), and let
F : Γ(U)→ Ω4(X,E) ∼= Γ(E) , s ↦→ πE((eˆxps)∗(τ)) , (4.6)
58
where τ ∈ Ω4(M,Λ27M) is defined as in (2.16). Then
(dF )0(s) = Ds , (4.7)
where D : Γ(νMX)→ Γ(E) is defined in (3.1). This follows from the proof of
Theorem 3.1 using that τ |X = 0 (so the result does not depend on the choice
of metric for the exponential map).
Let K be the subbundle of νMX|∂X consisting of all vectors that are ortho-
gonal to νW∂X, and let
πK : νMX|∂X → K (4.8)
denote the orthogonal projection. So if s ∈ Γ(νMX), then s|∂X ∈ Γ(νW∂X) if
and only if πK(s|∂X) = 0.
Let
πW : TM |W → TW (4.9)
and
πν : TM |∂X → νW∂X (4.10)
denote the orthogonal projections, let U∂X := U ∩ νMX|∂X , and let
H : Γ(U∂X)→ Ω3(∂X, νW∂X) ∼= Γ(νW∂X) , s ↦→ πν((eˆxpπν(s))∗(γ)) , (4.11)
where γ ∈ Ω3(W,TW ) is defined by
γ(a, b, c) := πW (a× b× c) (4.12)
for a, b, c ∈ TW . So if s ∈ Γ(U) defines a Cayley submanifold of M with
boundary onW (i.e., F (s) = 0 and πK(s|∂X) = 0), then this Cayley submanifold
meetsW orthogonally if and only if H(s|∂X) = 0. This follows because if (a, b, c)
is a local orthonormal frame of ∂X, then a× b× c is a unit normal vector field
of ∂X in X since X is Cayley. The projection onto νW∂X is enough since the
cross product of three vectors is orthogonal to these vectors.
59
So the moduli space of all local deformations of X as a Cayley submanifold
of M with boundary on W and meeting W orthogonally can be identified with
the moduli space of all solutions near the 0-section of the boundary problem
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
F (s) = 0 in X,
πK(s|∂X) = 0 on ∂X,
H(s|∂X) = 0 on ∂X.
(4.13)
Let D∗ : Γ(E)→ Γ(νMX) be the formal adjoint of D, and let
G : Γ(U)→ Γ(νMX) , s ↦→ D∗(F (s)) . (4.14)
Then
(dG)0(s) = D∗Ds (4.15)
by (4.7) since D∗ is linear. Let u ∈ Γ(νX∂X) be the inward-pointing unit
normal vector field of ∂X in X, and let
ρ : νMX|∂X → E|∂X , s ↦→ u× s . (4.16)
Then ρ is an isomorphism of vector bundles. Let
B : Γ(U)→ Γ(νW∂X) , s ↦→ πν(ρ−1(F (s)|∂X)) +H(s|∂X) . (4.17)
So solutions of (4.13) are also solutions of
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
G(s) = 0 in X,
πK(s|∂X) = 0 on ∂X,
B(s) = 0 on ∂X.
(4.18)
We will now prove that the boundary problem (4.18) is an elliptic boundary
problem with index 0.
60
Let P : Γ(νMX|∂X)→ Γ(νMX|∂X),
Ps :=
4∑
i=2
u× ei ×∇⊥eis−
8∑
i=5
(∇sΦ)(ei, e2, e3, e4)ei , (4.19)
where (ei)i=2,3,4 is any positive (i.e., u = e2 × e3 × e4) local orthonormal frame
of ∂X, (ei)i=5,...,8 is any local orthonormal frame of νMX|∂X , and ∇⊥ is the
induced connection on νMX. So if s ∈ Γ(νMX), then
ρ−1((Ds)|∂X) = ∇us|∂X + P (s|∂X) (4.20)
by (3.1) since
g(ρ−1(ei ×∇⊥eis), ej) = h(ei ×∇⊥eis, u× ej)
= −Φ(ei,∇⊥eis, u, ej)
= g(u× ei ×∇⊥eis, ej)
for i = 2, 3, 4, j = 5, . . . , 8 by (2.15), where h is the metric on Λ27M .
Lemma 4.6. Let s ∈ Γ(νMX). Then
(dB)0(s) = πν(∇us|∂X −∇πν(s|∂X)u+ P (πK(s|∂X))) . (4.21)
Proof. First let s ∈ Γ(νW∂X). Then
(dH)0(s) =
d
dtH(ts)
⏐⏐⏐⏐⏐
t=0
= ddtπν((eˆxpts)
∗(γ))
⏐⏐⏐⏐⏐
t=0
= πν((Lsγ)|∂X) . (4.22)
Let k be the dimension of W , and let (e1, . . . , e8) be a local orthonormal frame
of M such that (e1, . . . , e4) is a positive (i.e., e1 = e2 × e3 × e4) frame of X
with e1 = u and (e2, . . . , ek+1) is a frame of W . Then
γ =
k+1∑
i=2
(ei ⌟ Φ)|W ⊗ ei . (4.23)
61
So
(Lsγ)|∂X =
k+1∑
i=2
(Ls(ei ⌟ Φ))|∂X ⊗ ei +
k+1∑
i=2
(ei ⌟ Φ)|∂X ⊗ πW (∇sei)
=
k+1∑
i=2
(Ls(ei ⌟ Φ))|∂X ⊗ ei
since (ei ⌟ Φ)|∂X = 0 as Φ(ei, e2, e3, e4) = g(ei, e2 × e3 × e4) = g(ei, e1) = 0 for
i = 2, . . . , 8. We have Ls(ei ⌟ Φ) = Lsei ⌟ Φ+ ei ⌟ LsΦ. So
(Ls(ei ⌟ Φ))(e2, e3, e4) = Φ(Lsei, e2, e3, e4) + (LsΦ)(ei, e2, e3, e4)
= Φ(Lsei, e2, e3, e4) + (∇sΦ)(ei, e2, e3, e4)
+ Φ(∇eis, e2, e3, e4) + Φ(ei,∇e2s, e3, e4)
+ Φ(ei, e2,∇e3s, e4) + Φ(ei, e2, e3,∇e4s) .
Now
Φ(Lsei, e2, e3, e4) + Φ(∇eis, e2, e3, e4) = Φ(∇sei, e2, e3, e4)
= g(∇sei, e2 × e3 × e4) = g(∇sei, e1) = −g(∇su, ei)
since u = e1 = e2 × e3 × e4.
Recall from the proof of Theorem 3.1 that
8∑
i=5
(Φ(ei,∇e2s, e3, e4) + Φ(ei, e2,∇e3s, e4) + Φ(ei, e2, e3,∇e4s))(e1 × ei)
= −
4∑
i=2
8∑
j=5
g(∇eis, ej)(ei × ej) = −
4∑
i=2
ei ×∇⊥eis .
So
πν((Lsγ)(e2, e3, e4))
= −πν(∇su)−
4∑
i=2
πν(ρ−1(ei ×∇⊥eis)) +
k+1∑
i=5
(∇sΦ)(ei, e2, e3, e4)ei
= −πν(∇su+ Ps) .
62
For general s ∈ Γ(νMX), we have s|∂X = πν(s|∂X) + πK(s|∂X). Hence
(dB)0(s) = πν(ρ−1((Ds)|∂X)) + πν((Lπν(s|∂X)γ)(e2, e3, e4))
= πν(∇us|∂X + P (s|∂X)−∇πν(s|∂X)u− P (πν(s|∂X)))
= πν(∇us|∂X −∇πν(s|∂X)u+ P (πK(s|∂X)))
by (4.7), (4.17), and (4.20).
Lemma 4.7. The linearisation of the boundary problem (4.18) at the 0-section
is elliptic.
Proof. Let k be the dimension of W , and let (e1, . . . , e8) be a local orthonormal
frame of M such that (e1, . . . , e4) is a positive (i.e., e1 = e2 × e3 × e4) frame
of X with e1 = u and (e2, . . . , ek+1) is a frame of W . Let s ∈ Γ(νMX), and
write s = s5e5 + . . .+ s8e8. Then πK(s|∂X) = ∑8i=k+2 sjej. Hence
(dB)0(s) =
k+1∑
i=5
(e1.si)ei +
4∑
i=2
8∑
j=k+2
(ei.sj)πν(e1 × ei × ej) + l.o.t. (4.24)
by (4.19) and (4.21), where ‘l.o.t.’ stands for lower order terms (i.e., an operator
of order 0 in s).
So the symbol σ∂(x, ξ) (for x ∈ ∂X, ξ ∈ T ∗xX) of the boundary operator
B ⊕ πK is given by ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
iξ1 0 0
0 . . . 0 A
0 0 iξ1
0 0 0 1 0 0
0 . . . 0 0 . . . 0
0 0 0 0 0 1
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(4.25)
for some matrix A = A(x, ξ) in the frame (e5, . . . , e8) of νMX|∂X = νW∂X⊕K,
where ξ = ξ1e1+ · · ·+ ξ4e4. Here (e1, . . . , e4) is the dual coframe of (e1, . . . , e4).
The symbol σG(x, ξ) of the operator D∗D is given by
σG(x, ξ) = −|ξ|2 Id(νMX)x
63
for x ∈ X, ξ ∈ T ∗xX [BBW93, Lemma 3.3] since D + D∗ : Γ(νMX ⊕ E) →
Γ(νMX ⊕ E) is an operator of Dirac type. So σG(x, ξ − i∂te1)f(t) = 0 means
∂2t f(t) = |ξ|2f(t) for x ∈ ∂X, ξ ∈ T ∗x∂X ⊆ T ∗xX. The solutions are given
by f(t) = e−|ξ|ta + e|ξ|tb for a, b ∈ (νMX)x. So the set of solutions which are
bounded on R+ is
M+x,ξ := {e−|ξ|ta : a ∈ (νMX)x} .
Hence f ′(0) = −|ξ|f(0) for f ∈M+x,ξ. Therefore,
σ∂(x, ξ − i∂te1)f(0) =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
i|ξ| 0 0
0 . . . 0 A
0 0 i|ξ|
0 0 0 1 0 0
0 . . . 0 0 . . . 0
0 0 0 0 0 1
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
f(0) . (4.26)
This matrix is invertible for ξ ̸= 0. So
M+x,ξ → (νW∂X ⊕K)x , f ↦→ σ∂(x, ξ − i∂te1)f(0)
is bijective sinceM+x,ξ → (νMX)x, f ↦→ f(0) is bijective. Hence the linearisation
of the boundary problem (4.18) at the 0-section is elliptic (D∗D is clearly an
elliptic operator).
Lemma 4.8. The linearisation of the boundary problem (4.18) at the 0-section
has index 0.
Proof. We first show that the operator P defined in (4.19) has self-adjoint
symbol. Let s, t ∈ Γ(νMX|∂X), let (ei)i=2,3,4 be any positive (i.e., u = e2×e3×e4)
local orthonormal frame of ∂X, and let (ei)i=5,...,8 be any local orthonormal
frame of νMX|∂X . Then
g(Ps, t) =
4∑
i=2
g(u× ei ×∇eis, t)−
8∑
i=5
(∇sΦ)(ei, e2, e3, e4)g(ei, t)
=
4∑
i=2
Φ(t, u, ei,∇eis)− (∇sΦ)(t, e2, e3, e4)
64
by (4.19). We have
Φ(t, u, ei,∇eis) = ei.(Φ(t, u, ei, s))− (∇eiΦ)(t, u, ei, s)− Φ(∇eit, u, ei, s)
− Φ(t,∇eiu, ei, s)− Φ(t, u,∇eiei, s)
and
δ∂X(Φ(t, u, · , s)) = −
4∑
i=2
ei ⌟∇ei(Φ(t, u, · , s))
= −
4∑
i=2
(ei.(Φ(t, u, ei, s))− Φ(t, u,∇eiei, s)) .
Hence
g(Ps, t)− g(s, P t) = − δ∂X(Φ(t, u, · , s))
−
4∑
i=2
((∇eiΦ)(t, u, ei, s) + Φ(t,∇eiu, ei, s))
+ (∇tΦ)(s, e2, e3, e4)− (∇sΦ)(t, e2, e3, e4) .
Let P ∗ : Γ(νMX|∂X)→ Γ(νMX|∂X) be the formal adjoint of P . Since∫
∂X
δ∂X(Φ(t, u, · , s))vol∂X = −
∫
∂X
d∂X(∗∂X(Φ(t, u, · , s))) = 0
by Stokes’ Theorem, we therefore get
Ps− P ∗s = −
4∑
i=2
8∑
j=5
((∇eiΦ)(ej, u, ei, s) + Φ(ej,∇eiu, ei, s))ej
+
8∑
j=5
((∇ejΦ)(s, e2, e3, e4)− (∇sΦ)(ej, e2, e3, e4))ej ,
(4.27)
which is an operator of order 0 in s.
We will use Green’s formula [BBW93, Proposition 3.4 (b)]: If D˜ : Γ(S)→ Γ(S)
is an operator of Dirac type, then
⟨D˜s, t⟩L2(S) − ⟨s, D˜∗t⟩L2(S) = −⟨u · s, t⟩L2(S|∂X) for all s, t ∈ Γ(S). (4.28)
65
In particular, if s, t ∈ Γ(νMX), then
⟨D∗Ds, t⟩L2(νMX)
= ⟨Ds,Dt⟩L2(E) + ⟨Ds, u× t⟩L2(E|∂X)
= ⟨s,D∗Dt⟩L2(νMX) + ⟨Ds, u× t⟩L2(E|∂X) − ⟨u× s,Dt⟩L2(E|∂X)
= ⟨s,D∗Dt⟩L2(νMX) + ⟨∇us, t⟩L2(νMX|∂X) + ⟨P (s|∂X), t⟩L2(νMX|∂X)
− ⟨s,∇ut⟩L2(νMX|∂X) − ⟨s, P (t|∂X)⟩L2(νMX|∂X)
= ⟨D∗Dt, s⟩L2(νMX) + ⟨t,∇us⟩L2(νMX|∂X)
− ⟨∇ut+ (P − P ∗)(t|∂X), s⟩L2(νMX|∂X) .
If r ∈ Γ(νW∂X), then
g(∇us−∇πν(s|∂X)u, r) = g(∇us, r) + g(u,∇πν(s|∂X)r)
= g(∇us, r) + g(u,∇rπν(s|∂X)) = g(r,∇us)− g(∇ru, πν(s|∂X))
since g(u, [r, πν(s|∂X)]) = 0 as r, πν(s|∂X) ∈ Γ(TW |∂X) and u ∈ Γ(νMW |∂X).
Hence
⟨D∗Ds, t⟩L2(νMX) + ⟨πK(s|∂X), k⟩L2(K)
+ ⟨(dB)0(s) + 12πν((P − P ∗)(s|∂X)), r⟩L2(νW ∂X)
= ⟨D∗Ds, t⟩L2(νMX) + ⟨πK(s|∂X), k⟩L2(νMX|∂X)
+ ⟨∇us|∂X −∇πν(s|∂X)u+ P (πK(s|∂X)) + 12(P − P ∗)(s|∂X), r⟩L2(νMX|∂X)
= ⟨D∗Dt, s⟩L2(νMX) + ⟨t,∇us⟩L2(νMX|∂X)
− ⟨∇ut+ (P − P ∗)(t|∂X), s⟩L2(νMX|∂X) + ⟨k, πK(s|∂X)⟩L2(νMX|∂X)
+ ⟨r,∇us⟩L2(νMX|∂X) − ⟨∇ru, πν(s|∂X)⟩L2(νMX|∂X)
+ ⟨P ∗r, πK(s|∂X)⟩L2(νMX|∂X) − ⟨12(P − P ∗)(r), s|∂X⟩L2(νMX|∂X)
= ⟨D∗Dt, s⟩L2(νMX) + ⟨t+ r,∇us⟩L2(νMX|∂X)
− ⟨∇ut+∇ru+ (P − P ∗)(t|∂X) + 12(P − P ∗)(r), πν(s|∂X)⟩L2(νMX|∂X)
+ ⟨k −∇ut− (P − P ∗)(t|∂X) + P ∗r − 12(P − P ∗)(r), πK(s|∂X)⟩L2(νMX|∂X)
66
for all s, t ∈ Γ(νMX), k ∈ Γ(K), r ∈ Γ(νW∂X). So if (t, k, r) ∈ Γ(νMX) ⊕
Γ(K)⊕ Γ(νW∂X) is L2-orthogonal to the image of the operator
Γ(νMX)→ Γ(νMX)⊕ Γ(K)⊕ Γ(νW∂X) ,
s ↦→ (D∗Ds, πK(s|∂X), (dB)0(s) + 12πν((P − P ∗)(s|∂X))) ,
(4.29)
then ⟨D∗Dt, s⟩L2(νMX) = 0 for all s ∈ Γ(νMX) with compact support in the
interior of X. Hence D∗Dt = 0 in X. Using s ∈ Γ(νMX) with s|∂X = 0 (so
that ∇us|∂X ∈ Γ(νMX|∂X) is arbitrary), we get t|∂X + r = 0. In particular,
πK(t|∂X) = 0. Using s ∈ Γ(νMX) with πK(s|∂X) = 0, we get
πν(∇ut|∂X −∇πν(t|∂X)u+ 12(P − P ∗)(t|∂X)) = 0 .
Furthermore,
k = πK(∇ut|∂X + 12(P + P ∗)(t|∂X)) .
So the cokernel of the operator (4.29) is isomorphic to the kernel, and hence
the operator (4.29) has index 0. Since P − P ∗ is an operator of order 0 and
the index of an elliptic boundary problem depends only on the symbols of the
operators [Hör85, Theorem 20.1.8], this shows that the index of the linearisation
of the boundary problem (4.18) at the 0-section has index 0.
This finishes the proof of Proposition 4.5.
4.3 Remarks about the dimension of the
scaffold
If dimW = 3, then νW∂X has rank 0 andK = νMX|∂X . Hence the linearisation
of (4.18) at the 0-section becomes
⎧⎪⎨⎪⎩D
∗Ds = 0 in X,
s|∂X = 0 on ∂X.
(4.30)
67
If s ∈ Γ(νMX) satisfies (4.30), then
0 = ⟨D∗Ds, s⟩L2(νMX)
= ⟨Ds,Ds⟩L2(E) + ⟨(Ds)|∂X , u× (s|∂X)⟩L2(E|∂X)
= ∥Ds∥2L2(E)
by (4.28). Hence Ds = 0. So s = 0 by the unique continuation property
[BBW93, Corollary 8.3 and Remark 12.2]. This shows that X is rigid as a
Cayley submanifold of M with boundary on W and meeting W orthogonally.
If dimW = 7, then K has rank 0, and hence (4.13) becomes
⎧⎪⎨⎪⎩ F (s) = 0 in X,H(s|∂X) = 0 on ∂X. (4.31)
Suppose that W is orientable, and let φ := ∗W (Φ|W ). Then φ is a G2-structure
on W . Since g(γ, t) = t ⌟ (Φ|W ) for all t ∈ Γ(TW ), we have H(s|∂X) = 0 if and
only if γ|∂Xs = 0 if and only if (t ⌟ Φ)|∂Xs = 0 for all t ∈ Γ(TW ) if and only
if φ|∂Xs = ±vol∂Xs , where s ∈ Γ(νMX) and Xs := eˆxps(X). So s ∈ Γ(νMX)
satisfies (4.31) if and only if Xs is a Cayley submanifold of M and ∂Xs is an
associative submanifold of (W,φ).
The proof of [Gay14, Theorem 1.2] implies that associative submanifolds ofW
are rigid for a generic G2-structure. An argument like in the case dimW = 3
shows that this implies that also Cayley submanifolds of M with boundary
on W and meeting W orthogonally are rigid for a generic Spin(7)-structure
(here we use the fact that the map sending a Spin(7)-structure to its restriction
as a G2-structure on W has the property that the preimage of a residual set is
again a residual set). In the next section, we will show that this is also true for
the other possible dimensions of W , and in Section 4.5 we will see that Cayley
submanifolds are also rigid for a generic deformation of W .
68
4.4 Varying the Spin(7)-structure
In Section 3.4, we showed that for a generic Spin(7)-structure, closed Cayley
submanifolds form a smooth moduli space (see Theorems 3.10 and 3.13). Here
we prove this for compact, connected Cayley submanifolds with non-empty
boundary. Note that we use a second-order operator compared to a first-order
operator in the closed case. In the following theorem, we use the C∞-topology
for the space of all Spin(7)-structures.
Theorem 4.9 (Theorem 1.3). Let M be a smooth 8-manifold with a smooth
Spin(7)-structure Φ, let X be a smooth, compact, connected Cayley submanifold
of M with non-empty boundary, let W be a smooth submanifold of M with
∂X ⊆ W such that X and W meet orthogonally, and let 0 < α < 1.
Then for every generic smooth Spin(7)-structure Ψ that is C2,α-close to Φ
and inducing the same metric as Φ, the moduli space of all smooth Cayley
submanifolds of (M,Ψ) that are C2,α-close to X with boundary on W and
meeting W orthogonally is a finite set (possibly empty).
The statement of the above theorem remains true for the larger class of all
smooth Spin(7)-structures Ψ that are C2,α-close to Φ (not necessarily inducing
the same metric as Φ), where ‘meeting W orthogonally’ is understood with
respect to the metric induced by Ψ .
Proof. Recall from the proof of Theorem 3.10 that there are an open tubular
neighbourhood V ⊆ Λ41M ⊕Λ47M ⊕Λ435M of the 0-section and a smooth bundle
morphism Θ : V → Λ4M which parametrises Spin(7)-structures on M that are
C0-close to Φ. Let
πE : Λ2M |X → E (4.32)
denote the orthogonal projection, where the vector bundle E of rank 4 over X
is defined as in (2.23). Furthermore, let U ⊆ νMX and eˆxp be defined as in
the proof of Proposition 4.5, and let
F˜ : Γ(U)⊕ Γ(V )→ Γ(E) , (s, χ) ↦→ πE((eˆxps)∗(τΘ(χ))) , (4.33)
69
where τΘ(χ) ∈ Ω4(M,Λ2M) is defined as in (2.16) with respect to the Spin(7)-
structure Θ(χ). Then
(dF˜ )(0,0)(0, χ) = −πE(e|X) (4.34)
for e ∈ Γ(Λ27M) and χ = h(τΦ, e) by Lemma 3.12, where h is the met-
ric on Λ27M . Let D∗ : Γ(E) → Γ(νMX) be the formal adjoint of D, where
D : Γ(νMX)→ Γ(E) is defined in (3.1), and let
G˜ : Γ(U)⊕ Γ(V )→ Γ(νMX) , (s, χ) ↦→ D∗(F˜ (s, χ)) . (4.35)
Then
(dG˜)(0,0)(0, χ) = −D∗(πE(e|X)) (4.36)
for e ∈ Γ(Λ27M) and χ = h(τΦ, e) since D∗ is linear. Let U∂X ⊆ νMX|∂X be
defined as in the proof of Proposition 4.5, and let
H˜ : Γ(U∂X)⊕ Γ(V )→ Γ(νW∂X) , (s, χ) ↦→ πν((eˆxpπν(s))∗(γΘ(χ))) , (4.37)
where πν is defined in (4.10) and γΘ(χ) ∈ Ω3(W,TW ) is defined as in (4.12)
with respect to the Spin(7)-structure Θ(χ).
Lemma 4.10. Let e ∈ Γ(Λ27M), and let χ = h(τΦ, e), where h is the metric
on Λ27M (note that χ ∈ Ω47(M) by (2.17) and (2.18)). Then
(dH˜)(0,0)(0, χ) = πν(ρ−1(πE(e|∂X))) , (4.38)
where ρ is defined in (4.16).
Proof. Since χ ∈ Ω47(M), there is a path (Φt)t∈(−ε,ε) of Spin(7)-structures on M
with Φ0 = Φ and ddtΦt
⏐⏐⏐
t=0
= χ such that the metric induced by Φt is the same
metric as the metric induced by Φ for t ∈ (−ε, ε) [Kar05, Proposition 5.3.1].
Write Φt = Θ(χt) with ddtχt
⏐⏐⏐
t=0
= χ.
Let k be the dimension of W , and let (e1, . . . , e8) be a local orthonormal
frame of M such that (e1, . . . , e4) is a positive (i.e., e1 = e2 × e3 × e4) frame
70
of X with e1 = u (where u ∈ Γ(νX∂X) is the inward-pointing unit normal
vector field of ∂X in X) and (e2, . . . , ek+1) is a frame of W . Then
(dH˜)(0,0)(0, χ) =
d
dtH˜(0, χt)
⏐⏐⏐⏐⏐
t=0
= ddtπν(γΘ(χt))
⏐⏐⏐⏐⏐
t=0
= ddt
k+1∑
i=2
(ei ⌟ Φt)|∂X ⊗ πν(ei)
⏐⏐⏐⏐⏐
t=0
=
k+1∑
i=2
(ei ⌟ χ)|∂X ⊗ πν(ei)
=
k+1∑
i=5
χ(ei, e2, e3, e4)vol∂X ⊗ ei
=
k+1∑
i=5
h(τ(ei, e2, e3, e4), e)vol∂X ⊗ ei
= −
k+1∑
i=5
h(ei × (e2 × e3 × e4), e)vol∂X ⊗ ei
=
k+1∑
i=5
h(e, e1 × ei)vol∂X ⊗ ei
= πν(ρ−1(πE(e|∂X)))
by (2.16).
Let
B˜ : Γ(U)⊕ Γ(V )→ Γ(νW∂X) ,
(s, χ) ↦→ πν(ρ−1(F˜ (s, χ)|∂X)) + H˜(s|∂X , χ) .
(4.39)
Then
(dB˜)(0,0)(0, χ) = πν(ρ−1(−πE(e|∂X))) + πν(ρ−1(πE(e|∂X))) = 0 (4.40)
for e ∈ Γ(Λ27M) and χ = h(τΦ, e) by (4.34) and (4.38), where h is the metric
on Λ27M .
71
Lemma 4.11. The map
Γ(νMX)⊕ Γ(Λ41M ⊕ Λ47M ⊕ Λ435M)→ Γ(νMX)⊕ Γ(K)⊕ Γ(νW∂X) ,
(s, χ) ↦→ ((dG˜)(0,0)(s, χ), πK(s|∂X), (dB˜)(0,0)(s, χ))
(4.41)
is surjective, where the vector bundle K over ∂X is defined as in the proof of
Proposition 4.5 and πK is defined in (4.8).
Proof. We have seen that this map is given by
(s, χ) ↦→
(
D∗Ds−D∗(πE(e|X)), πK(s|∂X),
πν(∇us|∂X −∇πν(s|∂X)u+ P (πK(s|∂X)))
) (4.42)
for e ∈ Γ(Λ27M) such that χ = h(τΦ, e) (where h is the metric on Λ27M)
by (4.15), (4.21), (4.36), and (4.40) since πK is linear, where P is defined
in (4.19).
Let (t, k, r) ∈ Γ(νMX)⊕ Γ(K)⊕ Γ(νW∂X). Then there is some s ∈ Γ(νMX)
such that
πK(s|∂X) = k , πν(s|∂X) = 0 , and ∇us|∂X = r − Pk
since
Γ(νMX)→ Γ(νMX|∂X)⊕ Γ(νMX|∂X) , s ↦→ (s|∂X ,∇us|∂X)
is surjective. Furthermore, the operator D∗ : Γ(E)→ Γ(νMX) is surjective by
[BBW93, Theorem 9.1]. So there is some e ∈ Γ(Λ27M) such that
D∗(πE(e|X)) = D∗Ds− t .
Hence
(dG˜)(0,0)(s, χ) = t , πK(s|∂X) = k , and (dB˜)(0,0)(s, χ) = r
for χ = h(τΦ, e).
72
By Proposition 2.17, G˜ extends to a map
G˜2,α : C2,α(U)⊕ C2,α(V )→ C0,α(νMX) (4.43)
of class C1 since D∗ : C1,α(E)→ C0,α(νMX) is a linear first-order differential
operator, and B˜ extends to a map
B˜2,α : C2,α(U)⊕ C2,α(V )→ C1,α(νW∂X) (4.44)
of class C1. For fixed χ ∈ C2,α(V ), the equation
G˜2,α(s, χ) = D∗(πE((eˆxps)∗(τΘ(χ)))) = 0 (4.45)
is a nonlinear partial differential equation of order 2 in s, and the equation
B˜2,α(s, χ) = πν(ρ−1(πE((eˆxps)∗(τΘ(χ)))) + (eˆxpπν(s|∂X))
∗(γΘ(χ))) = 0 (4.46)
is a nonlinear partial differential equation of order 1 in s on the boundary.
Since the linearisation of
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
G˜2,α(s, χ) = 0 in X,
πK(s|∂X) = 0 on ∂X,
B˜2,α(s, χ) = 0 on ∂X
(4.47)
at 0 is elliptic for χ = 0, there is a C2,α-neighbourhood U˜1 ⊆ C2,α(U)⊕C2,α(V )
of (0, 0) such that ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
(dG˜2,α)(s,χ)(s′, 0) = 0 in X,
πK(s′|∂X) = 0 on ∂X,
(dB˜2,α)(s,χ)(s′, 0) = 0 on ∂X
73
is an elliptic boundary problem in s′ ∈ C2,α(νMX) for all (s, χ) ∈ U˜1. In
particular, the operator
((dG˜2,α)(s,χ), πK , (dB˜2,α)(s,χ))|C2,α(νMX) :
C2,α(νMX)→ C0,α(νMX)⊕ C2,α(K)⊕ C1,α(νW∂X)
(4.48)
is Fredholm for all (s, χ) ∈ U˜1, and hence its image is a closed subspace with
finite codimension. Furthermore, if (s, χ) ∈ U˜1 satisfies (4.47) and χ ∈ Ck,α(V )
(for some k ≥ 2), then s ∈ Ck+1,α(νMX) by elliptic regularity [Mor66, The-
orem 6.8.2] since τΘ(χ) ∈ Ck,α(Λ4M ⊗ Λ2M) and γΘ(χ) ∈ Ck,α(Λ3W ⊗ TW ).
The proofs of Lemmas 3.12 and 4.10 imply that if (s, χ) ∈ C2,α(U)⊕C2,α(V )
is C2,α-close to (0, 0) and satisfies πK(s|∂X) = 0, then (dB˜2,α)(s,χ)(0, χ′) = 0 for
all χ′ ∈ C2,α(Λ41M ⊕ Λ47M ⊕ Λ435M) since πν(s|∂X) = s|∂X if πK(s|∂X) = 0. So
there is a C2,α-neighbourhood U˜2 ⊆ C2,α(U) ⊕ C2,α(V ) of (0, 0) such that if
(s, χ) ∈ U˜2 satisfies (4.47) and χ ∈ Ck,α(V ) (for some k ≥ 2), then the image
of the operator
((dG˜2,α)(s,χ), πK , (dB˜2,α)(s,χ))|Ck,α(Λ41M⊕Λ47M⊕Λ435M) :
Ck,α(Λ41M ⊕ Λ47M ⊕ Λ435M)→ C0,α(νMX)⊕ C2,α(K)⊕ C1,α(νW∂X)
is exactly Ck−1,α(νMX)⊕0⊕0 since the operator D∗ : Ck,α(E)→ Ck−1,α(νMX)
is surjective by [BBW93, Theorem 9.1] and eˆxps : X → eˆxps(X) is a diffeo-
morphism, and hence the map Ck,α(Λ4Xs⊗Es)→ Ck,α(E), χ ↦→ πE((eˆxps)∗(χ))
is a linear isomorphism (note that s ∈ Ck+1,α(νMX) by above), where Es is
defined as in (2.23) with respect to the Cayley submanifold Xs := eˆxps(X)
(compare the proof of Theorem 3.10).
For k ≥ 2, let
G˜k,α := G˜2,α|C2,α(U)⊕Ck,α(V ) : C2,α(U)⊕ Ck,α(V )→ C0,α(νMX)
and
B˜k,α := B˜2,α|C2,α(U)⊕Ck,α(V ) : C2,α(U)⊕ Ck,α(V )→ C1,α(νW∂X) .
74
Then G˜k,α and B˜k,α are of class Ck−1 by Proposition 2.17. Let U˜ := U˜1 ∩ U˜2.
Then the above argumentation shows that if (s, χ) ∈ U˜ satisfies (4.47), then
((dG˜k,α)(s,χ), πK , (dB˜k,α)(s,χ)) : C2,α(νMX)⊕ Ck,α(Λ41M ⊕ Λ47M ⊕ Λ435M)
→ C0,α(νMX)⊕ C2,α(K)⊕ C1,α(νW∂X)
is surjective by Lemma 3.8 since Ck−1,α(νMX) is dense in C0,α(νMX) and the
map
(πK , (dB˜2,α)(s,χ))|C2,α(νMX) : C2,α(νMX)→ C2,α(K)⊕ C1,α(νW∂X)
is surjective for all (s, χ) ∈ C2,α(U)⊕ C2,α(V ) that are C2,α-close to (0, 0), and
((dG˜k,α)(s,χ), πK , (dB˜k,α)(s,χ))|C2,α(νMX) :
C2,α(νMX)→ C0,α(νMX)⊕ C2,α(K)⊕ C1,α(νW∂X)
is Fredholm. Furthermore, the Fredholm index is the same for all of these
operators as we may assume w.l.o.g. that U˜1 is connected. So the Fredholm
index is 0 by Lemma 4.8.
Let U˜3 ⊆ C2,α(U) and U˜4 ⊆ C2,α(V ) be open neighbourhoods of 0 such that
U˜3 × U˜4 ⊆ U˜ , let U2,α be the set of all χ ∈ U˜4 such that the operator (4.48) is
surjective for all s ∈ U˜3, and define Uk,α := U2,α ∩ Ck,α(Λ41M ⊕ Λ47M ⊕ Λ435M)
for k ≥ 2. Then Theorem 3.7 implies that Uk,α is the intersection of countably
many open dense subsets of Ck,α(Λ41M ⊕ Λ47M ⊕ Λ435M) for k ≥ 2. So if we
define U∞ :=
⋂∞
k=2 Uk,α, then U∞ is the intersection of countably many open
dense subsets of C∞(Λ41M ⊕ Λ47M ⊕ Λ435M) by Lemma 3.9.
So for every generic χ ∈ C∞(V ) that is C2,α-close to 0, the moduli space of
all solutions s ∈ C∞(U) of the boundary problem (4.47) that are C2,α-close to 0
is either empty or a smooth manifold of dimension 0 (i.e., a discrete set). This
discrete set is finite as it is compact and discrete in C2,β(νMX) for 0 < β < α by
Lemma 2.6 and since G, πK , and B are also continuous with respect to ∥ · ∥C2,β .
The result follows since the boundary problem (4.13) implies the boundary
problem (4.18) and a subset of a finite set is again a finite set.
75
Note that the proof also shows the following Ck,α-version, where we use the
Ck,α-topology for the space of all Spin(7)-structures.
Theorem 4.12. Let k ≥ 2, let 0 < α < 1, let M be an 8-manifold of
class Ck+1,α with a Spin(7)-structure Φ of class Ck,α, let X be a compact,
connected Cayley submanifold of M of class Ck+1,α with non-empty boundary,
and let W be a submanifold of M of class Ck+1,α with ∂X ⊆ W such that X
and W meet orthogonally.
Then for every generic Spin(7)-structure Ψ of class Ck,α that is C2,α-close to Φ
and inducing the same metric as Φ, the moduli space of all Cayley submanifolds
of (M,Ψ) of class Ck+1,α that are C2,α-close to X with boundary on W and
meeting W orthogonally is a finite set (possibly empty).
4.5 Varying the scaffold
In the last section, we proved that for a generic Spin(7)-structure, compact,
connected Cayley submanifolds with non-empty boundary are rigid (see The-
orems 4.9 and 4.12). Here we show that also for a generic deformation of the
scaffold (the submanifold containing the boundary of the Cayley submanifold),
compact, connected Cayley submanifolds with non-empty boundary are rigid.
In the following theorem, we use the C∞-topology for the space of all local
deformations of the scaffold W .
Theorem 4.13 (Theorem 1.4). Let M be a smooth 8-manifold with a smooth
Spin(7)-structure Φ, let X be a smooth, compact, connected Cayley submanifold
of M with non-empty boundary, let W be a smooth submanifold of M with
∂X ⊆ W such that X and W meet orthogonally, and let 0 < α < 1.
Then for every generic smooth local deformation W ′ of W that is C3,α-close
to W and with ∂X ⊆ W ′, the moduli space of all smooth Cayley submanifolds
ofM that are C2,α-close to X with boundary onW ′ and meetingW ′ orthogonally
is a finite set (possibly empty).
The statement of the above theorem remains true for the larger class of all
smooth local deformations W ′ of W that are C3,α-close to W (not necessarily
satisfying ∂X ⊆ W ′).
76
Proof. We start with the following lemma.
Lemma 4.14. There is a linear operator σ : Γ(νMW )→ Γ(TM) such that for
all t ∈ Γ(νMW ):
(i) σ(t)|W = t and
(ii) ∇r(σ(t))|W = 0 for all r ∈ Γ(νMW ).
Furthermore, σ extends to a bounded linear operator
σk,β : Ck,β(νMW )→ Ck,β(TM) (4.49)
for all k ≥ 0, 0 < β < 1.
Proof. By the Tubular Neighbourhood Theorem (Theorem 2.1), there are an
open tubular neighbourhood U˜ ⊆ νMW of the 0-section and a neighbourhood U ′
of W in M such that the exponential map exp |U˜ : U˜ → U ′ is a diffeomorphism.
For (x, v) ∈ U˜ , let
πx,v : TxM → Texpx(v)M
denote the parallel transport along the curve [0, 1]→M , t ↦→ expx(tv). Define
σ˜ : Γ(νMW )→ Γ(TM |U ′) by
σ˜(t)expx(v) := πx,v(tx)
for t ∈ Γ(νMW ) and (x, v) ∈ U˜ . Let χ : M → [0, 1] be a smooth function
such that χ ≡ 0 outside U ′ and χ ≡ 1 in some tubular neighbourhood of W
contained in U ′. For t ∈ Γ(νMW ), define
σ(t) := χ · σ˜(t) .
Then σ satisfies all the conditions.
Note that if t ∈ Γ(νMW ) has small C1-norm, then eˆxpσ(t) : M → M ,
x ↦→ eˆxpx((σ(t))x) is a diffeomorphism, where eˆxp is defined as in the proof of
Proposition 4.5. So ifW ′ = eˆxpt(W ) for some t ∈ Γ(νMW ) with small C1-norm,
then the submanifolds of M that are C1-close to X with boundary on W ′ are
of the form eˆxpσ(t)(eˆxps(X)) for some s ∈ Γ(νMX) with small C1-norm such
that s|∂X ∈ Γ(TW |∂X).
77
Let V ′ ⊆ νMW be an open tubular neighbourhood of the 0-section such that
the exponential map eˆxp|V ′ : V ′ → eˆxp(V ′) is a diffeomorphism, let U ⊆ νMX
be defined as in the proof of Proposition 4.5, and let
Fˆ : Γ(U)⊕ Γ(V ′)→ Ω4(X,E) ∼= Γ(E) ,
(s, t) ↦→ πE((eˆxps)∗((eˆxpσ(t))∗(τ))) ,
(4.50)
where the vector bundle E of rank 4 over X is defined as in (2.23), πE is defined
in (4.5), and τ ∈ Ω4(M,Λ27M) is defined as in (2.16). Then
(dFˆ )(0,0)(0, t) = D(σ(t)|X)⊥ , (4.51)
where D : Γ(νMX)→ Γ(E) is defined in (3.1) and (σ(t)|X)⊥ is the pointwise
orthogonal projection of σ(t)|X onto νMX. This follows from the proof of
Theorem 3.1 using that τ |X = 0. Let
Gˆ : Γ(U)⊕ Γ(V ′)→ Γ(νMX) , (s, t) ↦→ D∗(Fˆ (s, t)) , (4.52)
where D∗ : Γ(E)→ Γ(νMX) is the formal adjoint of D. Then
(dGˆ)(0,0)(0, t) = D∗D(σ(t)|X)⊥ (4.53)
since D∗ is linear.
Similarly to Lemma 4.14, there is also a linear operator
σˆ : Γ(νW∂X)→ Γ(TW ) (4.54)
such that σˆ(r)|∂X = r for all r ∈ Γ(νW∂X). Furthermore, σˆ extends to
a bounded linear operator σˆk,β : Ck,β(νW∂X) → Ck,β(TW ) for all k ≥ 0,
0 < β < 1. Note that if r ∈ Γ(νW∂X) has small C1-norm, then eˆxpσˆ(r) : W →W ,
x ↦→ eˆxpx((σˆ(r))x) is a diffeomorphism.
78
Define γˆ ∈ Ω3(M,TM) and µ ∈ Ω1(M,TM) by
γˆ(a, b, c) := a× b× c and µ(a) := a (4.55)
for a, b, c ∈ Γ(TM), let U∂X ⊆ νMX|∂X be defined as in the proof of Proposi-
tion 4.5, let
Hˆ1 : Γ(U∂X)⊕ Γ(V ′)→ Ω3(∂X, TM |∂X) ∼= Γ(TM |∂X) ,
(s, t) ↦→ (eˆxpπν(s))∗((eˆxpσ(t))∗(γˆ)) ,
(4.56)
where πν is defined in (4.10), let
Hˆ2 : Γ(U∂X)⊕ Γ(V ′)→ Ω1(W,TM |W ) ,
(s, t) ↦→ (eˆxpσˆ(πν(s)))∗((eˆxpσ(t))∗(µ)) ,
(4.57)
and let
Hˆ : Γ(U∂X)⊕ Γ(V ′)→ Γ(νW∂X) ,
(s, t) ↦→ πν((g(Hˆ1(s, t), Hˆ2(s, t)))♯) .
(4.58)
Here we view Hˆ2(s, t) as an element of Γ(T ∗W |∂X ⊗ TM |∂X) and use the inner
product with Hˆ1(s, t) on the TM |∂X-factor to get an element in Γ(T ∗W |∂X),
which we identify with an element of Γ(TW |∂X) using the metric isomorphism ♯.
Then
(dHˆ)(0,0)(s, 0) = (dH)0(s) . (4.59)
This follows similarly to the proofs of Lemmas 4.6 and 4.15 below. Let
Bˆ : Γ(U)⊕ Γ(V ′)→ Γ(νW∂X) ,
(s, t) ↦→ πν(ρ−1(Fˆ (s, t)|∂X)) + Hˆ(s|∂X , t) .
(4.60)
Lemma 4.15. Let t ∈ Γ(νMW ). Then
(dBˆ)(0,0)(0, t) = πν((g(∇t, u))♯) , (4.61)
79
where u ∈ Γ(νX∂X) is the inward-pointing unit normal vector field of ∂X in X.
Here we view ∇t as an element of Γ(T ∗W |∂X ⊗ TM |∂X) and use the inner
product with u on the TM |∂X-factor to get an element in Γ(T ∗W |∂X), which
we identify with an element of Γ(TW |∂X) using the metric isomorphism ♯.
Proof. Let k be the dimension of W , let (e1, . . . , e8) be a local orthonormal
frame of M such that (e1, . . . , e4) is a positive (i.e., e1 = e2 × e3 × e4) frame
of X with e1 = u and (e2, . . . , ek+1) is a frame of W , and let (e1, . . . , e8) be the
dual coframe. Then
γˆ =
8∑
i=1
(ei ⌟ Φ)⊗ ei and µ =
8∑
i=1
ei ⊗ ei . (4.62)
Now
(dHˆ)(0,0)(0, t)
= πν((g((dHˆ1)(0,0)(0, t), Hˆ2(0, 0)))♯) + πν((g(Hˆ1(0, 0), (dHˆ2)(0,0)(0, t)))♯)
= πν((dHˆ1)(0,0)(0, t)) + πν((g((dHˆ2)(0,0)(0, t), u))♯)
since (g(v, Hˆ2(0, 0)))♯ = v for v ∈ Γ(TM |W ) and Hˆ1(0, 0) = u as (v ⌟ Φ)|∂X =
g(v, u)vol∂X for v ∈ Γ(TM). We have
(dHˆ1)(0,0)(0, t) =
d
dr Hˆ1(0, rt)
⏐⏐⏐⏐⏐
r=0
= ddr (eˆxpσ(rt))
∗(γˆ)
⏐⏐⏐⏐⏐
r=0
= (Lσ(t)γˆ)|∂X
=
8∑
i=1
(Lσ(t)(ei ⌟ Φ))|∂X ⊗ ei +
8∑
i=1
(ei ⌟ Φ)|∂X ⊗∇σ(t)ei
= vol∂X ⊗
⎛⎝ 8∑
i=1
(Lσ(t)(ei ⌟ Φ))(e2, e3, e4)ei +∇σ(t)e1
⎞⎠
since Φ(ei, e2, e3, e4) = g(ei, e2×e3×e4) = g(ei, e1) for i = 1, . . . , 8. So similarly
to the proof of Lemma 4.6, we get
πν((dHˆ1)(0,0)(0, t))
= −πν(P (σ(t)|∂X)⊥) +
k+1∑
i=5
g(∇σ(t)ei, e1)ei +∇σ(t)e1
80
= −πν(ρ−1((D(σ(t)|X)⊥)|∂X)) +
k+1∑
i=5
(g(∇σ(t)ei, e1) + g(ei,∇σ(t)e1))ei
= −(dFˆ )(0,0)(0, t)|∂X
since (∇e1σ(t))|∂X = 0 by condition (ii) in Lemma 4.14. Further,
(dHˆ2)(0,0)(0, t) =
d
dr Hˆ2(0, rt)
⏐⏐⏐⏐⏐
r=0
= ddr (eˆxpσ(t))
∗(µ)
⏐⏐⏐⏐⏐
r=0
= Lσ(t)µ
=
8∑
i=1
Lσ(t)ei ⊗ ei +
8∑
i=1
ei ⊗∇σ(t)ei
= −
8∑
i,j=1
g(Lσ(t)ei, ej)ei ⊗ ej +
8∑
i,j=1
g(∇σ(t)ei, ej)ei ⊗ ej
=
8∑
i,j=1
g(∇eiσ(t), ej)ei ⊗ ej
=
8∑
i=1
ei ⊗∇eiσ(t)
= ∇σ(t) .
So together we get
(dBˆ)(0,0)(0, t) = (dHˆ)(0,0)(0, t) + (dFˆ )(0,0)(0, t) = πν((g(∇t, u))♯) .
Lemma 4.16. The map
Γ(νMX)⊕ Γ(νMW )→ Γ(νMX)⊕ Γ(K)⊕ Γ(νW∂X) ,
(s, t) ↦→ ((dGˆ)(0,0)(s, t), πK(s|∂X), (dBˆ)(0,0)(s, t))
(4.63)
is surjective, where the vector bundle K over ∂X is defined as in the proof of
Proposition 4.5 and πK is defined in (4.8).
Proof. We have seen that this map is given by
(s, t) ↦→ (D∗Ds+D∗D(σ(t)|X)⊥,πK(s|∂X),(dB)0(s|∂X)+πν((g(∇t,u))♯)) (4.64)
by (4.15), (4.53), (4.59), and (4.61) since πK is linear.
81
Let (t′, k, r) ∈ Γ(νMX)⊕Γ(K)⊕Γ(νW∂X). Then there is some s′ ∈ Γ(νMX)
such that
D∗Ds′ = t′ and πK(s′|∂X) = k
since
Γ(νMX)→ Γ(νMX)⊕ Γ(νMX|∂X) , s ↦→ (D∗Ds, s|∂X)
is surjective as the boundary problem⎧⎪⎨⎪⎩D
∗Ds = 0 in X,
s|∂X = 0 on ∂X
has index 0 (compare the proof of Lemma 4.8) and 0-dimensional kernel by
the unique continuation property [BBW93, Corollary 8.3 and Remark 12.2]
(compare the discussion of dimW = 3 in Section 4.3). Furthermore, there is
some t ∈ Γ(νMW ) such that
t|∂X = 0 and πν((g(∇t, u))♯) = r − (dB)0(s′|∂X)
since
Γ(νMW )→ Γ(νMW |∂X)⊕ Γ(νW∂X) , t ↦→ (t|∂X , πν((g(∇t, u))♯))
is surjective. Note that
(dB)0((σ(t)|X)⊥) = 0
since (dB)0 is a first-order linear operator and
(i) σ(t)|∂X = t|∂X = 0 and
(ii) (∇u(σ(t)|X)⊥)⊥|∂X = (∇u(σ(t)|X))⊥|∂X = 0
by Lemma 4.14 (for (ii), note that (∇u(σ(t)|X)∥)⊥|∂X depends only on (σ(t)|X)∥|∂X ,
which is 0 by (i)). So if s ∈ Γ(νMX) is defined by
s := s′ − (σ(t)|X)⊥ ,
then
(dGˆ)(0,0)(s, t) = t′ , πK(s|∂X) = k , and (dBˆ)(0,0)(s, t) = r .
82
By Proposition 2.17, Gˆ extends to a map
Gˆ3,α : C2,α(U)⊕ C3,α(V ′)→ C0,α(νMX) (4.65)
of class C1 since D∗ : C1,α(E)→ C0,α(νMX) is a linear first-order differential
operator, and Bˆ extends to a map
Bˆ3,α : C2,α(U)⊕ C3,α(V ′)→ C1,α(νW∂X) (4.66)
of class C1. For fixed t ∈ C3,α(V ′), the equation
Gˆ3,α(s, t) = D∗(πE((eˆxps)∗((eˆxpσ(t))∗(τ)))) = 0 (4.67)
is a nonlinear partial differential equation of order 2 in s, and the equation
Bˆ3,α(s, t)
= πν(ρ−1(πE((eˆxps)∗((eˆxpσ(t))∗(τ)))))
+ πν((g((eˆxpπν(s))
∗((eˆxpσ(t))∗(γˆ)), (eˆxpσˆ(πν(s)))
∗((eˆxpσ(t))∗(µ))))♯)
= 0
(4.68)
is a nonlinear partial differential equation of order 1 in s on the boundary.
Since the linearisation of ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
Gˆ3,α(s, t) = 0 in X,
πK(s|∂X) = 0 on ∂X,
Bˆ3,α(s, t) = 0 on ∂X
(4.69)
at 0 is elliptic for t = 0, there is a (C2,α⊕C3,α)-neighbourhood U˜1 ⊆ C2,α(U)⊕
C3,α(V ′) of (0, 0) such that
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
(dGˆ3,α)(s,t)(s′, 0) = 0 in X,
πK(s′|∂X) = 0 on ∂X,
(dBˆ3,α)(s,t)(s′, 0) = 0 on ∂X
83
is an elliptic boundary problem in s′ ∈ C2,α(νMX) for all (s, t) ∈ U˜1. In
particular, the operator
((dGˆ3,α)(s,t), πK , (dBˆ3,α)(s,t))|C2,α(νMX) :
C2,α(νMX)→ C0,α(νMX)⊕ C2,α(K)⊕ C1,α(νW∂X)
(4.70)
is Fredholm for all (s, t) ∈ U˜1, and hence its image is a closed subspace with finite
codimension. Furthermore, if (s, t) ∈ U˜1 satisfies (4.69) and t ∈ Ck,α(V ′) (for
some k ≥ 3), then s ∈ Ck,α(νMX) by elliptic regularity [Mor66, Theorem 6.8.2]
since (eˆxpσ(t))∗(τ) ∈ Ck−1,α(Λ4M ⊗Λ27M), (eˆxpσ(t))∗(γˆ) ∈ Ck−1,α(Λ3M ⊗ TM),
and (eˆxpσ(t))∗(µ) ∈ Ck−1,α(T ∗M ⊗ TM).
The proof of Theorem 3.1 implies that if (s, t) ∈ C2,α(U) ⊕ C3,α(V ′) is
(C2,α ⊕ C3,α)-close to (0, 0) and satisfies (4.69), then
(dGˆ3,α)(s,t)(0, t′) = (dGˆ3,α)(s,t)((((d exp)s)−1(σ(t′))|X)⊥, 0)
since τ |Xs,t = 0, where Xs,t := eˆxpσ(t)(eˆxps(X)). So there is a (C2,α ⊕
C3,α)-neighbourhood U˜2 ⊆ C2,α(U)⊕ C3,α(V ′) of (0, 0) such that if (s, χ) ∈ U˜2
satisfies (4.69) and t ∈ Ck,α(V ′) (for some k ≥ 3), then the image of the
operator
(dGˆ3,α)(s,t)|Tk,α : Tk,α → C0,α(νMX)
is contained in the image of the operator
(dGˆ3,α)(s,t)|S2,α : S2,α → C0,α(νMX) ,
where
S2,α := {s′ ∈ C2,α(νMX) : (dBˆ3,α)(s,t)(s′) = 0} ,
Tk,α := {t′ ∈ Ck,α(νMW ) : t′|∂Xs = 0} ,
and Xs := eˆxps(X), since
(dBˆ3,α)(s,t)((((d exp)s)−1(σ(t′))|X)⊥, 0) = 0
if t′|∂Xs = 0 (compare the proof of Lemma 4.16).
84
Furthermore, there is a (C2,α⊕C3,α)-neighbourhood U˜3 ⊆ C2,α(U)⊕C3,α(V ′)
of (0, 0) such that if (s, χ) ∈ U˜3 satisfies (4.69) and t ∈ Ck,α(V ′) (for some
k ≥ 3), then
((dGˆ3,α)(s,t), πK)|C2,α(νMX) : C2,α(νMX)→ C0,α(νMX)⊕ C2,α(K)
is surjective, and the image of the operator
(dBˆk,α)(s,t)|Tk,α : Tk,α → C1,α(νW∂X)
is a dense subspace of C1,α(νW∂X) (compare the proof of Lemma 4.16).
For k ≥ 3, let
Gˆk,α := Gˆ3,α|C2,α(U)⊕Ck,α(V ′) : C2,α(U)⊕ Ck,α(V ′)→ C0,α(νMX)
and
Bˆk,α := Bˆ3,α|C2,α(U)⊕Ck,α(V ′) : C2,α(U)⊕ Ck,α(V ′)→ C1,α(νW∂X) .
Then Gˆk,α and Bˆk,α are of class Ck−2 by Proposition 2.17. Let U˜ := U˜1∩U˜2∩U˜3.
Then the above argumentation shows that if (s, t) ∈ U˜ satisfies (4.69), then
((dGˆk,α)(s,t), πK , (dBˆk,α)(s,t)) : C2,α(νMX)⊕ Ck,α(νMW )
→ C0,α(νMX)⊕ C2,α(K)⊕ C1,α(νW∂X)
is surjective by Lemma 3.8 (compare the proof of Lemma 4.16), and
((dGˆk,α)(s,t), πK , (dBˆk,α)(s,t))|C2,α(νMX) :
C2,α(νMX)→ C0,α(νMX)⊕ C2,α(K)⊕ C1,α(νW∂X)
is Fredholm. Furthermore, the Fredholm index is the same for all of these
operators as we may assume w.l.o.g. that U˜1 is connected. So the Fredholm
index is 0 by Lemma 4.8.
85
Let U˜3 ⊆ C2,α(U) and U˜4 ⊆ C3,α(V ′) be open neighbourhoods of 0 such that
U˜3 × U˜4 ⊆ U˜ , let U3,α be the set of all t ∈ U˜4 such that the operator (4.70)
is surjective for all s ∈ U˜3, and define Uk,α := U3,α ∩ Ck,α(νMW ) for k ≥ 3.
Then Theorem 3.7 implies that Uk,α is the intersection of countably many open
dense subsets of Ck,α(νMW ) for k ≥ 3. So if we define U∞ := ⋂∞k=3 Uk,α, then
U∞ is the intersection of countably many open dense subsets of C∞(νMW ) by
Lemma 3.9.
So for every generic t ∈ C∞(V ′) that is C3,α-close to 0, the moduli space of
all solutions s ∈ C∞(U) of the boundary problem (4.69) that are C2,α-close to 0
is either empty or a smooth manifold of dimension 0 (i.e., a discrete set). This
discrete set is finite as it is compact and discrete in C2,β(νMX) for 0 < β < α by
Lemma 2.6 and since G, πK , and B are also continuous with respect to ∥ · ∥C2,β .
The result follows since the boundary problem (4.13) implies the boundary
problem (4.18) and a subset of a finite set is again a finite set.
Note that the proof also shows the following Ck,α-version, where we use the
Ck,α-topology for the space of local deformations of W .
Theorem 4.17. Let k ≥ 3, let 0 < α < 1, let M be an 8-manifold of class Ck,α
with a Spin(7)-structure Φ of class Ck,α, let X be a compact, connected Cayley
submanifold of M of class Ck,α with non-empty boundary, and let W be a
submanifold of M of class Ck,α with ∂X ⊆ W such that X and W meet
orthogonally.
Then for every generic local deformation W ′ of W of class Ck,α that is
C3,α-close to W and with ∂X ⊆ W ′, the moduli space of all Cayley submanifolds
ofM of class Ck,α that are C2,α-close to X with boundary onW ′ and meetingW ′
orthogonally is a finite set (possibly empty).
4.6 Remark about torsion-free Spin(7)-structures
In contrast to the case of closed Cayley submanifolds (see Section 3.5), we get
the following genericity statement for torsion-free Spin(7)-structures (rather
86
than Spin(7)-structures that are not necessarily torsion-free in Theorem 4.9) in
the case of a compact, connected Cayley submanifold with non-empty boundary.
The main point is the following: If M is an 8-manifold with a torsion-free
Spin(7)-structure Φ and φ : M →M is a diffeomorphism, then φ∗(Φ) is again a
torsion-free Spin(7)-structure. But we may use diffeomorphisms that keep the
Cayley submanifold X fixed and vary the scaffold W (subject to the condition
∂X ⊆ W ), and we have already seen that for a generic deformation of W , the
moduli space is a finite set (Theorem 4.13). In the following theorem, we use
the C∞-topology for the space of all torsion-free Spin(7)-structures.
Theorem 4.18. Let M be a smooth 8-manifold with a smooth, torsion-free
Spin(7)-structure Φ, let X be a smooth, compact, connected Cayley submanifold
of M with non-empty boundary, let W be a smooth submanifold of M with
∂X ⊆ W such that X and W meet orthogonally, and let 0 < α < 1.
Then for every generic smooth, torsion-free Spin(7)-structure Ψ that is
C2,α-close to Φ, the moduli space of all smooth Cayley submanifolds of (M,Ψ)
that are C2,α-close to X with boundary on W and meeting W orthogonally
(with respect to the metric induced by Ψ) is a finite set (possibly empty).
Proof. The following lemma replaces Lemma 4.10 and Equation (4.40) in the
proof of Theorem 4.9.
Lemma 4.19. Let M be an 8-manifold with a Spin(7)-structure Φ, let X be a
compact Cayley submanifold with boundary, and let v ∈ Γ(TM). Then
(dB˜)(0,0)(0,LvΦ) = πν(∇uv|∂X) + πν((g(∇(v|W ), u))♯) , (4.71)
where πν is defined in (4.10) and u ∈ Γ(νX∂X) is the inward-pointing unit
normal vector field of ∂X in X. Here we view ∇(v|W ) as an element of
Γ(T ∗W |∂X ⊗TM |∂X) and use the inner product with u on the TM |∂X-factor to
get an element in Γ(T ∗W |∂X), which we identify with an element of Γ(TW |∂X)
using the metric isomorphism ♯.
87
Proof. In the proof of Lemma 4.10, if we let the orthonormal frame (e1(t), . . . , e8(t))
depend on t such that (e1(t), . . . , e4(t)) is a frame of X, (e2(t), . . . , e4(t)) is a
frame of ∂X, and (e2(t), . . . , ek+1(t)) is a frame of W , then
(dH˜)(0,0)(0, χ) =
d
dt
k+1∑
i=2
(ei(t) ⌟ Φt)|∂X ⊗ πν(ei(t))
⏐⏐⏐⏐⏐
t=0
=
k+1∑
i=2
((ei)′ ⌟ Φ)|∂X ⊗ πν(ei) +
k+1∑
i=2
(ei ⌟ χ)|∂X ⊗ πν(ei)
+
k+1∑
i=2
(ei ⌟ Φ)|∂X ⊗ ddtπν(ei(t))
⏐⏐⏐⏐⏐
t=0
=
k+1∑
i=5
(ei ⌟ χ)|∂X ⊗ ei
since
((ei)′ ⌟ Φ)|∂X = g((ei)′, e1)vol∂X = 0 and
(ei ⌟ Φ)|∂X = g(ei, e1)vol∂X = 0
for i = 5, . . . , k + 1 as (ei)′ is tangent to W because ei(t) is tangent to W for
all t. So
(dH˜)(0,0)(0,LvΦ) =
k+1∑
i=5
(ei ⌟ LvΦ)|∂X ⊗ ei
=
k+1∑
i=5
(Lv(ei ⌟ Φ)− Lvei ⌟ Φ)|∂X ⊗ ei .
The proof of Lemma 4.6 shows that
k+1∑
i=5
(Lv(ei ⌟ Φ))|∂X ⊗ ei = −πν(ρ−1((D(v|X)⊥)|∂X)) + πν(∇uv|∂X)
+
k+1∑
i=5
g(∇vei, e1)vol∂X ⊗ ei .
88
Hence
(dB˜)(0,0)(0,LvΦ) = πν(ρ−1(((dF˜ )(0,0)(0,LvΦ))|∂X)) + (dH˜)(0,0)(0,LvΦ)
= πν(∇uv|∂X) +
k+1∑
i=5
(g(∇vei, e1)− g(Lvei, e1))vol∂X ⊗ ei
= πν(∇uv|∂X) +
k+1∑
i=5
g(∇eiv, e1)vol∂X ⊗ ei
= πν(∇uv|∂X) + πν((g(∇(v|W ), u))♯)
by (3.32).
So if X denotes the space of all torsion-free Spin(7)-structures on M , then
Γ(νMX)⊕ Γ(TΦX )→ Γ(νMX)⊕ Γ(K)⊕ Γ(νW∂X)
(s, χ) ↦→ ((dG˜)(0,0)(s, χ), πK(s|∂X), (dB˜)(0,0)(s, χ))
is surjective by the proof of Lemma 4.16, and hence if (s, χ) ∈ C2,α(U)⊕C2,α(V )
is C2,α-close to (0, 0), satisfies (4.47), and χ ∈ Ck,α(V ) (for some k ≥ 2), then
C2,α(νMX)⊕Ck,α(TΘ(χ)X )→ C0,α(νMX)⊕C2,α(K)⊕C1,α(νW∂X) ,
(s′,χ′) ↦→ ((dG˜k,α)(s,χ)(s′,χ′),πK(s′|∂X),(dB˜k,α)(s,χ)(s′,χ′))
is surjective by similar arguments as in the proof of Theorem 4.13. The rest of
the proof is analogous to the proof of Theorem 4.9.
Remark 4.20. A similar result holds in the case of associative submanifolds inside
a G2-manifold (this can be proved by the methods of [Gay14] although Gayet
did not prove it): Let M be a 7-manifold with a torsion-free G2-structure φ,
let X be a compact, connected associative submanifold of M with non-empty
boundary, and let W be a coassociative submanifold of M with ∂X ⊆ W .
Then for every generic torsion-free G2-structure ψ that is close to φ, the
moduli space of all associative submanifolds of (M,ψ) that are close to X with
boundary on W is either empty or a smooth manifold of dimension equal to
the index.
89
4.7 Non-ellipticity of boundary conditions for
first-order boundary problem
For the deformations of compact Cayley submanifolds with boundary, we
considered a second-order boundary problem instead of a first-order boundary
problem. The reason is that the operator D from (3.1) has no local elliptic
boundary conditions. In this section, we prove this fact in the case when the
boundary operator is given by an operator of order 0.
Proposition 4.21. Let M be an 8-manifold with a Spin(7)-structure Φ, and
let X be a compact Cayley submanifold of M with non-empty boundary. Define
the vector bundle E of rank 4 over X and the operator D : Γ(νMX)→ Γ(E) as
in (2.23) and (3.1), respectively. Furthermore, let V be a vector bundle over ∂X,
and let B′ : Γ(νMX)→ Γ(V ) be a (real) linear differential operator of order 0.
Then the boundary problem⎧⎪⎨⎪⎩Ds = 0 in X,B′s = 0 on ∂X (4.72)
is not elliptic.
Note that the operator D is an elliptic operator. So the failure of ellipticity
lies in the boundary condition.
Proof. Let (e1, . . . , e8) be a local Spin(7)-frame such that (e1, . . . , e4) is a
positive orthonormal frame of X. Then (e5, . . . , e8) is an orthonormal frame
of νMX, and (e1×e5, . . . , e1×e8) is an orthonormal frame of E. By (3.1), we have
Ds = e1×∇⊥e1s+· · ·+e4×∇⊥e4s+A(s), where A : Γ(νMX)→ Γ(E) is an operator
of order 0. Here ∇⊥ is the induced connection on νMX. Using (2.20), the
symbol σD(x, ξ) : (νMX)x → Ex for x ∈ X, ξ ∈ T ∗xX is given in these frames by⎛⎜⎜⎜⎜⎜⎜⎝
ξ1 ξ2 ξ3 ξ4
−ξ2 ξ1 −ξ4 ξ3
−ξ3 ξ4 ξ1 −ξ2
−ξ4 −ξ3 ξ2 ξ1
⎞⎟⎟⎟⎟⎟⎟⎠ ,
where ξ = ξ1e1+ · · ·+ ξ4e4. Here (e1, . . . , e4) is the dual co-frame of (e1, . . . , e4).
90
So the system σD(x, ξ − i∂te1)f(t) = 0 (w.l.o.g. we can assume that e1 is
the inward-pointing unit normal vector field) of linear ordinary differential
equations is ⎛⎜⎜⎜⎜⎜⎜⎝
−i∂t ξ2 ξ3 ξ4
−ξ2 −i∂t −ξ4 ξ3
−ξ3 ξ4 −i∂t −ξ2
−ξ4 −ξ3 ξ2 −i∂t
⎞⎟⎟⎟⎟⎟⎟⎠
⎛⎜⎜⎜⎜⎜⎜⎝
f1
f2
f3
f4
⎞⎟⎟⎟⎟⎟⎟⎠ = 0 .
So we get ⎛⎜⎜⎜⎜⎜⎜⎝
f ′1
f ′2
f ′3
f ′4
⎞⎟⎟⎟⎟⎟⎟⎠ =
⎛⎜⎜⎜⎜⎜⎜⎝
0 −iξ2 −iξ3 −iξ4
iξ2 0 iξ4 −iξ3
iξ3 −iξ4 0 iξ2
iξ4 iξ3 −iξ2 0
⎞⎟⎟⎟⎟⎟⎟⎠
⎛⎜⎜⎜⎜⎜⎜⎝
f1
f2
f3
f4
⎞⎟⎟⎟⎟⎟⎟⎠ .
This matrix has eigenvalues ±µ, where
µ :=
√
ξ22 + ξ23 + ξ24 .
The eigenspace of the negative eigenvalue −µ is spanned by⎛⎜⎜⎜⎜⎜⎜⎝
iµ
ξ2
ξ3
ξ4
⎞⎟⎟⎟⎟⎟⎟⎠ and
⎛⎜⎜⎜⎜⎜⎜⎝
−ξ2
iµ
−ξ4
ξ3
⎞⎟⎟⎟⎟⎟⎟⎠
if (ξ3, ξ4) ̸= (0, 0), and it is spanned by⎛⎜⎜⎜⎜⎜⎜⎝
iµ
ξ2
0
0
⎞⎟⎟⎟⎟⎟⎟⎠ and
⎛⎜⎜⎜⎜⎜⎜⎝
0
0
ξ2
iµ
⎞⎟⎟⎟⎟⎟⎟⎠
if (ξ3, ξ4) = (0, 0). So M+x,ξ is 2-dimensional.
91
Now assume that the boundary problem (4.72) is elliptic. Then M+x,ξ → Vx,
f ↦→ σB′(x, ξ − i∂te1)f(0) would be bijective. In particular, V needs to be
2-dimensional. So let
B′ =
⎛⎝b11 b12 b13 b14
b21 b22 b23 b24
⎞⎠ .
Then the symbol σB′(x, ξ) is given by the same linear map. Let
x := b11b22 − b12b21 − b13b24 + b14b23 ,
y := b11b23 + b12b24 − b13b21 − b14b22 ,
z := b11b24 − b12b23 + b13b22 − b14b21 .
We want to show that the system
⎧⎨⎩ b11(αiµ− βξ2) + b12(αξ2 + βiµ) + b13(αξ3 − βξ4) + b14(αξ4 + βξ3) = 0b21(αiµ− βξ2) + b22(αξ2 + βiµ) + b23(αξ3 − βξ4) + b24(αξ4 + βξ3) = 0
has a solution (α, β) ̸= (0, 0) for some choice of (ξ2, ξ3, ξ4) with (ξ3, ξ4) ̸= (0, 0),
or that the system
⎧⎨⎩ b11αiµ+ b12αξ2 + b13βξ2 + b14βiµ = 0b21αiµ+ b22αξ2 + b23βξ2 + b24βiµ = 0
has a solution (α, β) ̸= (0, 0) for some choice of ξ2 ̸= 0.
The existence of a non-zero solution is equivalent to the vanishing of the
determinant. A calculation shows that the determinant of the first system is
−(ξ23 + ξ24)x+ (ξ2ξ3 − iξ4µ)y + (ξ2ξ4 + iξ3µ)z ,
and the determinant of the second system is
iξ2µy − ξ22z .
92
So if (x, y, z) = (0, 0, 0), then the determinants will always vanish, and we can
take any (ξ2, ξ3, ξ4) ̸= (0, 0, 0). Otherwise we can choose (ξ2, ξ3, ξ4) = (x, y, z).
Then the first determinant will always vanish, and the second determinant will
vanish if (y, z) = (0, 0). This shows that one of the systems above yields a
non-trivial element of the eigenspace in the kernel of B′, which contradicts the
elliptic boundary condition.
93

5 Examples of compact Cayley
submanifolds with boundary
In this chapter, we present and discuss some examples for the deformation theory
of compact Cayley submanifolds with boundary. We first show, in Section 5.1,
versions of the volume minimising property for this class of submanifolds. In
Section 5.2, we discuss examples with a smooth k-dimensional moduli space
(for 0 ≤ k ≤ 4) inside a manifold with holonomy SU(2) × SU(2). Then in
Section 5.3, we construct a rigid compact Cayley submanifold with boundary
inside the complete Riemannian 8-manifold with holonomy Spin(7) constructed
by Bryant and Salamon in [BS89]. We further relate the deformation theory of
compact Cayley submanifolds with boundary to the deformation theories of
compact special Lagrangian, coassociative, and associative submanifolds with
boundary in Sections 5.4–5.6. Throughout this section, any Spin(7)-structure
will be assumed to be torsion-free.
5.1 Volume minimising property
Harvey and Lawson [HL82] showed that closed calibrated submanifolds are
volume-minimising in their homology class (Proposition 2.8). This applies, in
particular, to Cayley submanifolds of a Spin(7)-manifold. Note that the condi-
tion that the Spin(7)-structure is torsion-free is crucial for this. Gayet [Gay14]
proved that a compact associative submanifold of a G2-manifold with boundary
in a coassociative submanifold is volume-minimising in its relative homology
class. Here we show versions of the volume minimising property for compact
95
Cayley submanifolds with boundary. There are different versions for different
dimensions of the scaffold W : Proposition 5.1 for dimW = 4, Proposition 5.4
for dimW = 5, and Proposition 5.8 for dimW = 6. The proof for dimW = 4
is completely analogous to [Gay14], but we need additional arguments for
dimW ≥ 5 (see Remark 5.2).
Proposition 5.1. Let M be an 8-manifold with a torsion-free Spin(7)-struc-
ture Φ, let X be a compact Cayley submanifold of M with boundary, let W be
a 4-dimensional submanifold of M with ∂X ⊆ W such that Φ|W = 0 (which
implies that X and W meet orthogonally), and let Y be a compact, oriented
4-dimensional submanifold of M with ∂Y ⊆ W which lies in the same relative
homology class as X in H4(M,W ).
Then the volume of Y is greater than or equal to the volume of X, and
equality holds if and only if Y is Cayley.
Remark 5.2. The condition Φ|W = 0 forcesW to have dimension at most 4 since
every 5-dimensional subspace of (R8, Φ0) contains a (unique) Cayley subspace.
Proof. Since X and Y lie in the same relative homology class, there are a
5-chain S in M and a 4-chain Z in W such that X − Y = ∂S + Z. So∫
X
Φ−
∫
Y
Φ =
∫
∂S
Φ+
∫
Z
Φ =
∫
S
dΦ+
∫
Z
Φ = 0
by Stokes’ Theorem since dΦ = 0 as Φ is torsion-free and Φ|Z = 0 as Φ|W = 0
and Z ⊆ W . Hence
vol(X) =
∫
X
volX =
∫
X
Φ =
∫
Y
Φ ≤
∫
Y
volY = vol(Y )
since X is Cayley, and equality holds if and only if Φ|Y = volY , that is, if and
only if Y is Cayley.
As a preparation for dimension 5, we have the following lemma.
Lemma 5.3. Let M be an 8-manifold with a Spin(7)-structure Φ, let X be a
Cayley submanifold of M with boundary, let W be an oriented 5-dimensional
submanifold of M with ∂X ⊆ W , and let n := (∗W (Φ|W ))♯. Then X and W
meet orthogonally if and only if n|∂X ∈ Γ(T∂X).
96
Proof. Let u ∈ Γ(νX∂X) be the inward-pointing unit normal vector field of ∂X
inX, let x ∈ ∂X, let a, b ∈ Tx∂X be orthonormal and orthogonal to nx, let V be
the subspace of TxM consisting of all vectors that are orthogonal to a and b, and
let Z be the subspace of V ∩ TxW consisting of all vectors that are orthogonal
to nx (note that Z is 2-dimensional). Then V → V , v ↦→ a× b×v is orthogonal.
Furthermore, Z is invariant under this map since the orthogonal complement
of nx in TxW is a Cayley subspace. Hence ux ∈ Z⊥ if and only if a×b×ux ∈ Z⊥.
Now suppose that ux is orthogonal to TxW . Then ux ∈ Z⊥ since Z ⊆ TxW .
So a× b× ux ∈ Z⊥. But a× b× ux ∈ Tx∂X ⊆ TxW since X is Cayley. Hence
a× b× ux = ±nx since a× b× ux ⊥ a, b. So nx ∈ Tx∂X.
Conversely, suppose that nx ∈ Tx∂X. Then nx = ±a × b × ux since X is
Cayley. So ux ∈ Z⊥ since nx ∈ Z⊥. But also ux ⊥ nx since ux = ∓a× b× nx.
Hence ux is orthogonal to TxW since ux ⊥ a, b.
Proposition 5.4. Let M be an 8-manifold with a torsion-free Spin(7)-struc-
ture Φ, let X be a compact Cayley submanifold of M with boundary, let W
be an oriented 5-dimensional submanifold of M with ∂X ⊆ W such that X
and W meet orthogonally, and let n := (∗W (Φ|W ))♯ (note that n|∂X ∈ Γ(T∂X)
by Lemma 5.3). Suppose that H4(W ) = 0 and that n is parallel with respect to
the Levi-Civita connection of W . Furthermore, let Y be a compact, oriented
4-dimensional submanifold of M with ∂Y ⊆ W which lies in the same relative
homology class as X in H4(M,W ) such that ∂Y is a local deformation of ∂X
with n|∂Y ∈ Γ(T∂Y ).
Then the volume of Y is greater than or equal to the volume of X, and
equality holds if and only if Y is Cayley.
Later, we will apply this proposition and Proposition 5.8 below in the
situation that Y is a local deformation of X. In that case, we can restrict W
to an open tubular neighbourhood W ′ of ∂X in W . Then H4(W ′) = 0 as ∂X
is a deformation retract of W ′ and ∂X is 3-dimensional.
Remark 5.5. The orientability of W is not necessary, but the vector field n may
not be well-defined if W is not orientable. However, all conditions involving n
are essentially local, and locally W is orientable.
97
Proof. Since X and Y lie in the same relative homology class, there are a
5-chain S in M and a 4-chain Z in W such that X − Y = ∂S + Z. So∫
X
Φ−
∫
Y
Φ =
∫
∂S
Φ+
∫
Z
Φ =
∫
S
dΦ+
∫
Z
Φ =
∫
Z
Φ
by Stokes’ Theorem since dΦ = 0 as Φ is torsion-free. We have Φ|W = dΨ for
some Ψ ∈ Ω3(W ) since H4(W ) = 0. Hence if Z ′ is another 4-chain in W such
that ∂Z ′ = ∂Z = ∂X − ∂Y , then∫
Z′
Φ =
∫
Z′
dΨ =
∫
∂Z′
Ψ =
∫
∂Z
Ψ =
∫
Z
dΨ =
∫
Z
Φ
by Stokes’ Theorem.
Write ∂Y = (exps)(∂X) for some s ∈ Γ(νW∂X) with small ∥s∥C0 , where exp
is the exponential map of W . Let x ∈ ∂X, let ε > 0 be small, and let
f : [0, 1]× [0, ε]→ W , (t, r) ↦→ expexpx(tsx)(rn(expx(tsx))) .
Note that f(0, r) ∈ ∂X and f(1, r) ∈ ∂Y for all r ∈ [0, ε] since n|∂X ∈ Γ(T∂X),
n|∂Y ∈ Γ(T∂Y ), and n is parallel. Furthermore, R(a, b)n = 0 for all a, b ∈ Γ(TW )
since n is parallel, where R denotes the curvature tensor of W . Hence
f(t, r) = expexpx(rnx)(ts˜(r))
for all t ∈ [0, 1], r ∈ [0, ε] since n is parallel, where s˜ is the parallel transport of sx
along the curve [0, ε]→ ∂X, r ↦→ f(0, r). Note that s˜(ε) is normal to ∂X as a
similar argument to above shows that Tf(0,ε)∂X is given by the parallel transport
of Tx∂X along the curve [0, ε] → ∂X, r ↦→ f(0, r). Hence s(f(0, ε)) = s˜(ε).
This shows that n(expx(tsx)) ∈ Texpx(tsx)(expts)(∂X) for all t ∈ [0, 1]. In
particular, ∂Y is isotopic to ∂X through submanifolds such that n is tangent to
them, and we may assume that n is tangent to Z. But this implies that Φ|Z = 0.
Hence
vol(X) =
∫
X
volX =
∫
X
Φ =
∫
Y
Φ ≤
∫
Y
volY = vol(Y )
since X is Cayley, and equality holds if and only if Φ|Y = volY , that is, if and
only if Y is Cayley.
98
As a preparation for dimension 6, we have the following two lemmas.
Lemma 5.6. Let M be an 8-manifold with a Spin(7)-structure Φ, let W be an
oriented 6-dimensional submanifold of M , and let ω := −∗W (Φ|W ). Then
1
2ω
2 = −Φ|W and 16ω3 = volW . (5.1)
Proof. Let (e1, . . . , e8) be a local Spin(7)-frame such that (e1, . . . , e6) is a
positive frame of W . Then
Φ|W = e1234 + e1256 − e3456
by (2.19). So
ω = −∗W (Φ|W ) = e12 − e34 − e56 .
Hence
1
2ω
2 = −e1234 − e1256 + e3456 = −Φ|W
and
1
6ω
3 = e123456 = volW .
Lemma 5.7. Let M be an 8-manifold with a Spin(7)-structure Φ, let X be a
Cayley submanifold of M with boundary, let W be an oriented 6-dimensional
submanifold of M with ∂X ⊆ W , and let ω := −∗W (Φ|W ). Then X and W
meet orthogonally if and only if ω|∂X = 0.
Proof. Let u ∈ Γ(νX∂X) be the interior unit normal vector field of ∂X in X.
First suppose that X and W meet orthogonally. Then there is a local
Spin(7)-frame (e1, . . . , e8) such that e1 = u, (e2, e3, e4) is a frame of ∂X, and
(e2, e6, e3, e7, e4, e8) is a positive frame of W . Then
Φ|W = e2367 + e2468 + e3478
by (2.19). So
ω = −∗W (Φ|W ) = e26 + e37 + e48 .
Hence ω|∂X = 0.
99
Conversely, suppose that ω|∂X = 0. Let (a, b, c) be a local orthonormal frame
of ∂X, and let d ∈ Γ(TW ). Then u = ±a×b×c and ω(a, b) = ω(a, c) = ω(b, c) = 0.
So
g(d, u) = ±g(d, a× b× c) = ±Φ(d, a, b, c) = ∓12(ω ∧ ω)(d, a, b, c)
= ∓12(ω(d, a)ω(b, c)− ω(d, b)ω(a, c) + ω(d, c)ω(a, b)) = 0
by (5.1). Hence X and W meet orthogonally.
Proposition 5.8. Let M be an 8-manifold with a torsion-free Spin(7)-struc-
ture Φ, let X be a compact Cayley submanifold of M with boundary, let W
be an oriented 6-dimensional submanifold of M with ∂X ⊆ W such that X
and W meet orthogonally, and let ω := −∗W (Φ|W ). Suppose that H4(W ) = 0
and that (W,ω) is a symplectic manifold (note that ∂X is a Lagrangian sub-
manifold of (W,ω) by Lemma 5.7). Furthermore, let Y be a compact, oriented
4-dimensional submanifold of M with ∂Y ⊆ W which lies in the same relative
homology class as X in H4(M,W ) such that ∂Y is a Lagrangian submanifold
of (W,ω) which is a local deformation of ∂X.
Then the volume of Y is greater than or equal to the volume of X, and
equality holds if and only if Y is Cayley.
Note that dω = 0 suffices for (W,ω) to be a symplectic manifold as ω is
non-degenerate by (5.1).
Proof. Since X and Y lie in the same relative homology class, there are a
5-chain S in M and a 4-chain Z in W such that X − Y = ∂S + Z. So∫
X
Φ−
∫
Y
Φ =
∫
∂S
Φ+
∫
Z
Φ =
∫
S
dΦ+
∫
Z
Φ =
∫
Z
Φ
by Stokes’ Theorem since dΦ = 0 as Φ is torsion-free. We have Φ|W = dΨ for
some Ψ ∈ Ω3(W ) since H4(W ) = 0. Hence if Z ′ is another 4-chain in W such
that ∂Z ′ = ∂Z = ∂X − ∂Y , then∫
Z′
Φ =
∫
Z′
dΨ =
∫
∂Z′
Ψ =
∫
∂Z
Ψ =
∫
Z
dΨ =
∫
Z
Φ
by Stokes’ Theorem.
100
Since (W,ω) is a symplectic manifold and ∂X is a Lagrangian submanifold
of (W,ω), there is an open neighbourhood of ∂X in W which is symplecto-
morphic to an open tubular neighbourhood of the 0-section in T ∗∂X (Weinstein
Tubular Neighbourhood Theorem). Furthermore, local deformations of ∂X
that are Lagrangian submanifolds of (W,ω) correspond to closed 1-forms (with
small C0-norm) under this identification. In particular, ∂Y is isotopic to ∂X
through Lagrangian submanifolds, and we may assume that for every x ∈ Z,
the tangent space TxZ contains a Lagrangian subspace of TxW . This implies
that Φ|Z = 0 since Φ|W = −12ω ∧ ω by (5.1).
Hence
vol(X) =
∫
X
volX =
∫
X
Φ =
∫
Y
Φ ≤
∫
Y
volY = vol(Y )
since X is Cayley, and equality holds if and only if Φ|Y = volY , that is, if and
only if Y is Cayley.
Remark 5.9. For other calibrations, there are also versions of the volume
minimising property for compact calibrated submanifolds with boundary. For
special Lagrangian submanifolds, see Lemma 5.14 in Section 5.4. A version for
coassociative submanifolds follows from Proposition 5.8 and Proposition 5.19
in Section 5.5.
5.2 Holonomy contained in SU(2)× SU(2)
Let Z1 and Z2 be two Riemannian 4-manifolds with holonomy contained
in SU(2), and letM := Z1×Z2 be their product (endowed with the product met-
ric). Then the holonomy ofM is contained in SU(2)×SU(2) ⊆ SU(4) ⊆ Spin(7).
In fact, if M is a simply-connected, complete Riemannian 8-manifold with
holonomy contained in SU(2) × SU(2), then there are Riemannian 4-mani-
folds Z1 and Z2 as above by the de Rham Decomposition Theorem.
Let X˜ be a compact, connected 4-dimensional submanifold of Z1 with non-
empty boundary, let W˜ be a k-dimensional submanifold of Z2 (0 ≤ k ≤ 4), and
let p ∈ W˜ . Define X := X˜ ×{p} and W := ∂X × W˜ . Then X is a Cayley sub-
manifold ofM since X is a complex surface. Furthermore, W is a (k+3)-dimen-
sional submanifold of M with ∂X ⊆ W such that X and W meet orthogonally.
101
Proposition 5.10. Let M , X˜, X, W˜ , W , and k be as above, and let Y be a
local deformation of X such that ∂Y ⊆ W .
Then Y is a Cayley submanifold of M such that Y and W meet orthogonally
if and only if Y = X˜ × {q} for some q ∈ W˜ . So the moduli space of all local
deformations of X as a Cayley submanifold of M with boundary on W and
meeting W orthogonally is a smooth k-dimensional manifold.
Proof. The submanifolds X˜ × {q} for q ∈ W˜ are clearly Cayley submanifolds
of M with boundary on W and meeting W orthogonally. So the moduli
space contains a smooth k-dimensional manifold. We will now show that the
dimension of the kernel of the linearisation of the boundary problem (4.18) at
the 0-section is at most k. The claim then follows from Lemma 2.5.
First note that the normal bundle νMX is flat (with respect to the induced
connection ∇⊥). Let (e5, . . . , e8) be a parallel orthonormal frame of νMX such
that e5, . . . , ek+4 ∈ Γ(νW∂X) and ek+5, . . . , e8 ∈ Γ(νMW |∂X), and let
s = s1e5 + · · ·+ s4e8 ∈ Γ(νMX)
be in the kernel of the linearisation of (4.18) at the 0-section, where s1, . . . ,s4∈C∞(X).
Let (e1, . . . , e4) be a local orthonormal frame of X such that (e1, . . . , e8) is a
Spin(7)-frame (w.l.o.g. we may assume that (e5, . . . , e8) is positively oriented).
We have D∗Ds = 0 by (4.15), where D : Γ(νMX)→ Γ(E) is the Dirac operator
defined in (3.1) and the vector bundle E of rank 4 over X is defined as in (2.23).
Since νMX and E are flat, this means⎛⎜⎜⎜⎜⎜⎜⎝
∂1 −∂2 −∂3 −∂4
∂2 ∂1 ∂4 −∂3
∂3 −∂4 ∂1 ∂2
∂4 ∂3 −∂2 ∂1
⎞⎟⎟⎟⎟⎟⎟⎠
⎛⎜⎜⎜⎜⎜⎜⎝
∂1 ∂2 ∂3 ∂4
−∂2 ∂1 −∂4 ∂3
−∂3 ∂4 ∂1 −∂2
−∂4 −∂3 ∂2 ∂1
⎞⎟⎟⎟⎟⎟⎟⎠
⎛⎜⎜⎜⎜⎜⎜⎝
s1
s2
s3
s4
⎞⎟⎟⎟⎟⎟⎟⎠
=
⎛⎜⎜⎜⎜⎜⎜⎝
∆ 0 0 0
0 ∆ 0 0
0 0 ∆ 0
0 0 0 ∆
⎞⎟⎟⎟⎟⎟⎟⎠
⎛⎜⎜⎜⎜⎜⎜⎝
s1
s2
s3
s4
⎞⎟⎟⎟⎟⎟⎟⎠ = 0
102
in X by (3.1) and (2.20) since ∇Φ = 0, where we used (e1 × e5, . . . , e1 × e8) as
a frame of E. So s1, . . . , s4 are harmonic functions.
We further have πK(s|∂X) = 0, where πK is defined in (4.8). So sk+1, . . . , s4 = 0
on ∂X. Hence sk+1 = · · · = s4 = 0 in X since the solution of the Laplace
equation with Dirichlet boundary condition is unique.
Furthermore, πν(∇us|∂X − ∇πν(s|∂X)u) = 0 by (4.21) since πK(s|∂X) = 0,
where πν is defined in (4.10) and u ∈ Γ(νX∂X) is the inward-pointing unit
normal vector field of ∂X in X. Note that πν(∇πν(s|∂X)u) = 0 since M is
endowed with the product metric and u ∈ Γ(TZ1|∂X), πν(s|∂X) ∈ Γ(TZ2|∂X).
So πν(∇us|∂X) = 0. Hence ∂us1 = · · · = ∂usk = 0 on ∂X, where ∂u is
the (interior) normal derivative. So s1, . . . , sk satisfy the Laplace equation
with Neumann boundary condition, and hence are constant. Therefore, the
dimension of the kernel of the linearisation of (4.18) at the 0-section is at
most k.
5.3 Bryant–Salamon construction
Bryant and Salamon [BS89, Theorem 2 of Section 4] constructed a Spin(7)-
structure Φ on the total space of the spin bundle S− over S4 (with the round
metric) such that the resulting manifold is a complete Riemannian manifold
with holonomy equal to Spin(7) (in particular, ∇Φ = 0). As noted in [McL98,
Section 6], the 0-section of S− is a (rigid) closed Cayley submanifold of S− with
respect to Φ. The metric has the form
g = fs(r)π∗gs + fν(r)gν ,
where gs is the round metric on S4, gν is the flat metric on the fibres of S−
induced by gs, r is its associated norm, π : S− → S4 is the natural projection,
and fs, fν : [0,∞)→ R are given by
fs(r) = 5(1 + r2)
3
5 and fν(r) = 4(1 + r2)−
2
5 .
103
The particular choice of fs and fν will not be important in the following. In
particular, one can also use the solution fs(r) = −5(1−r2) 35 , fν(r) = 4(1−r2)− 25
(case (i) in [BS89]), which is only defined on the open subset of S− where r < 1.
The associated metric is not complete.
Let M := S−, let X be a compact 4-dimensional submanifold of S4 with
boundary (where we view S4 as the 0-section of S−), and let W be a subbundle
of S−|∂X of rank k (0 ≤ k ≤ 4). Then X is a Cayley submanifold of M with
boundary, and W is a (k + 3)-dimensional submanifold of M with ∂X ⊆ W
such that X and W meet orthogonally.
Proposition 5.11. Let M , X, and W be as above. Then X is rigid as a
Cayley submanifold of M with boundary on W and meeting W orthogonally.
Proof. We will show that the kernel of the linearisation of the boundary
problem (4.18) at the 0-section contains only the 0-section. So let s ∈ Γ(νMX)
be in the kernel of the linearisation of (4.18) at the 0-section. Then D∗Ds = 0
by (4.15), where D : Γ(νMX)→ Γ(E) is the Dirac operator defined in (3.1) and
the vector bundle E of rank 4 over X is defined as in (2.23). Also πK(s|∂X) = 0,
where πK is defined in (4.8). Furthermore, πν(∇us|∂X − ∇πν(s|∂X)u) = 0
by (4.21) since πK(s|∂X) = 0, where πν is defined in (4.10) and u ∈ Γ(νX∂X)
is the inward-pointing unit normal vector field of ∂X in X. Lemma 5.12 below
implies that πν(∇πν(s|∂X)u) = 0. So πν(∇us|∂X) = 0.
We have
D∗Ds = (∇⊥)∗∇⊥s+ 3s
by the Lichnerowicz formula [LM89, Theorem II.8.8], where ∇⊥ is the induced
connection on νMX. Hence
0 = ⟨D∗Ds, s⟩L2(νMX)
= ⟨(∇⊥)∗∇⊥s, s⟩L2(νMX) + 3∥s∥
2
L2(νMX)
= ∥∇⊥s∥2L2(T ∗X⊗νMX) + ⟨∇us, s⟩L2(νMX|∂X) + 3∥s∥
2
L2(νMX)
= ∥∇⊥s∥2L2(T ∗X⊗νMX) + 3∥s∥
2
L2(νMX)
104
since ∇us and s are pointwise orthogonal on ∂X as πν(∇us|∂X) = 0 and
πK(s|∂X) = 0. So s = 0.
Lemma 5.12. Let s, t ∈ Γ(TM) such that (dπ)(s) = (dπ)(t) = 0. Then
∇st|X ∈ Γ(νMX).
Proof. Let (x1, . . . , x4) be coordinates on S4, let (x5, . . . , x8) be linear coordin-
ates on the fibres of S−, and let ∇˜ be the spin connection on S−. Then
g
(
∂
∂xi
, ∂
∂xj
)
= fν(r) gν
(
∂
∂xi
, ∂
∂xj
)
for i, j = 5, . . . , 8 and
g
(
∂
∂xi
, ∂
∂xj
)
= fν(r)
8∑
k=5
xk gν
(
∇˜ ∂
∂xi
∂
∂xk
, ∂
∂xj
)
for i = 1, . . . , 4 and j = 5, . . . , 8. Let
∇ ∂
∂xi
∂
∂xj
=
8∑
k=1
Γ kij
∂
∂xk
and gij = g
(
∂
∂xi
, ∂
∂xj
)
for i, j = 1, . . . , 8. Then Γ kij = 12(∂igjk + ∂jgik − ∂kgij) for i, j, k = 1, . . . , 8.
Hence
Γ kij =
1
2fν(r)
(
gν
(
∇˜ ∂
∂xk
∂
∂xi
, ∂
∂xj
)
+ gν
(
∇˜ ∂
∂xk
∂
∂xj
, ∂
∂xi
)
− ∂k
(
gν
(
∂
∂xi
, ∂
∂xj
)))
+ 12(∂ifν)(r)
8∑
ℓ=5
xℓ gν
(
∇˜ ∂
∂xk
∂
∂xℓ
, ∂
∂xj
)
+ 12(∂jfν)(r)
8∑
ℓ=5
xℓ gν
(
∇˜ ∂
∂xk
∂
∂xℓ
, ∂
∂xi
)
for i, j = 5, . . . , 8 and k = 1, . . . , 4. So Γ kij = 0 at r = 0 for i, j = 5, . . . , 8 and
k = 1, . . . , 4 since gν is compatible with ∇˜.
105
5.4 Cayley deformations of special Lagrangian
submanifolds
Let M be a Calabi–Yau 4-fold with Kähler form ω and holomorphic volume
form Ω, which we assume to be normalised, that is, ω4 = 32Ω ∧ Ω¯. Then
Φ := −12 ω ∧ ω +ReΩ
defines a Spin(7)-structure on M (see Sections 2.2.5 and 2.2.6). This Spin(7)-
structure is torsion-free since ω and Ω are closed.
An orientable 4-dimensional submanifold X of M is called special Lagrangian
if (ReΩ)|X = volX for some orientation of X. This is equivalent to ω|X = 0,
(ImΩ)|X = 0 (see Sections 2.2.5 and 2.2.6). So every special Lagrangian
submanifold is Cayley, but not every Cayley submanifold is special Lagrangian
(for example, complex surfaces are also Cayley).
Proposition 5.13. Let M be a Calabi–Yau 4-fold, let X be a compact special
Lagrangian submanifold of M with boundary, let W be a complex 3-dimensional
submanifold of M such that ∂X ⊆ W (which implies that X and W meet
orthogonally by Lemma 5.15 below), and let Y be a local deformation of X with
∂Y ⊆ W .
Then Y is a Cayley submanifold of M such that Y and W meet orthogonally
if and only if Y is a special Lagrangian submanifold of M . So the moduli space
of all local deformations of X as a Cayley submanifold of M with boundary
on W and meeting W orthogonally can be identified with the moduli space of
all local deformations of X as a special Lagrangian submanifold of M with
boundary on W .
The moduli space of all local deformations of X as a special Lagrangian sub-
manifold of M with boundary on W is a smooth manifold of dimension b1(X).
This follows from Butscher’s work [But03], as we will see after the proof. The
following lemma is analogous to Proposition 5.1.
106
Lemma 5.14. Let M be a Calabi–Yau n-fold, let X be a compact special Lag-
rangian submanifold ofM with boundary, letW be a complex (n−1)-dimensional
submanifold of M with ∂X ⊆ W , and let Y be an oriented real n-dimensional
submanifold of M with ∂Y ⊆ W which lies in the same relative homology class
as X in Hn(M,W ).
Then the volume of Y is greater than or equal to the volume of X, and
equality holds if and only if Y is special Lagrangian.
Proof. The proof is similar to the proof of Proposition 5.1 using ReΩ instead
of Φ, where Ω is the holomorphic volume form of M . Here (ReΩ)|W = 0 as W
is complex.
As noted in [But03, Remark after Definition 1], we have the following.
Lemma 5.15. Let M be a Kähler manifold, let X be a Lagrangian submanifold
of M with boundary, and let W be a complex submanifold of M with complex
codimension 1 such that ∂X ⊆ W . Then X and W meet orthogonally.
For the reader’s convenience, here is a proof of this lemma.
Proof. Let J : TM → TM be the complex structure of M , let u ∈ Γ(νX∂X) be
the inward-pointing unit normal vector field of ∂X inX, and let (e1, . . . , en−1) be
a local orthonormal frame of ∂X (where n := dimCM). Then Ju, Je1, . . . , Jen−1
are normal toX sinceX is Lagrangian andM is Kähler. Moreover, Je1, . . . , Jen−1
are tangent toW sinceW is complex and ∂X ⊆W . Furthermore, (u, e1, . . . , en−1,
Ju, Je1, . . . , Jen−1) is an orthonormal frame of TM |∂X since J is orthogonal.
So (e1, . . . , en−1, Je1, . . . , Jen−1) is an orthonormal frame of W . Hence X and
W meet orthogonally.
Proof (Proposition 5.13). If Y is a special Lagrangian submanifold of M , then
Y is clearly also a Cayley submanifold of M . Furthermore, Y and W meet
orthogonally by Lemma 5.15.
Conversely, suppose that Y is a Cayley submanifold of M such that Y and
W meet orthogonally. Then vol(Y ) = vol(X) by Proposition 5.8. So Y is
special Lagrangian by Lemma 5.14.
107
Butscher proved the following theorem about minimal Lagrangian deforma-
tions of compact special Lagrangian submanifolds with boundary.
Theorem 5.16 ([But03, Main Theorem]). Let M be a Calabi–Yau manifold
with Kähler form ω, let X be a compact special Lagrangian submanifold of M
with boundary, let u ∈ Γ(νX∂X) be the inward-pointing unit normal vector field
of ∂X in X, and let W be a submanifold of M with real codimension 2 such
that
(i) (W,ω|W ) is a symplectic manifold,
(ii) ∂X ⊆ W ,
(iii) u ∈ Γ((TW |∂X)ω), and
(iv) the bundle (TW )ω is trivial.
Here Sω denotes the symplectic orthogonal complement of a subspace S of a
symplectic vector space V , defined as Sω := {v ∈ V : ω(v, s) = 0 ∀ s ∈ S}.
Then the moduli space of all local deformations of X as a minimal Lagrangian
submanifold with boundary on W is a smooth finite-dimensional manifold which
is parametrised over
H1N(X) := {η ∈ Ω1(X) : dη = 0, δη = 0, u ⌟ η|∂X = 0} .
The space H1N (X) of harmonic 1-fields on X with Neumann boundary condi-
tion is isomorphic to H1(X;R) [CDGM06, page 927]. So it has dimension b1(X).
Butscher stated the theorem for minimal Lagrangian submanifolds but we have
the following.
Lemma 5.17. Let M be a Calabi–Yau n-fold, let X be a compact special Lag-
rangian submanifold ofM with boundary, letW be a complex (n−1)-dimensional
submanifold of M such that ∂X ⊆ W , and let Y be a local deformation of X
with ∂Y ⊆ W . Suppose that Y is a minimal Lagrangian submanifold. Then Y
is special Lagrangian.
Proof. We have vol(Y ) ≥ vol(X) by Lemma 5.14. Furthermore, Y is special
Lagrangian with phase angle θ for some θ ∈ R by Proposition 2.15 (if Y is
not connected, then θ can be viewed as a locally constant function). So it is
108
calibrated with respect to Re(eiθΩ), where Ω is the holomorphic volume form
of M . Therefore, vol(X) ≥ vol(Y ) by a lemma analogous to Lemma 5.14 for
special Lagrangian submanifolds with phase angle θ. So vol(Y ) = vol(X), and
hence Y is special Lagrangian by Lemma 5.14.
Corollary 5.18. Let M be a Calabi–Yau n-fold, let X be a compact spe-
cial Lagrangian submanifold of M with boundary, and let W be a complex
(n− 1)-dimensional submanifold of M such that ∂X ⊆ W .
Then the moduli space of all local deformations of X as a special Lagrangian
submanifold ofM with boundary onW is a smooth manifold of dimension b1(X).
Proof. First note that the proof of Lemma 5.15 also shows that (u, Ju) is an
orthonormal frame of νMW |∂X , where u ∈ Γ(νX∂X) is the inward-pointing unit
normal vector field of ∂X in X. In particular, u ∈ Γ((TW |∂X)ω). Furthermore,
νMW |∂X is trivial. Since we are only interested in local deformations of X and
∂X is compact, we therefore may assume that νMW is trivial. So we can apply
Theorem 5.16, which gives the desired result by Lemma 5.17.
5.5 Cayley deformations of coassociative
submanifolds
Let M˜ be a 7-manifold with a G2-structure φ˜, let ψ˜ be the Hodge-dual of φ˜
(with respect to the metric and orientation induced by φ˜), let M := R× M˜ ,
and let t denote the coordinate on the R-factor. Then
Φ := dt ∧ φ˜+ ψ˜
defines a Spin(7)-structure on M (see Section 2.2.2). The Spin(7)-structure Φ
is torsion-free if and only if the G2-structure φ˜ is torsion-free.
An orientable 4-dimensional submanifold X˜ of M˜ is called coassociative if
ψ˜|X˜ = volX˜ for some orientation of X˜. This is equivalent to φ˜|X˜ = 0 [HL82,
Corollary 1.20 in Chapter IV]. So X˜ is a coassociative submanifold of M˜ if and
only if X := {0} × X˜ is a Cayley submanifold of M .
109
Proposition 5.19. Let M˜ be a 7-manifold with a torsion-free G2-structure φ˜,
let ψ˜ be the Hodge-dual of φ˜, let X˜ be a compact, connected coassociative
submanifold of M˜ with non-empty boundary, let W˜ be an oriented 6-dimensional
submanifold of M˜ such that (W˜ , ∗W˜ (ψ˜|W˜ )) is a symplectic manifold, and suppose
that ∂X˜ ⊆ W˜ and that X˜ and W˜ meet orthogonally. Define M := R × M˜ ,
W := {0} × W˜ , and X := {0} × X˜. Furthermore, let Y be a local deformation
of X such that ∂Y ⊆ W .
Then Y is a Cayley submanifold of M such that Y and W meet orthogonally
if and only if Y = {0} × Y˜ for some coassociative submanifold Y˜ of M˜ with
∂Y˜ ⊆ W˜ such that Y˜ and W˜ meet orthogonally. So the moduli space of all
local deformations of X as a Cayley submanifold of M with boundary on W
and meeting W orthogonally can be identified with the moduli space of all local
deformations of X˜ as a coassociative submanifold of M˜ with boundary on W˜
and meeting W˜ orthogonally.
Kovalev and Lotay [KL09] showed that the moduli space of all local de-
formations of X˜ as a coassociative submanifold of M˜ with boundary on W˜
and meeting W˜ orthogonally is a smooth manifold of dimension not greater
than b1(∂X˜).
Remark 5.20. Using [Bär97], one can show that the dimension is not greater
than the minimum of the first Betti numbers of the connected components of
the boundary. Indeed, by [KL09, Theorem 4.10 and Corollary 3.8], the moduli
space is parametrised by
(H2+)bc := {α ∈ Ω2+(X˜) : dα = 0 in X˜ and d∂X˜(u ⌟ α|∂X˜) = 0 on ∂X˜} .
LetZ be a connected component of the boundary ∂X˜, and suppose that α∈(H2+)bc
satisfies α|Z = 0. Then also u ⌟ α|Z = ∗Z(α|Z) = 0 since α is self-dual. Now
[Bär97, Main Theorem] implies that α = 0 since Ω2+(X˜)⊕ Ω4(X˜)→ Ω3(X˜),
(α, β) ↦→ dα + δβ is a Dirac operator and Z has Hausdorff dimension 3. Note
further that α ∈ (H2+)bc implies that α|Z is a harmonic form on the boundary
since ∗Z(α|Z) = u⌟α|Z . Hence the map (H2+)bc → H1(Z), α ↦→ α|Z is injective.
110
Proof. If Y˜ is a coassociative submanifold of M˜ with ∂Y˜ ⊆ W˜ such that
Y˜ and W˜ meet orthogonally, then Y := {0}× Y˜ is clearly a Cayley submanifold
of M with ∂Y ⊆ W such that Y and W meet orthogonally.
Conversely, suppose that Y is a Cayley submanifold ofM such that Y andW
meet orthogonally. Let π : M = R× M˜ → M˜ be the projection. If Z is a local
deformation of X, then the volume of Z is greater than or equal to the volume
of π(Z), and equality holds if and only if Z is a submanifold of {t} × M˜ for
some t ∈ R. So Y = {t}× Y˜ for some t ∈ R and a 4-dimensional submanifold Y˜
of M˜ since the volume of Y is equal to the volume of X and the volume of π(Y )
is greater than or equal to the volume of X by Proposition 5.8 (note that
{0} × π(Y ) is a submanifold of M with ∂({0} × π(Y )) = {0} × ∂(π(Y )) =
{0}× π(∂Y ) = ∂Y since ∂Y ⊆ {0}× W˜ ). We have t = 0 since ∂Y ⊆ {0}× W˜ .
Furthermore, Y˜ is coassociative since Y is Cayley.
5.6 Cayley deformations of associative
submanifolds
Recall from the last section that if M˜ is a 7-manifold with a G2-structure φ˜,
ψ˜ is the Hodge-dual of φ˜, M := S1 × M˜ , and t denotes the coordinate on the
S1-factor, then Φ := dt∧ φ˜+ ψ˜ defines a Spin(7)-structure on M . Furthermore,
the Spin(7)-structure Φ is torsion-free if and only if the G2-structure φ˜ is
torsion-free.
An orientable 3-dimensional submanifold X˜ of M˜ is called associative if
φ˜|X˜ = volX˜ for some orientation of X˜. So X˜ is an associative submanifold of M˜
if and only if X := S1 × X˜ is a Cayley submanifold of M . Also recall from the
last section that a 4-dimensional submanifold W˜ of M˜ is called coassociative
if φ˜|W˜ = 0. Note that if X˜ is associative, W˜ is coassociative, and ∂X˜ ⊆ W˜ ,
then X˜ and W˜ meet orthogonally as the 3-form φ˜ defines a cross product
TM˜ × TM˜ → TM˜ , (v, w) ↦→ v × w such that g(u, v × w) = φ˜(u, v, w), and
φ˜|X˜ = volX˜ , φ˜|W˜ = 0 imply that if v, w ∈ T∂X˜, then v × w is both tangent
to X˜ and orthogonal to W˜ .
111
Proposition 5.21. Let M˜ be a 7-manifold with a torsion-free G2-structure,
let X˜ be a compact associative submanifold of M˜ with boundary, and let W˜ be
a coassociative submanifold of M˜ with ∂X˜ ⊆ W˜ (which implies that X˜ and W˜
meet orthogonally). Define M := S1 × M˜ , X := S1 × X˜, and W := S1 × W˜ .
Furthermore, let Y be a local deformation of X such that ∂Y ⊆ W .
Then Y is a Cayley submanifold of M such that Y and W meet orthogonally
if and only if Y = S1 × Y˜ for some associative submanifold Y˜ of M˜ with
∂Y˜ ⊆ W˜ . So the moduli space of all local deformations of X as a Cayley
submanifold of M with boundary on W and meeting W orthogonally can be
identified with the moduli space of all local deformations of X˜ as an associative
submanifold of M˜ with boundary on W˜ .
Gayet and Witt [GW11] proved that the boundary problem for associative
submanifolds of a G2-manifold with boundary in a coassociative submanifold is
an elliptic boundary problem. They also computed the index of this boundary
problem.
Proof. If Y˜ is an associative submanifold of M˜ with ∂Y˜ ⊆ W˜ (which implies
that Y˜ and W˜ meet orthogonally), then Y := S1 × Y˜ is clearly a Cayley
submanifold of M with ∂Y ⊆ W such that Y and W meet orthogonally.
Conversely, suppose that Y is a Cayley submanifold ofM such that Y andW
meet orthogonally. Write Yt := Y ∩ ({t} × M˜) for t ∈ S1. Note that Yt is a
submanifold of M˜ for all t ∈ S1 as Y is a local deformation of X. Then
vol(Y ) ≥
∫
S1
vol(Yt) dt
with equality if and only if Y = S1 × Y0. Furthermore, vol(Yt) ≥ vol(X˜) since
Yt is a local deformation of X˜ and X˜ is volume-minimising in its homology class
as it is associative. Here we have equality if and only if Yt is associative. So
vol(Y ) ≥
∫
S1
vol(Yt) dt ≥
∫
S1
vol(X˜) dt = vol(X) .
But vol(Y ) = vol(X) by Proposition 5.4. Hence we get equality for both
inequalities, which implies that Y = S1 × Y0 with Y0 associative. Clearly,
∂Y0 ⊆ W˜ as ∂Y ⊆ W .
112
6 Asymptotically cylindrical
Cayley submanifolds
In this chapter, we present the deformation theory of asymptotically cylindrical
Cayley submanifolds. We first review, in Section 6.1, the Fredholm theory of
elliptic operators on asymptotically cylindrical manifolds, the Atiyah–Patodi–
Singer Index Theorem, the relative Euler class, and the generalised Gauss–
Bonnet–Chern Theorem, and prove an extension of the volume-minimising
property to asymptotically cylindrical calibrated submanifolds. Then in Sec-
tion 6.2, we extend the results of Section 3.1 about the deformations of closed
Cayley submanifolds to the asymptotically cylindrical setting. We proceed
in Section 6.3.1 by generalising the index formula from Section 3.2 for closed
Cayley submanifolds to the asymptotically cylindrical case (Theorem 1.5). In
Section 6.3.2, we investigate the relation between the two operators Bev and D˜,
whose η-invariants appear in the index formula of Theorem 1.5, resulting in an
alternative index formula that involves the spectral flow (Proposition 6.22).
In specific examples of asymptotically cylindrical Cayley submanifolds (which
can usually be found if the holonomy of the cross-section at infinity of the
Spin(7)-manifold is a proper subgroup of G2), the cross-section at infinity of
the Cayley submanifold has certain properties that help to simplify the general
index formula. In Section 6.3.3, we prove simplified index formulae (which
only involve topological quantities) under such special assumptions. We finish
this chapter by showing in Section 6.4 that for a generic asymptotically cyl-
indrical Spin(7)-structure, asymptotically cylindrical Cayley submanifolds form
a smooth moduli space (Theorem 1.6 and two variations, where the asymptotic
limits are not necessarily fixed).
113
6.1 Asymptotically cylindrical manifolds
In this section, we review the Fredholm theory of elliptic operators on asymp-
totically cylindrical manifolds, the Atiyah–Patodi–Singer Index Theorem, the
relative Euler class, and the generalised Gauss–Bonnet–Chern Theorem, and
prove an extension of the volume-minimising property to asymptotically cyl-
indrical calibrated submanifolds.
6.1.1 Analysis on asymptotically cylindrical manifolds
Here we give the definition of asymptotically cylindrical manifolds and present
the Fredholm theory of asymptotically cylindrical linear elliptic operators,
taken from [LM85].
Definition 6.1. Let (M, g) be a complete Riemannian manifold. Then (M, g)
is called asymptotically cylindrical if there are a compact subset K ⊆ M , a
closed Riemannian manifold (N, h) (the cross-section), and a diffeomorphism
Ψ : (0,∞)×N →M \K such that
|∇k∞(Ψ ∗(g)− g∞)| = O(e−δt) for all k ∈ N
for some δ > 0, where t denotes the projection onto the (0,∞)-factor and
∇∞ is the Levi-Civita connection of (0,∞)×N with respect to the product
metric g∞ = dt2 + h.
A submanifold X of M is called asymptotically cylindrical with rate λ > 0
if (X, g|X) is complete and there are a compact subset K ′ ⊆ X, a closed
submanifold Y of N , and a section v of the normal bundle of (R,∞)× Y in
(R,∞)×N (for some R ≥ 0) with
|∇k∞v| = O(e−λt) for all k ∈ N
such that
(R,∞)× Y →M , (t, x) ↦→ Ψ(exp(t,x)(v(t, x)))
is a diffeomorphism onto X \K ′, where exp is the exponential map with respect
to the metric g∞.
114
Let (M, g) be an asymptotically cylindrical manifold, let t denote the pro-
jection onto the (0,∞)-factor extended to M , let E be a vector bundle over M
equipped with an asymptotically cylindrical metric connection ∇, let k ≥ 0 be
an integer, let 0 < α < 1, and let λ ∈ R. For a Ck-section s of E, we define
∥s∥Ck,α
λ
:= ∥eλts∥Ck,α . (6.1)
The weighted Hölder space Ck,αλ (E) with weight λ consists of all Ck-sections
with finite ∥ · ∥Ck,α
λ
-norm. If we define
C∞λ (E) := {s ∈ C∞(E) : |∇ks| = O(e−λt) for all k ∈ N} , (6.2)
then
C∞λ (E) =
∞⋂
k=0
Ck,αλ (E) . (6.3)
The C∞λ -topology on C∞λ (E) is the initial topology induced by the inclusions
C∞λ (E) ↪→ Ck,αλ (E). Note that C∞λ (E) is a Fréchet space with the C∞λ -topology,
and hence a complete metric space. So a residual set is dense by the Baire
Category Theorem.
Theorem 6.2 ([LM85, Theorem 6.2 and Lemmas 7.1 and 7.3]). Let (M, g)
be an asymptotically cylindrical manifold, and let P : Γ(E) → Γ(F ) be an
asymptotically cylindrical linear elliptic differential operator of order k that is
asymptotic to the translation-invariant operator P∞ : Γ(E∞)→ Γ(F∞).
Then P extends to a bounded linear map P ℓ,αλ : C
ℓ+k,α
λ (E)→ Cℓ,αλ (F ) for all
ℓ ∈ N, α ∈ (0, 1), and λ ∈ R. For λ ∈ R, let dP (λ) be the complex dimension
of the space spanned by the solutions s ∈ Γ(E∞) of P∞s = 0 on the cylinder
R×N such that e(λ+iγ)ts is polynomial in t for some γ ∈ R, and let
DP := {λ ∈ R : dP (λ) ̸= 0} . (6.4)
Then DP is a discrete set such that P ℓ,αλ : Cℓ+k,αλ (E) → Cℓ,αλ (F ) is Fredholm
if and only if λ /∈ DP . For λ ∈ R \ DP , let indλ P := indP ℓ,αλ , which is
independent of ℓ and α, and hence well-defined.
115
Then
indλ P − indδ P =
∑
γ∈DP
λ<γ<δ
dP (γ) (6.5)
for all λ, δ ∈ R \ DP with λ < δ. Moreover, if also [λ, δ] ∩ DP = ∅, then
kerP ℓ,αλ = kerP
ℓ,α
δ , which is independent of ℓ and α.
Lemma 6.3 (cf. [LM85, Theorem 7.4]). Let M be an asymptotically cylindrical
manifold, let P : Γ(E)→ Γ(F ) be an asymptotically cylindrical linear elliptic
differential operator, and let P ∗ : Γ(F )→ Γ(E) be the formal adjoint. Then
indλ P ∗ = − indλ P − dP (0) (6.6)
for all λ > 0 such that [−λ, λ] ∩ DP ⊆ {0}.
Lemma 6.4 (cf. [LM85, page 436]). Let M be an asymptotically cylindrical
manifold, and let P : Γ(E) → Γ(F ) and Q : Γ(E) → Γ(F ) be asymptotically
cylindrical linear elliptic differential operators of the same order. If P and Q
have the same symbol and are asymptotic to the same translation-invariant
operator, then indλ P = indλQ for all λ /∈ DP = DQ.
Let M be an asymptotically cylindrical manifold with cross-section N , let
E be a vector bundle over M equipped with an asymptotically cylindrical
metric connection ∇, let k ≥ 0 be an integer, let 0 < α < 1, and let λ > 0.
Write E|(0,∞)×N ∼= (0,∞) × F using the connection ∇, where F is a vector
bundle over N . Then
Cˆk,αλ (E) := C
k,α
λ (E)⊕ Ck,α(F ) (6.7)
is the extended weighted Hölder space with weight λ. Note that Cˆk,αλ (E) can
be viewed as a subspace of Ck,α(E) as follows. Let ρ : R→ [0, 1] be a smooth
function with ρ(t) = 0 for t ≤ 1 and ρ(t) = 1 for t ≥ 2. For sˆ ∈ Ck,α(F ),
view it as a translation-invariant section of R × F (i.e., ∇tsˆ = 0), and con-
sider ρ(t)sˆ ∈ Ck,α(E). Furthermore, if s ∈ Ck,αλ (E), then s ∈ Ck,α(E) since
e−λt ∈ Ck,α(E) as λ > 0. Note that the norm on Cˆk,αλ (E) is not Lipschitz
equivalent to ∥ · ∥Ck,α .
116
For asymptotically cylindrical submanifolds, we have the following version of
the Tubular Neighbourhood Theorem (see Theorem 2.1).
Lemma 6.5. Let (M, g) be an asymptotically cylindrical manifold with cross-
section N , and let X be an asymptotically cylindrical submanifold of M with
cross-section Y .
Then there are open tubular neighbourhoods U ⊆ νMX and V ⊆ νNY of
the 0-sections with (Ψ |(R,∞)×N)−1(U) = (R,∞) × V for some R > 0, where
Ψ : (0,∞) × N → M is as in Definition 6.1, such that the exponential map
(with respect to the metric g) defines an isomorphism from U onto an open
neighbourhood of X in M .
So any submanifold that is C1-close to X can be parametrised by a section of
the normal bundle νMX with small C1-norm. Furthermore, that submanifold
is asymptotically cylindrical if and only if the corresponding section of the
normal bundle decays to 0 at an exponential rate.
We further have the following version of Lemma 3.9 for asymptotically
cylindrical manifolds (the proof is almost literally the same).
Lemma 6.6. Let M be a smooth, asymptotically cylindrical manifold, let E
be a vector bundle over M equipped with a smooth, asymptotically cylindrical
metric connection, let k0 ≥ 0 be an integer, let 0 < α < 1, let λ ∈ R,
let Mk0 ⊆ Ck0,αλ (E), let Mk := Mk0 ∩ Ck,αλ (E) for k ≥ k0 + 1, and let
M∞ :=Mk0 ∩C∞λ (E). Suppose that Mk is the intersection of countably many
open dense subsets of Ck,αλ (E) for all k ≥ k0. Then M∞ is the intersection of
countably many open dense subsets of C∞λ (E).
Also Proposition 2.17 generalises to asymptotically cylindrical manifolds as
follows since the proof of [Bai01, Theorem 2.2.15] extends to asymptotically cyl-
indrical manifolds and for any vector bundle over an asymptotically cylindrical
manifold, there are finitely many open subsets that cover the asymptotically
cylindrical manifold on which this vector bundle is trivial.
117
Proposition 6.7. Let M be an asymptotically cylindrical manifold, let X be
an asymptotically cylindrical submanifold of M , let TX := ⨁i(T ∗X)⊗i and
TM := ⨁i(T ∗M)⊗i be the tensor algebras of X and M , respectively, let E be a
vector bundle over M equipped with an asymptotically cylindrical connection ∇,
let U be an open tubular neighbourhood of the 0-section in νMX as in Lemma 6.5
such that the exponential map defines a diffeomorphism from U to an open
neighbourhood V of X in M , let k ≥ 1, ℓ ≥ 0 be integers, let 0 < α < 1, and
let λ > 0. For (x, v) ∈ U , let πx,v : Eexpx(v) → Ex denote the parallel transport
along the curve [0, 1]→M , t ↦→ expx((1− t)v) (so for all s ∈ Γ(U), the map
πs : (exps)∗E → E|X is an isomorphism of vector bundles).
Then the map
Ψ : Ck,αλ (U)⊕ Ck−1+ℓ,αλ ((TM ⊗ E)|V )→ Ck−1,αλ (TX ⊗ E|X) ,
(s,Θ) ↦→ (idTX⊗πs)((exps)∗Θ)
(6.8)
is of class Cℓ.
6.1.2 Atiyah–Patodi–Singer Index Theorem
The main tool to compute the index of the operator of Dirac type that arises as
the linearisation of the deformation map will be the Atiyah–Patodi–Singer Index
Theorem [APS75a]. Here we recall the Atiyah–Patodi–Singer Index Theorem
and also the signature theorem for asymptotically cylindrical manifolds.
Definition 6.8 ([APS75a] and [APS76, Proposition (2·8) and Theorem (4·5)]).
Let M be an odd-dimensional closed manifold, and let P : Γ(E)→ Γ(E) be a
self-adjoint linear elliptic differential operator. Define
ηP (z) :=
∑
λ ̸=0
(sgn λ)|λ|−z (6.9)
for z ∈ C with Re z large, where λ runs over all eigenvalues of P (counted with
multiplicity). This function has a meromorphic extension to C, and z = 0 is
not a pole. The η-invariant of P is defined by η(P ) := ηP (0).
118
Theorem 6.9 (Atiyah–Patodi–Singer Index Theorem [APS75a, (4·3)] (see also
[Mel93, Theorem 9.1])). Let M be an n-dimensional asymptotically cylindrical
manifold with cross-section N , let S+ ⊗ F and S− ⊗ F be positive and negative
(twisted) spinor bundles, respectively, and let D : Γ(S+ ⊗ F )→ Γ(S− ⊗ F ) be
the positive (twisted) Dirac operator. Then
indλD =
∫
M
(Aˆ(M) ch(F ))n − dim ker D˜ + η(D˜)2 , (6.10)
where λ > 0 is such that (0, λ] ∩ DD = ∅, D˜ : Γ((S+ ⊗ F )|N)→ Γ((S+ ⊗ F )|N)
is defined by the equation
Ds = u · (∇us+ D˜s)
for all s ∈ Γ(S+ ⊗ F ), where u := − ∂
∂t
, and η(D˜) is the η-invariant of D˜.
Note that the index of a negative (twisted) Dirac operator D : Γ(S− ⊗ F )→
Γ(S+ ⊗ F ) is therefore given by
indλD = −
∫
M
(Aˆ(M) ch(F ))n − dim ker D˜ − η(D˜)2 (6.11)
by Lemma 6.3, where D˜ : Γ((S− ⊗ F )|N) → Γ((S− ⊗ F )|N) is defined by the
equation
Ds = −u · (−∇us+ D˜s) .
Theorem 6.10 ([APS75a, Theorem (4·14)] (see also [Mel93, Theorem 9.4])).
Let M be an n-dimensional, oriented, asymptotically cylindrical manifold with
cross-section N . Suppose that n = 4k. Then
σ(M) =
∫
M
Lk(p)− η(Bev) , (6.12)
where
(i) σ(M) is the signature of M (the signature of the non-degenerate quadratic
form induced by the cup-product on the image of H2kcs (M) in H2k(M)),
(ii) Lk(p) is the k-th Hirzebruch L-polynomial in the Pontryagin forms, and
119
(iii) Bev : Ωev(N)→ Ωev(N) is the odd signature operator on N , defined by
Bev(φ) := (−1)k+p(∗d− d∗)φ
for φ ∈ Ω2p(N).
Note that this is the negative of Bev in [APS75a] since we use the opposite
orientation on N . An immediate consequence of this signature formula is
Novikov additivity.
Lemma 6.11 (Novikov additivity). Let M1 and M2 be n-dimensional, oriented
manifolds with cylindrical ends with cross-sections N1 and N2, respectively.
Suppose that n = 4k, and let N be an oriented (n−1)-dimensional manifold that
is diffeomorphic to a connected component of N1 and a connected component
of N2 by orientation-preserving diffeomorphisms. Then
σ(M1 ∪N M2) = σ(M1) + σ(M2) .
The definitions of Bev and the η-invariant imply the following.
Lemma 6.12. Let M be an oriented, odd-dimensional, closed Riemannian
manifold, and let Bev : Ωev(M)→ Ωev(M) be defined as in (iii) of Theorem 6.10.
Suppose that M has an orientation-reversing isometry. Then η(Bev) = 0.
This happens, in particular, if M = S1 ×N for some oriented Riemannian
manifold N .
6.1.3 Relative Euler class and generalised
Gauss–Bonnet–Chern Theorem
In the index formulae that we prove, we will use the relative Euler class. Here we
define the relative Euler class and present the generalised Gauss–Bonnet–Chern
Theorem for the computation of the relative Euler class (see [Sha73]).
120
Definition 6.13. Let M be a manifold, let N be a submanifold of M , let E
be an oriented vector bundle over M of rank n, and let s ∈ Γ(E|N) be a non-
vanishing section of E over N . The relative Euler class e(E, s) ∈ Hn(M,N)
is defined as follows. Let E0 ⊆ E be the complement of the zero section, let
S := s(N) (note that S ⊆ E0 since s is non-vanishing), and let u ∈ Hn(E,E0)
be the orientation class. Furthermore, let p : (E, S) → (M,N) denote the
projection (note that p is a homotopy equivalence), and let i : (E, S)→ (E,E0)
denote the inclusion. Then e(E, s) := (p∗)−1i∗u.
If M is a connected, oriented, n-dimensional, asymptotically cylindrical
manifold with cross-section N , E is an oriented vector bundle of rank n
over M , and s ∈ Γ(E|N) is a non-vanishing section, then Hn(M,N) ∼= Z, and
we can interpret the relative Euler class e(E, s) as an integer. Here we interpret
N as {t} ×N ↪→ (0,∞)×N ↪→ M for t > 0. If we talk about the metric or
connection on E|N , we always mean the asymptotic limit as t→∞.
Recall the definition of the Pfaffian: If A = (aij) is a 2k× 2k skew-symmetric
matrix, then
Pf(A) := 12kk!
∑
σ∈S2k
sgn(σ)
k∏
i=1
aσ(2i−1),σ(2i) ,
where S2k is the symmetric group and sgn(σ) is the signature of σ. It satisfies
(Pf(A))2 = detA.
Theorem 6.14 (‘Generalised Gauss–Bonnet–Chern Theorem’ (cf. [Sha73])).
Let M be a connected, oriented, n-dimensional, asymptotically cylindrical
manifold with cross-section N , let E be an oriented vector bundle over M of
rank n, let ∇ be an asymptotically cylindrical metric connection on E, and let
s ∈ Γ(E|N) be a non-vanishing section with point-wise norm 1. Suppose that
n = 2k is even.
Then there exists a differential form Θ(F∇|N ,∇s) ∈ Ωn−1(N) such that
e(E, s) =
∫
M
1
(2π)k Pf(F∇) +
∫
N
Θ(F∇|N ,∇s) , (6.13)
121
where F∇ ∈ Ω2(M,End(E)) is the curvature of ∇ and Pf denotes the Pfaffian
on End(E). Moreover, Θ(F∇|N ,∇s)
(i) depends only on F∇|N ∈ Ω2(N,End(E|N)) and ∇s ∈ Ω1(N,E|N),
(ii) satisfies Θ(F∇|N ,∇s) = 0 if ∇s = 0, and
(iii) satisfies Θ(F∇|N ,∇s) = 0 if v ⌟ F∇ = 0 and ∇vs = 0 for some non-
vanishing v ∈ Γ(TN).
The term 1(2π)k Pf(F∇) in the formula (6.13) is the Euler density, which (by
abuse of notation) we will only denote by e(E) later.
Proof. A proof of an analogous statement for compact manifolds with boundary
works almost literally like in [Sha73, §3], except that we use s instead of the
exterior normal vector field. The version for asymptotically cylindrical manifolds
can then be deduced by going to the limit t→∞.
6.1.4 Volume minimising property
Harvey and Lawson [HL82] proved that closed calibrated submanifolds are
volume-minimising in their homology class (see Proposition 2.8). In fact,
they proved this for compactly supported deformations. Here we present an
extension of this property to asymptotically cylindrical calibrated submanifolds.
Proposition 6.15 (cf. [HL82, Theorem 4.2 in Chapter II]). Let M be an
asymptotically cylindrical manifold with cross-section N , let φ ∈ Ωn(M) be a
calibration on M that is asymptotically cylindrical, let X be an asymptotically
cylindrical, calibrated submanifold of M , and let Y be an asymptotically cyl-
indrical, oriented, n-dimensional submanifold of M with the same asymptotic
limit as X which lies in the same relative homology class as X in Hn(M,N).
Furthermore, fix a diffeomorphism M \K ∼= (0,∞)×N as in Definition 6.1,
where K is a compact subset of M , and let t denote the projection onto the
(0,∞)-factor.
Then
lim
T→∞
(
vol({y ∈ Y : t ≤ T})− vol({x ∈ X : t ≤ T})
)
≥ 0 , (6.14)
and equality holds if and only if Y is calibrated.
122
Proof. Let T > 0 be large enough, and let XT := {x ∈ X : t ≤ T} and
YT := {y ∈ Y : t ≤ T}. Since X and Y lie in the same relative homology
class, there are an (n+ 1)-chain ST in M and an n-chain ZT in N such that
YT −XT = ∂ST + ZT . So∫
YT
φ−
∫
XT
φ =
∫
∂ST
φ+
∫
ZT
φ =
∫
ST
dφ+
∫
ZT
φ =
∫
ZT
φ
by Stokes’ Theorem since dφ = 0 as φ is a calibration. Furthermore,
lim
T→∞
∫
ZT
φ = 0
since X and Y have the same asymptotic limit. Hence
lim
T→∞
(
vol(YT )− vol(XT )
)
= lim
T→∞
⎛⎝∫
YT
volY −
∫
XT
volX
⎞⎠
≥ lim
T→∞
⎛⎝∫
YT
φ−
∫
XT
φ
⎞⎠
= lim
T→∞
∫
ZT
φ = 0
since φ is a calibration and X is calibrated. Equality holds if and only if
φ|Y = volY , that is, if and only if Y is calibrated.
6.2 Deformations of asymptotically cylindrical
Cayley submanifolds
Here we present the basic setup, that is, the deformation map and its linearisa-
tion in the asymptotically cylindrical setting. For more details, see Section 3.1.
Using the Tubular Neighbourhood Theorem for asymptotically cylindrical
manifolds (Lemma 6.5), we can define the deformation map as in Section 3.1,
F : Γλ(U)→ Γλ(E) , s ↦→ πE((exps)∗(τ)) ,
123
where U ⊆ νMX is an appropriate tubular neighbourhood of the 0-section as
in Lemma 6.5, the vector bundle E of rank 4 over X is defined as in (2.23),
and τ ∈ Ω4(M,Λ27M) is defined as in (2.16). As in Section 3.1, we get
(dF )0(s) = Ds ,
where D : Γ(νMX)→ Γ(E) is defined as in (3.1).
So similarly to Theorem 3.1 and Proposition 3.3, we have the following
theorem and proposition.
Theorem 6.16. Let M be an asymptotically cylindrical 8-manifold with an
asymptotically cylindrical Spin(7)-structure asymptotic to (0,∞)×N , where
N is a 7-manifold with a G2-structure, let X be an asymptotically cylindrical
Cayley submanifold of M asymptotic to (0,∞) × Y , where Y is a closed
associative submanifold of N , and let λ > 0.
Then the Zariski tangent space to the moduli space of all local deformations
of X as an asymptotically cylindrical Cayley submanifold of M with rate λ and
with the same asymptotic limit as X can be identified with the kernel of the
operator Dλ : Γλ(νMX)→ Γλ(E), where D is defined as in (3.1).
Proposition 6.17. Let M be a smooth, asymptotically cylindrical 8-mani-
fold with a smooth, asymptotically cylindrical Spin(7)-structure asymptotic to
(0,∞)×N , where N is a 7-manifold with a G2-structure, let X be a smooth,
asymptotically cylindrical Cayley submanifold of M asymptotic to (0,∞)× Y ,
where Y is a closed associative submanifold of N , let 0 < α < 1, and let λ > 0.
If the operator Dλ : Γλ(νMX)→ Γλ(E) is surjective, where D is defined as
in (3.1), then the moduli space of all smooth, asymptotically cylindrical Cayley
submanifolds of M with rate λ that are C2,αλ -close to X is a smooth manifold
of dimension dim kerDλ = indλD.
124
6.3 Index formula for the operator of Dirac type
In this section, we derive various formulae for the index of the operator of
Dirac type that arises as the linearisation of the deformation map under general
and special assumptions. In Section 6.3.1, we generalise the index formula for
closed Cayley submanifolds from Section 3.2 to the asymptotically cylindrical
case. The formula we get (Theorem 6.18) involves the η-invariants of two
operators. In Section 6.3.2, we investigate the relation between these two
operators, resulting in an alternative index formula involving the spectral flow
(Proposition 6.22). We finish this section by deriving simplified formulae in
Section 6.3.3 under special assumptions on the cross-section at infinity.
6.3.1 Main index formula
Here we prove a general index formula containing η-invariants.
Theorem 6.18 (Theorem 1.5). Let M be an asymptotically cylindrical 8-mani-
fold with an asymptotically cylindrical Spin(7)-structure asymptotic to (0,∞)×N ,
where N is a 7-manifold with torsion-free G2-structure, let X be an asymptot-
ically cylindrical Cayley submanifold of M asymptotic to (0,∞) × Y , where
Y is a closed associative submanifold of N , and let D : Γ(νMX) → Γ(E) be
defined as in (3.1).
Then
indλD =
1
2χ(X)−
1
2σ(X)−
∫
X
e(νMX)− dim ker D˜2 +
η(D˜)− η(Bev)
2 , (6.15)
where λ > 0 is such that [−λ, 0) contains no eigenvalue of D˜. Here
(i) χ(X) is the Euler characteristic of X,
(ii) σ(X) is the signature of X (the signature of the non-degenerate quadratic
form induced by the cup-product on the image of H2cs(X) in H2(X)),
(iii) e(νMX) is the Euler density of the normal bundle νMX (see Section 6.1.3),
125
(iv) D˜ : Γ(νNY )→ Γ(νNY ) is the (twisted) Dirac operator that arises as the
linearisation of the deformation map for associative submanifolds, defined
by
D˜s :=
3∑
i=1
ei ×∇⊥eis ,
where (ei)i=1,2,3 is any local orthonormal frame of Y and ∇⊥ is the induced
connection on νNY , and
(v) Bev : Ω0(Y )⊕Ω2(Y )→ Ω0(Y )⊕Ω2(Y ) is the odd signature operator on Y ,
defined by
Bev(f, α) := (∗dα,−∗df − d∗α) .
Proof. Like in the closed case (see the proof of Proposition 3.4), D is a negative
twisted Dirac operator. Furthermore, D˜ : Γ(νNY )→ Γ(νNY ) satisfies
Ds = −u× (−∇us+ D˜s) ,
where u := − ∂
∂t
. So the Atiyah–Patodi–Singer Index Theorem (Theorem 6.9,
(6.11)) implies
indλD = −
∫
X
(Aˆ(X) ch(F ))4 − dim ker D˜ − η(D˜)2 . (6.16)
Like in the closed case (see the proof of Proposition 3.4), we get
−
∫
X
(Aˆ(X) ch(F ))4 =
1
2
∫
X
e(X)− 16
∫
X
p1(X)−
∫
X
e(νMX) . (6.17)
Now the signature theorem for asymptotically cylindrical manifolds (The-
orem 6.10) implies
1
3
∫
X
p1(X) = σ(X) + η(Bev) . (6.18)
Furthermore, the generalised Gauss–Bonnet–Chern Theorem (Theorem 6.14,
see also [Mel93, Lemma 9.2]) yields
∫
X
e(X) = χ(X) . (6.19)
Now (6.15) follows from the above formulae.
126
6.3.2 Relation between Bev and D˜ and alternative index
formula
Here we investigate the relation between the two operators Bev and D˜ that
appear in the index formula of Theorem 6.18. We derive an alternative index
formula involving the spectral flow (Proposition 6.22).
Lemma 6.19. The operator Bev is the Dirac operator associated to the Dirac
bundle Λ0Y ⊕ Λ2Y with the Clifford multiplication
v · (f, α) = (v ⌟ ∗α,−f∗v♭ − v♭ ∧ ∗α)
for v ∈ TY , f ∈ Λ0Y , α ∈ Λ2Y and the Levi-Civita connection.
Proof. We have
3∑
i=1
ei · (∇eif,∇eiα) =
3∑
i=1
(ei ⌟ ∗∇eiα,−∇eif∗ei − ei ∧ ∗∇eiα)
=
⎛⎝∗ 3∑
i=1
ei ∧∇eiα,−∗
3∑
i=1
ei ∧∇eif + ∗
3∑
i=1
ei ⌟∇eiα
⎞⎠
= (∗dα,−∗df − ∗δα)
= Bev(f, α) .
Lemma 6.20. Let s ∈ Γ(νNY ) be a non-vanishing section with pointwise
norm 1 (which always exists by [Ste51, Corollary 29.3] since Y is 3-dimensional
and νNY has rank 4), and let
h : Λ0Y ⊕ Λ2Y → νNY , (f, α) ↦→ fs+ s× (∗α)♯ , (6.20)
where ‘×’ is the cross product of the G2-structure φ := ∂∂t ⌟ Φ∞ on N . Then
h(v · (f, α)) = v × h(f, α)
for v ∈ TY , f ∈ Λ0Y , and α ∈ Λ2Y . So h defines an isomorphism of bundles
of Clifford modules.
127
Proof. There is a vector-valued 3-form χ ∈ Ω3(N, TN) such that [AS08, (6)]
χ(u, v, w) = −u× (v × w)− g(u, v)w + g(u,w)v (6.21)
for all u, v, w ∈ Γ(TN). Hence
v × (s× w) = g(v, w)s− s× (v × w)
for all v, w ∈ Γ(TY ) since s ∈ Γ(νNY ). So
h(v · (f, α)) = h((v ⌟ ∗α,−f∗v♭ − v♭ ∧ ∗α))
= (v ⌟ ∗α)s− s× (∗(f∗v♭ + v♭ ∧ ∗α))♯
= g(v, (∗α)♯)s+ v × fs− s× (v × (∗α)♯)
= v × (fs+ s× (∗α)♯)
= v × h(f, α)
since
(∗(v♭ ∧ w♭))♯ = v × w
for all v, w ∈ Γ(TY ) as Y is associative.
Corollary 6.21. Let s ∈ Γ(νNY ) be a non-vanishing section with pointwise
norm 1, and let
Dˆ : Γ(νNY )→ Γ(νNY ) , z ↦→ h(Bev(h−1(z))) , (6.22)
where h : Λ0Y ⊕Λ2Y → νNY is defined in (6.20). Then the operators Dˆ and D˜
have the same symbol. Moreover, if s is parallel, then Dˆ = D˜.
Proposition 6.22. Let s ∈ Γ(νNY ) be a non-vanishing section with point-
wise norm 1, let Dˆ : Γ(νNY ) → Γ(νNY ) be defined as in (6.22), and let
Dt : Γ(νNY ) → Γ(νNY ) (0 ≤ t ≤ 1) be a smooth family of linear differen-
tial operators with the same symbol such that D0 = Dˆ and D1 = D˜.
128
Then
indλD =
1
2χ(X)−
1
2σ(X)− e(νMX, s) + sf(Dt)−
dim ker D˜
2 , (6.23)
where sf(Dt) is the spectral flow of the family (Dt), that is, the number of
eigenvalues that go from < 0 to ≥ 0 minus the number of eigenvalues that go
from ≥ 0 to < 0.
Proof. First note that the spectral flow does not depend on the choice of the
smooth family [APS76, Theorem (7·4)]. So let µ : R → [0, 1] be a smooth
function such that µ(t) = 0 for t ≤ −1 and µ(t) = 1 for t ≥ 1, and let
P : Γ(R× νNY )→ Γ(R× νNY ) be defined (over the manifold R× Y ) by
Ps := −∇ts+ µ(t)D˜s+ (1− µ(t))Dˆs . (6.24)
Then indλ P + 12(dim ker D˜ + dimker Dˆ) is equal to the spectral flow from Dˆ
to D˜ [APS76, Section 7], where λ > 0 is such that [−λ, 0) contains no eigenvalue
of Bev or D˜.
Note that P is the negative Dirac operator with respect to the connection
µ(t)∇+ (1− µ(t))∇ˆ, where ∇ˆ := (h−1)∗∇. So the Atiyah–Patodi–Singer Index
Theorem (Theorem 6.9, (6.11)) implies
indλ P = −
∫
R×Y
Aˆ(R× Y ) ch(F |Y )− dim ker D˜ − η(D˜)2
− dim ker Dˆ + η(Dˆ)2 .
(6.25)
Furthermore,
η(Dˆ) = η(Bev) . (6.26)
As in the proof of Proposition 3.4, we get
−
∫
R×Y
(Aˆ(R× Y ) ch(F |Y ))4
= 12
∫
R×Y
e(R× Y )− 16
∫
R×Y
p1(R× Y )−
∫
R×Y
e(R× νNY ) .
(6.27)
129
Here e(R × Y ) = p1(R × Y ) = 0. The generalised Gauss–Bonnet–Chern
Theorem (Theorem 6.14) implies
∫
X
e(νMX) = e(νMX, s)−
∫
Y
Θ(F∇|Y ,∇s)
= e(νMX, s) +
∫
R×Y
e(R× νNY ) . (6.28)
Now (6.23) follows from (6.15) using the above formulae.
Remark 6.23. Note that both e(νMX, s) and sf(Dt) depend on the choice
of s but their difference does not. In particular, if s and s′ are two non-
vanishing sections of νNY (w.l.o.g. with pointwise norm 1) that are homotopic
through non-vanishing sections, then e(νMX, s) = e(νMX, s′). Hence also
sf(Dt) = sf(D′t).
6.3.3 Additional assumptions
Here we prove simplified versions of the index formula under special assumptions
on the cross-section at infinity. Examples where these assumptions are satisfied
can usually be found if the holonomy of the cross-section at infinity of the
Spin(7)-manifold is a proper subgroup of G2. In particular, we will apply
the results of this section in Section 7.1 to asymptotically cylindrical Cayley
submanifolds inside the asymptotically cylindrical Riemannian 8-manifolds
with holonomy Spin(7) constructed by Kovalev in [Kov13].
Proposition 6.24. Let M be an asymptotically cylindrical manifold with an
asymptotically cylindrical Spin(7)-structure asymptotic to (0,∞)×N , where
N is a 7-manifold with torsion-free G2-structure, let X be an asymptotically
cylindrical Cayley submanifold asymptotic to (0,∞)× Y , where Y is a closed
associative submanifold of N , and let D : Γ(νMX) → Γ(E) be defined as
in (3.1). Suppose that the normal bundle νNY of Y in N has a non-trivial
parallel section s ∈ Γ(νNY ).
130
Then
indλD =
1
2χ(X)−
1
2σ(X)− e(νMX, s)−
b0(Y ) + b1(Y )
2 , (6.29)
where λ > 0 is such that [−λ, 0) contains no eigenvalue of Bev.
Proof. We have Dˆ = D˜ by Corollary 6.21 since s is parallel. Hence η(D˜) = η(Bev)
and dim ker D˜ = dimkerBev = b0(Y ) + b1(Y ). So (6.29) follows from (6.15)
using the generalised Gauss–Bonnet–Chern Theorem (Theorem 6.14).
Proposition 6.25. Let M be an asymptotically cylindrical manifold with an
asymptotically cylindrical Spin(7)-structure asymptotic to (0,∞)×N , where
N is a 7-manifold with torsion-free G2-structure, let X be an asymptotically
cylindrical Cayley submanifold asymptotic to (0,∞)× Y , where Y is a closed
associative submanifold of N , let D : Γ(νMX) → Γ(E) be defined as in (3.1),
and let X be the compactification of X so that ∂X = Y . Suppose that there
are a 4-dimensional manifold X˜ and a free involution ρ : X˜ → X˜ such that
X ∼= X˜/⟨ρ⟩. Let π : X˜ → X˜/⟨ρ⟩ ∼= X be the projection, and let Y˜ := π−1(Y )
(so that ∂X˜ = Y˜ ). Suppose further that (π|Y˜ )∗(νNY ) has a non-trivial parallel
section s ∈ Γ((π|Y˜ )∗(νNY )) such that s ◦ ρ = −s.
Then
indλD =
1
2χ(X) +
1
2σ(X)−
1
2σ(X˜)−
1
2e(π
∗(νMX), s)
+ b
0(Y ) + b1(Y )
2 −
b0(Y˜ ) + b1(Y˜ )
2 ,
(6.30)
where λ > 0 is such that [−λ, 0) contains no eigenvalue of D˜.
Proof. The section s defines a flat subbundle K of νNY of rank 1. Now
Lemma 6.20 defines an isomorphism νNX ∼= (Λ0Y ⊕ Λ2Y )⊗K. Furthermore,
the connections are identified via this isomorphism since s is parallel and
the G2-structure is torsion-free. In particular, under this isomorphism, D˜ is
identified with the operator
Ω0(Y,K)⊕ Ω2(Y,K)→ Ω0(Y,K)→ Ω2(Y,K) ,
(f, α) ↦→ (∗dα,−∗df − d∗α) .
131
Hence
η(D˜)− η(Bev) = ρξ(Y ) , (6.31)
where ξ : π(Y ) → {±1} is the holonomy representation of K and ρξ(Y ) is
defined in [APS75b, Theorem (2·4)]. Now [APS75b, Theorem (2·4)] and
[APS75b, Lemma (2·5)] imply
ρξ(Y ) = 2σ(X)− σ(X˜) . (6.32)
Also note that
dim ker D˜ = b0(Y˜ )−ρ + b1(Y˜ )−ρ = (b0(Y˜ )− b0(Y )) + (b1(Y˜ )− b1(Y )) . (6.33)
Furthermore,
∫
X
e(νMX) =
1
2
∫
X˜
e(π∗(νMX)) =
1
2e(π
∗(νMX), s) (6.34)
by the generalised Gauss–Bonnet–Chern Theorem (Theorem 6.14) since π : X˜ → X
is a 2-fold cover and s is a non-trivial parallel section of π∗(νMX)|∂X˜ . Now
(6.30) follows from (6.15) using the above formulae.
Proposition 6.26. Let M be an asymptotically cylindrical manifold with an
asymptotically cylindrical Spin(7)-structure asymptotic to (0,∞)×N , where
N is a 7-manifold with torsion-free G2-structure, let X be an asymptotically
cylindrical Cayley submanifold asymptotic to (0,∞)× Y , where Y is a closed
associative submanifold of N , and let D : Γ(νMX) → Γ(E) be defined as
in (3.1). Suppose that N ∼= S1 × C, where C is a Calabi–Yau 3-fold, and that
Y ∼= S1 × Z, where Z is a complex curve in C. Furthermore, let s ∈ Γ(νNY )
be a non-vanishing section that is invariant under rotations of S1.
Then
indλD =
1
2χ(X)−
1
2σ(X)− e(νMX, s)− dimCH
0(Z, νCZ) , (6.35)
where λ > 0 is such that [−λ, 0) contains no eigenvalue of D˜.
132
Remark 6.27. Under the above hypotheses, consider the map
(0,∞)× S1 × C → C× C , (t, eiθ, x) ↦→ (e−t+iθ, x) .
This is a diffeomorphism onto {z ∈ C : 0 < |z| < 1} × C. Let M be the
compactification of M that is obtained by using this diffeomorphism and
extending it to {0} × C (so M is a closed manifold with M \ M ∼= C).
Extend X similarly to X with X \X ∼= Z. Then we can identify νMX on the
cylindrical end with {z ∈ C : |z| < 1} × νCZ. Now the hypothesis that s is
invariant under rotations of S1 means that s induces a non-vanishing section
of νCZ. Hence e(νMX, s) = e(νMX), which is equal to the self-intersection
number of X in M .
Proof. First note that
η(Bev) = 0 (6.36)
by Lemma 6.12. Let J : νNY → νNY be the parallel, orthogonal almost complex
structure on νNY induced by the complex structure on νCZ, and let
v := − ∂
∂θ
∈ Γ(TY ) ,
where θ is the coordinate on the S1-factor. Then J(r) = v×r for all r ∈ Γ(νNY ).
Furthermore,
w × Jr + J(w × r) = w × (v × r) + v × (w × r) = −2g(v, w)r
for all w ∈ Γ(TY ) by [AS08, (6)]. So
D˜(Jr) + J(D˜r) = −2∇vr .
Let ρ : Y → Y , ρ(eiθ, x) := (e−iθ, x) for (eiθ, x) ∈ S1×Z ∼= Y . Then v ◦ ρ = −v.
So ∇v(r ◦ ρ) = −(∇vr) ◦ ρ. Hence
D˜(r ◦ ρ)− (D˜r) ◦ ρ = −2(J∇vr) ◦ ρ .
133
Thus, if
J˜ : Γ(νNY )→ Γ(νNY ) , r ↦→ (Jr) ◦ ρ ,
then
D˜(J˜r) = D˜((Jr) ◦ ρ) = (D˜(Jr)) ◦ ρ+ 2(∇vr) ◦ ρ = −(J(D˜r)) ◦ ρ = −J˜(D˜r) .
This implies that the spectrum of D˜ is symmetric. Hence also
η(D˜) = 0 . (6.37)
Note that ker D˜ is isomorphic to the space of holomorphic sections of the
normal bundle νCZ [CHNP15, Lemma 5.11], and hence
dim ker D˜ = dimRH0(Z, νCZ) = 2 dimCH0(Z, νCZ) . (6.38)
Furthermore, ∇vs = 0 since s is invariant under rotations of S1. So the
generalised Gauss–Bonnet–Chern Theorem (Theorem 6.14) implies
∫
X
e(νMX) = e(νMX, s) (6.39)
since N = S1 × C is endowed with the product metric. Now (6.35) follows
from (6.15) using the above formulae.
6.4 Varying the Spin(7)-structure
In this section, we prove that for a generic asymptotically cylindrical Spin(7)-
structure, the moduli space of asymptotically cylindrical Cayley submanifolds
is a finite-dimensional smooth manifold. There are various versions, depending
on whether the asymptotic limits of the Spin(7)-structures and the Cayley
submanifolds are fixed. In the following theorems, we use the C∞λ -topology
(Theorems 6.28 and 6.30) or the Cˆ∞λ -topology (Theorem 6.31) for the space of
all Spin(7)-structures. We start with the version where the asymptotic limits
are fixed.
134
Theorem 6.28 (Theorem 1.6). Let M be a smooth, asymptotically cylindrical
8-manifold with a smooth, asymptotically cylindrical Spin(7)-structure asymp-
totic to (0,∞) × N , where N is a 7-manifold with a G2-structure, let X be
a smooth, asymptotically cylindrical Cayley submanifold of M asymptotic to
(0,∞)×Y , where Y is a closed associative submanifold of N , and let 0 < α < 1.
If λ > 0 is small enough, then for every generic, smooth, asymptotically
cylindrical Spin(7)-structure Ψ with rate λ that is C2,αλ -close to Φ (with the
same asymptotic limit as Φ) and inducing the same metric as Φ, the moduli
space of all smooth, asymptotically cylindrical Cayley submanifolds of (M,Ψ)
with rate λ that are C2,αλ -close to X (with the same asymptotic limit as X) is
either empty or a smooth manifold of dimension indλD, where D is defined
in (3.1).
The statement of the above theorem remains true for the larger class of all
smooth, asymptotically cylindrical Spin(7)-structures Ψ with rate λ that are
C2,αλ -close to Φ (not necessarily inducing the same metric as Φ).
Proof. Recall from the proof of Theorem 3.10 that there are an open tubular
neighbourhood V ⊆ Λ41M ⊕Λ47M ⊕Λ435M of the 0-section and a smooth bundle
morphism Θ : V → Λ4M which parametrises Spin(7)-structures on M that are
C0-close to Φ. Let
πE : Λ2M |X → E (6.40)
denote the orthogonal projection, where the vector bundle E of rank 4 over X
is defined as in (2.23). Furthermore, let U ⊆ νMX be defined as in Lemma 6.5,
and let
F˜ : Γλ(U)⊕ Γλ(V )→ Γλ(E) , (s, χ) ↦→ πE((exps)∗(τΘ(χ))) , (6.41)
where τΘ(χ) ∈ Ω4(M,Λ2M) is defined as in (2.16) with respect to the Spin(7)-
structure Θ(χ). Then
(dF˜ )(0,0)(s, 0) = Ds (6.42)
for s ∈ Γλ(νMX) by the proof of Theorem 3.1, and
(dF˜ )(0,0)(0, χ) = −πE(e|X) (6.43)
135
by Lemma 3.12, where e ∈ Γλ(Λ27M) and χ := g(τΦ, e) ∈ Γλ(Λ47M). By
Proposition 6.7, F˜ extends to a map
F˜ 2,αλ : C
2,α
λ (U)⊕ C2,αλ (V )→ C1,αλ (E) (6.44)
of class C1. For fixed χ ∈ C2,αλ (V ), the equation
F˜ 2,αλ (s, χ) = πE((exps)∗(τΘ(χ))) = 0 (6.45)
is a nonlinear partial differential equation of order 1 in s. Furthermore, the
linearisation at 0 is elliptic for χ = 0. Hence there is a C2,αλ -neighbourhood
U˜1 ⊆ C2,αλ (U)⊕ C2,αλ (V ) of (0, 0) such that
(dF˜ 2,αλ )(s,χ)|C2,α
λ
(νMX) : C
2,α
λ (νMX)→ C1,αλ (E) (6.46)
is an asymptotically cylindrical elliptic differential operator of order 1 for all
(s, χ) ∈ U˜1. Note further that the operators (6.46) and D2,αλ are C1,αλ -asymptotic
to the same translation-invariant operator. In particular, if λ > 0 is small
enough, then the operator (6.46) is Fredholm for all (s, χ) ∈ U˜1, and hence its
image is a closed subspace with finite codimension.
Lemma 6.29. Suppose that (s, χ) ∈ U˜1 satisfies (6.45) and χ ∈ Ck,αλ (V ) (for
some k ≥ 2). Then s ∈ Ck+1,αλ (νMX).
Proof. Consider the equation
D∗F˜ 2,αλ (s, χ) = 0 , (6.47)
which is a partial differential equation of order 2 in s. This equation is quasi-
linear, that is, we can write the equation in the form L(s,χ)s = 0, where L(s,χ) is
a linear elliptic differential operator whose coefficients depend on s only up to
first derivatives. So L(s,χ) is C1,αλ -asymptotic to a translation-invariant operator
whose coefficients are in C1,α. Hence s ∈ C3,αλ (νMX) [MP78, Theorem 6.3] (see
also [Nor08, Theorem 4.2.22]). By induction, we get s ∈ Ck+1,αλ (νMX) since
136
L(s,χ) also depends on χ up to first derivatives so that the maximum regularity
we can expect are Ck−1,αλ -coefficients.
Furthermore, there is a C2,αλ -neighbourhood U˜2 ⊆ C2,αλ (U)⊕C2,αλ (V ) of (0, 0)
such that if (s, χ) ∈ U˜2 satisfies (6.45) and χ ∈ Ck,αλ (V ) (for some k ≥ 2), then
the image of the operator
(dF˜ 2,αλ )(s,χ)|Ck,α
λ
(Λ41M⊕Λ47M⊕Λ435M) : C
k,α
λ (Λ41M ⊕ Λ47M ⊕ Λ435M)→ C1,αλ (E)
is exactly Ck,αλ (E) since eˆxps : X → eˆxps(X) is an asymptotically cylindrical diffeo-
morphism, and hence the map Ck,αλ (Λ4Xs⊗Es)→ Ck,αλ (E), χ ↦→ πE((eˆxps)∗(χ))
is a linear isomorphism (note that s ∈ Ck+1,αλ (νMX) by Lemma 6.29), where Es
is defined as in (2.23) with respect to the Cayley submanifold Xs := eˆxps(X)
(compare the proof of Theorem 3.10).
For k ≥ 2, let
F˜ k,αλ := F˜
2,α
λ |C2,α
λ
(U)⊕Ck,α
λ
(V ) : C
2,α
λ (U)⊕ Ck,αλ (V )→ C1,αλ (E) .
Then F˜ k,αλ is of class Ck−1 by Proposition 6.7. Let U˜ := U˜1∩U˜2. Then the above
argumentation shows that if (s, χ) ∈ U˜ ∩ (C2,αλ (U)⊕ Ck,αλ (V )) satisfies (6.45),
then
(dF˜ k,αλ )(s,χ) : C
2,α
λ (νMX)⊕ Ck,αλ (Λ41M ⊕ Λ47M ⊕ Λ435M)→ C1,αλ (E)
is surjective by Lemma 3.8 since Ck,αλ (E) is dense in C
1,α
λ (E), and
(dF˜ k,αλ )(s,χ)|C2,α
λ
(νMX) = (dF˜
2,α
λ )(s,χ)|C2,α
λ
(νMX) : C
2,α
λ (νMX)→ C1,αλ (E)
is Fredholm. Furthermore, the Fredholm index is the same for all of these
operators as we may assume w.l.o.g. that U˜1 is connected. So the Fredholm
index is indλD since D = (dF˜ )(0,0)|Γλ(νMX).
Let U˜3 ⊆ C2,αλ (U) and U˜4 ⊆ C2,αλ (V ) be open neighbourhoods of 0 such that
U˜3 × U˜4 ⊆ U˜ , let U2,αλ be the set of all χ ∈ U˜4 such that the operator (6.46) is
137
surjective for all s ∈ U˜3, and define Uk,αλ := U2,αλ ∩Ck,αλ (Λ41M⊕Λ47M⊕Λ435M) for
k ≥ 2. Let k0 := max{indλD+1, 2}. Then Theorem 3.7 implies that Uk,αλ is the
intersection of countably many open dense subsets of Ck,αλ (Λ41M⊕Λ47M⊕Λ435M)
for k ≥ k0. So if we define U∞λ :=
⋂∞
k=k0 U
k,α
λ , then U∞λ is the intersection of
countably many open dense subsets of C∞λ (Λ41M⊕Λ47M⊕Λ435M) by Lemma 6.6.
So for every generic χ ∈ C∞λ (V ) that is C2,αλ -close to 0, the moduli space of
all solutions s ∈ C∞λ (U) of (6.45) that are C2,αλ -close to 0 is either empty or a
smooth manifold of dimension indλD.
In the following version, we fix the asymptotic limits of the Spin(7)-structures
but allow the asymptotic limits of the Cayley submanifolds to vary.
Theorem 6.30. Let M be a smooth, asymptotically cylindrical 8-manifold with
a smooth, asymptotically cylindrical Spin(7)-structure asymptotic to (0,∞)×N ,
where N is a 7-manifold with a G2-structure, let X be a smooth, asymptotically
cylindrical Cayley submanifold of M asymptotic to (0,∞) × Y , where Y is
a closed associative submanifold of N , and let 0 < α < 1. Suppose that the
moduli space of all smooth associative submanifolds of N that are C2,α-close
to Y contains a smooth manifold Y with Y ∈ Y.
If λ > 0 is small enough, then for every generic, smooth, asymptotically
cylindrical Spin(7)-structure Ψ with rate λ that is C2,αλ -close to Φ (with the
same asymptotic limit as Φ) and inducing the same metric as Φ, the moduli
space of all smooth, asymptotically cylindrical Cayley submanifolds of (M,Ψ)
that are Cˆ2,αλ -close to X and asymptotic to (0,∞) × Y ′ with rate λ for some
Y ′ ∈ Y is either empty or a smooth manifold of dimension indλD + dimY,
where D is defined in (3.1).
Here Cˆ2,αλ (νMX) := C
2,α
λ (νMX) ⊕ C2,α(νNY ) is extended weighted Hölder
space with weight λ (see Section 6.1.1). The statement of the above theorem
remains true for the larger class of all smooth, asymptotically cylindrical
Spin(7)-structures Ψ with rate λ that are C2,αλ -close to Φ (not necessarily
inducing the same metric as Φ).
138
Proof. Let ρ : R → [0, 1] be a smooth function with ρ(t) = 0 for t ≤ 1 and
ρ(t) = 1 for t ≥ 2. For sˆ ∈ Γ(νNY ), view it as a translation-invariant section
of R × νNY (i.e., ∇tsˆ = 0), and consider ρ(t)sˆ ∈ Γ(νMX). Furthermore,
consider Y as a submanifold of C2,α(νNY ), and let
F˜ : Γλ(U)⊕ Y ⊕ Γλ(V )→ Γλ(E) ,
(s, sˆ, χ) ↦→ πE((exps+ρ(t)sˆ)∗(τΘ(χ))) .
(6.48)
Note that the image is indeed in Γλ(E) since Y consists of associative sub-
manifolds (which implies that (expsˆ)∗(τΦ∞) = 0 on R × Y for sˆ ∈ Y). We
have
(dF˜ )(0,0,0)(s, sˆ, 0) = Ds+ ρ′(t)sˆ (6.49)
for s ∈ Γλ(νMX) and sˆ ∈ T0Y by the proof of Theorem 3.1 since D˜sˆ = 0 as Y
consists of associative submanifolds and hence sˆ is an infinitesimal associative
deformation. We further have
(dF˜ )(0,0,0)(0, 0, χ) = −πE(e|X) (6.50)
by Lemma 3.12, where e ∈ Γλ(Λ27M) and χ := g(τΦ, e) ∈ Γλ(Λ47M). By
Proposition 6.7, F˜ extends to a map
F˜ 2,αλ : C
2,α
λ (U)⊕ Y ⊕ C2,αλ (V )→ C1,αλ (E) (6.51)
of class C1. Note that the operator
(dF˜ 2,αλ )(0,0,0)|C2,α
λ
(νMX)⊕T0Y : C
2,α
λ (νMX)⊕ T0Y → C1,αλ (E) (6.52)
is Fredholm with index indλD + dimY since the operator
C2,αλ (νMX)⊕ T0Y → C1,αλ (E) , (s, sˆ) ↦→ Ds
is Fredholm with index indλD + dimY and the operator
C2,αλ (νMX)⊕ T0Y → C1,αλ (E) , (s, sˆ) ↦→ ρ′(t)sˆ
139
is compact as the embedding operator C2,α(νNY ) → C1,α(νNY ) is compact
[RS82, Proposition 1 in Section 2.3.2.6]. Hence the operator
(dF˜ 2,αλ )(s,sˆ,χ)|C2,α
λ
(νMX)⊕TsˆY : C
2,α
λ (νMX)⊕ TsˆY → C1,αλ (E) (6.53)
is Fredholm for all (s, sˆ,χ)∈C2,αλ (U)⊕Y⊕C2,αλ (V ) that are (C2,αλ ⊕C2,α⊕C2,αλ )-close
to (0, 0, 0). The remaining arguments are analogous to the proof of Theorem 6.28
above.
In the following version, the asymptotic limits of both the Spin(7)-structures
and the Cayley submanifolds are allowed to vary.
Theorem 6.31. Let M be a smooth, asymptotically cylindrical 8-manifold with
a smooth, asymptotically cylindrical Spin(7)-structure asymptotic to (0,∞)×N ,
where N is a 7-manifold with a G2-structure, let X be a smooth, asymptotically
cylindrical Cayley submanifold of M asymptotic to (0,∞)× Y , where Y is a
closed associative submanifold of N , and let 0 < α < 1.
If λ > 0 is small enough, then for every generic, smooth, asymptotically
cylindrical Spin(7)-structure Ψ with rate λ that is Cˆ2,αλ -close to Φ (not ne-
cessarily with the same asymptotic limit as Φ) and inducing the same metric
as Φ, the moduli space of all smooth, asymptotically cylindrical Cayley sub-
manifolds of (M,Ψ) with rate λ that are Cˆ2,αλ -close to X (not necessarily with
the same asymptotic limit as X) is either empty or a smooth manifold of
dimension indλD, where D is defined in (3.1).
The statement of the above theorem remains true for the larger class of all
smooth, asymptotically cylindrical Spin(7)-structures Ψ with rate λ that are
Cˆ2,αλ -close to Φ (not necessarily inducing the same metric as Φ).
Proof. The set of all G2-structures is open [Joy00, page 243]. In particular,
there is an open tubular neighbourhood Vˆ ⊆ Λ3N of the 0-section such that
φ+ϕ is a G2-structure for all ϕ ∈ Γ(Vˆ ), where φ := ∂∂t ⌟Φ∞ is the G2-structure
on N . Let Uˆ ⊆ νNY be an open tubular neighbourhood of the 0-section
140
such that the exponential map exp |Uˆ : Uˆ → exp(Uˆ) is a diffeomorphism, let
ρ : R→ [0, 1] be as in the proof of Theorem 6.30 above, and let
F˜ : Γλ(U)⊕ Γ(Uˆ)⊕ Γλ(V )⊕ Γ(Vˆ )→ Γˆλ(E) ∼= Γλ(E)⊕ Γ(νNY ) ,
(s, sˆ, χ, ϕ) ↦→ πE((exps+ρ(t)sˆ)∗(τΘ(χ+ρ(t)χˆ(ϕ)))) ,
(6.54)
where χˆ(ϕ) ∈ Ω41(R×N)⊕ Ω47(R×N)⊕ Ω435(R×N) is such that
Θ(χˆ(ϕ)) = dt ∧ (φ+ ϕ) + ∗g(φ+ϕ)(φ+ ϕ)
for ϕ ∈ Γ(Vˆ ). Note that the isomorphism Γˆλ(E) ∼= Γλ(E)⊕ Γ(νNY ) is induced
by the isomorphism νNY → E|Y , sˆ ↦→ ∂∂t × sˆ of vector bundles. We have
(dF˜ )(0,0,0,0)(s, sˆ, 0, 0) = (Ds+ ρ′(t)sˆ, D˜sˆ)
for s ∈ Γλ(νMX) and sˆ ∈ Γ(νNY ) by the proof of Theorem 3.1. We further
have
(dF˜ )(0,0,0,0)(0, 0, χ, ϕ) = (−πE(e|X), πν(v|Y ))
by Lemma 3.12 and [Gay14, (10) and (12)], where e ∈ Γλ(Λ27M), v ∈ Γ(TN),
χ := g(τΦ, e) ∈ Γλ(Λ47M), and ϕ := ∗N(φ ∧ v♭) ∈ Γ(Λ37N). By Proposition 6.7,
F˜ extends to a map
F˜ 2,αλ : C
2,α
λ (U)⊕ C2,α(Uˆ)⊕ C2,αλ (V )⊕ C2,α(Vˆ )→ C1,αλ (E)⊕ C1,α(νNY )
of class C1. Note that the operator
(dF˜ 2,αλ )(0,0,0,0)|C2,α
λ
(νMX)⊕C2,α(νNY ) :
C2,αλ (νMX)⊕ C2,α(νNY )→ C1,αλ (E)⊕ C1,α(νNY )
(6.55)
is Fredholm with index indλD since the operators D2,αλ : C
2,α
λ (νMX)→ C1,αλ (E)
and D˜2,α : C2,α(νNY )→ C1,α(νNY ) are Fredholm, ind D˜ = 0, and the operator
C2,α(νNY )→ C1,αλ (E) , sˆ ↦→ ρ′(t)sˆ
141
is compact as the embedding operator C2,α(νNY ) → C1,α(νNY ) is compact
[RS82, Proposition 1 in Section 2.3.2.6]. Hence the operator
(dF˜ 2,αλ )(s,sˆ,χ,ϕ)|C2,α
λ
(νMX)⊕C2,α(νNY ) :
C2,αλ (νMX)⊕ C2,α(νNY )→ C1,αλ (E)⊕ C1,α(νNY )
(6.56)
is Fredholm for all (s, sˆ, χ, ϕ) ∈ C2,αλ (U) ⊕ C2,α(Uˆ) ⊕ C2,αλ (V ) ⊕ C2,α(Vˆ ) that
are (C2,αλ ⊕ C2,α ⊕ C2,αλ ⊕ C2,α)-close to (0, 0, 0, 0). The remaining arguments
are analogous to the proof of Theorem 6.28 above.
142
7 Examples of asymptotically
cylindrical Cayley submanifolds
In this chapter, we present and discuss some examples for the deformation theory
of asymptotically cylindrical Cayley submanifolds. In particular, in Section 7.1,
we provide examples of asymptotically cylindrical Cayley submanifolds inside
the asymptotically cylindrical Riemannian 8-manifolds with holonomy Spin(7)
constructed by Kovalev in [Kov13] and calculate the indices of these Cayley
submanifolds. We further relate the deformation theory of asymptotically
cylindrical Cayley submanifolds to the deformation theories of asymptotically
cylindrical coassociative and special Lagrangian submanifolds in Section 7.2.
7.1 Examples in manifolds with holonomy Spin(7)
In this section, we provide examples of asymptotically cylindrical Cayley
submanifolds inside the asymptotically cylindrical Riemannian 8-manifolds
with holonomy Spin(7) constructed by Kovalev in [Kov13] and calculate the
indices of these Cayley submanifolds.
7.1.1 Weighted projective spaces and connected sum
Let a0, . . . , an be positive integers with highest common factor 1. The weighted
projective space CPna0,...,an is defined as the quotient of Cn+1 \ {0} by the
equivalence relation
(z0, . . . , zn) ∼ (λa0z0, . . . , λanzn) for all λ ∈ C \ {0}.
143
We denote by [z0, . . . , zn] the equivalence class of (z0, . . . , zn). A weighted
projective space is a complex orbifold.
The definition of an orbifold V is analogous to that of a manifold except
that a neighbourhood Ux of a point x ∈ V is homeomorphic to Rn/Γx, where
Γx is a finite group acting faithfully on Rn. The smooth locus consists of all
points that have a neighbourhood homeomorphic to Rn. All other points are
singular points.
IfM is a manifold and n is a positive integer, we write nM for the connected
sum of n copies of M . The connected sum does not depend on the embeddings
of the discs along which the manifolds are glued. Furthermore, the connected
sum is commutative and associative up to orientation preserving diffeomorph-
ism. Note that nM is cobordant to ∐ni=1M , the disjoint union of n copies
of M . In particular, if M is null-cobordant, so is nM , and a null-cobordism
of M determines a null-cobordism of nM by composing with the cobordism
between nM and ∐ni=1M .
7.1.2 Cayley submanifold in a Spin(7)-manifold
constructed from CP41,1,1,1,4
Here we construct an asymptotically cylindrical Cayley submanifold inside the
asymptotically cylindrical Spin(7)-manifold constructed in [Kov13, Section 6.2]
and compute the index. Let V := CP41,1,1,1,4. This is an orbifold with unique
singular point p0 := [0, 0, 0, 0, 1]. Let
C := {[z0, . . . , z4] ∈ V : z80 + z81 + z82 + z83 + z24 = 0} ,
C ′ := {[z0, . . . , z4] ∈ V : z80 − z81 + 2z82 − 2z83 + iz24 = 0} , and
Σ := C ∩ C ′ .
Note that both C and C ′ are smooth and that Σ is a complete intersection.
Let V˜ be the blow-up of V along Σ. Then C lifts to a submanifold C˜ of V˜
which is isomorphic to C.
144
Define two antiholomorphic involutions
ρ1 : V → V , [z0, . . . , z4] ↦→ [z1,−z0, z3,−z2, z4] and
ρ2 : V → V , [z0, . . . , z4] ↦→ [z1, z0, z3, z2, z4] .
Note that ρ1ρ2 = ρ2ρ1. Furthermore, both ρ1 and ρ2 preserve p0, C, and C ′,
and hence also Σ. So they lift to antiholomorphic involutions ρ˜1 and ρ˜2 of V˜
such that ρ˜1ρ˜2 = ρ˜2ρ˜1. Note further that ρ1 fixes only p0.
Let M1 := (V˜ \ (C˜ ∪ {p0}))/⟨ρ˜1⟩, let W be the blow-up of C4/Z4 at 0, where
Z4 acts on C4 by multiplication by i, let
ρ′2 : C4/Z4 → C4/Z4 , [z1, . . . , z4] ↦→ [z1, z2, z3, z4] ,
ρ′3 : C4/Z4 → C4/Z4 , [z1, . . . , z4] ↦→ [z2,−z1, z4,−z3] ,
and let ρ˜′2 and ρ˜′3 be the lifts to W . Then ρ˜′3 is an involution of W that
acts freely. Let M2 := W/⟨ρ˜′3⟩, and let M := M1 ∪f M2, where we glue M1
and M2 together using the following diffeomorphism f of S7/Q8, where Q8 is
the quaternion group. Let f ′ : C2 → C2,
f ′(x1 + ix2, x3 + ix4) := (−x1 + ix3, x2 + ix4) ,
and define f := (f ′ ⊕ f ′)|S7/Q8 . Then M admits an asymptotically cylindrical
Riemannian metric with holonomy Spin(7) [Kov13, Theorem 5.9]. Note that
the cross-section at infinity of M is diffeomorphic to N := (S1×C)/⟨ρ⟩, where
ρ(eiθ, z) := (e−iθ, ρ1(z)).
We may assume that the Kähler metric on V˜ is ρ˜1- and ρ˜2-invariant. Then
the holomorphic volume form Ω on V˜ satisfies w.l.o.g. (ρ˜1)∗(Ω) = Ω and
(ρ˜2)∗(Ω) = eiθΩ for some θ ∈ R. Now ρ˜1ρ˜2 = ρ˜2ρ˜1 implies e2iθ = 1. So either
(ρ˜2)∗(Ω) = Ω or (ρ˜2)∗(Ω) = −Ω. In the first case, the fixed-point set of ρ˜2 is
calibrated with respect to ReΩ (see [Joy00, Method 2 in Section 11.9]), and in
the second case, the fixed-point set of ρ˜2 is calibrated with respect to ImΩ.
In particular, ρ˜1 acts orientation-preserving on the fixed-point set of ρ˜2 in the
first case and orientation-reversing in the second case.
145
So consider the fixed-point set of ρ2 in V ,
Vρ2 := {[u, u, v, v, τ ] : u, v ∈ C, τ ∈ R, (u, v, τ) ̸= (0, 0, 0)} .
The action of ρ1 on Vρ2 is given by
ρ1([u, u, v, v, τ ]) = [iu, iu, iv, iv, τ ] .
In particular, ρ1 acts orientation-preserving on Vρ2 (i.e., the quotient Vρ2/⟨ρ1⟩ is
orientable). This shows that (ρ˜2)∗(Ω) = Ω as seen above. Hence the asymptotic
limit of the asymptotically cylindrical Spin(7)-structure Φ is ρ˜2-invariant.
Note that 12(Φ+ (ρ˜2)
−1(Φ)) is also an asymptotically cylindrical, torsion-free
Spin(7)-structure on M with the same asymptotic limit as Φ. Furthermore, the
holonomy of the associated metric is still equal to Spin(7) as the holonomy is
determined topologically [Kov13, Theorem 3.2]. So by resolving the singularity
in this ρ˜2-equivariant way, ρ˜2 yields an involution of M that preserves the
Spin(7)-structure. Hence its fixed-point set is a Cayley submanifold of M by
Lemma 2.14. Denote it by X. We will now deduce the topological type of X
and calculate the index of the deformation map.
Consider the fixed-point set of ρ1ρ2 in V ,
Vρ1ρ2 := {[u, 0, v, 0, w] : u, v, w ∈ C, (u, v, w) ̸= (0, 0, 0)} and
V ′ρ1ρ2 := {[0, u, 0, v, w] : u, v, w ∈ C, (u, v, w) ̸= (0, 0, 0)} .
Note that ρ1 interchanges these two components. Let X˜1 be the fixed-point set
of ρ˜2 in V˜ \ (C˜∪{p0}), let X˜2 be the fixed-point set of ρ˜1ρ˜2 in V˜ \ (C˜∪{p0}), let
X˜3 be the fixed-point set of ρ˜′2ρ˜′3 in W , and let X1 := X˜1/⟨ρ˜1⟩, X2 := X˜2/⟨ρ˜1⟩,
and X3 := X˜3/⟨ρ˜3⟩.
Lemma 7.1. We have
X ∼= X1 ∪L(4;1) X2 ∪L(4;1) X3 ,
146
where L(4; 1) is a lens space, defined by considering S3 ⊆ C2 and taking
the quotient with respect to the cyclic group Z4 whose action is induced by
multiplication by i.
Proof. We have
f ′(u, u¯) = ((−1 + i)Reu, (1− i) Im u) ,
f ′(iu, u¯) = (iu¯, u¯) , and
f ′(u, 0) = (−Reu, Im u)
for all u ∈ C. So under the diffeomorphism f ,
{[u, u¯, v, v¯, 1] : u, v ∈ C} ⊆ Vρ2
is identified with
{[(1− i)t1, (1− i)t2, (1− i)t3, (1− i)t4] : t1, . . . , t4 ∈ R} ⊆ C4/Z4 ,
and
{[u, u¯, v, v¯,−1] : u, v ∈ C} = {[iu, u¯, iv, v¯, 1] : u, v ∈ C} ⊆ Vρ2
is identified with
{[iu, u, iv, v] : u, v ∈ C} ⊆ C4/Z4 ,
and
{[u, 0, v, 0, 1] : u, v ∈ C} ⊆ Vρ1ρ2
is identified with
{[t1, t2, t3, t4] : t1, . . . , t4 ∈ R} ⊆ C4/Z4 .
Now [t1, t2, t3, t4] and [(1 − i)t1, (1 − i)t2, (1 − i)t3, (1 − i)t4] lie on the same
complex curve. So under the blow-up of C4/Z4 at 0, they are glued together.
Furthermore, the component {[iu, u, iv, v] : u, v ∈ C} is blown up at 0 (this is a
complex surface in C4/Z4 with unique singular point 0).
147
Let Y˜ be the fixed-point set of ρ2 in C, let Z˜ be the fixed-point set of ρ1ρ2
in C, and let Y := Y˜ /⟨ρ1⟩ and Z := Z˜/⟨ρ1⟩. Then the cross-section at infinity
of X consists of S1 × Z and two copies of Y .
Note that Y˜ is a special Lagrangian submanifold of C since X˜1 is a special
Lagrangian submanifold of V˜ \ (C˜ ∪ {p0}) as seen above. Furthermore, the
pull-back of νNY under the map Y˜ → Y is νS1×C Y˜ . So s := ∂∂θ is a parallel
normal vector field of Y˜ in S1 × C, where θ denotes the coordinate on the
S1-factor. Furthermore, s ◦ ρ = −s. So we can apply parts of Proposition 6.25
for the index calculation.
Lemma 7.2. We have
Y ∼= 13 (S1 × S2) .
Proof. Let
X˜4 := {[u, u¯, v, v¯, τ ] ∈ Vρ2 : Reu8 +Re v8 + 12τ 2 ≤ 0}
and X4 := X˜4/⟨ρ1⟩. Note that
Y˜ = {[u, u¯, v, v¯, τ ] ∈ Vρ2 : Reu8 +Re v8 + 12τ 2 = 0} .
The map
X˜4 × [0, 1]→ X˜4 , ([u, u¯, v, v¯, τ ], t) ↦→ [u, u¯, v, v¯, τ t])
defines a deformation retraction of X˜4 onto
Y˜ ′ := {[u, u¯, v, v¯, 0] ∈ Vρ2 : Reu8 +Re v8 ≤ 0} .
Note that Y˜ is the closed double of Y˜ ′. Furthermore, there exists a deformation
retraction f : C× [0, 1]→ C onto
{z ∈ C : z8 = −|z|8}
148
such that
(i) f(e 14πiz, t) = e 14πif(z, t),
(ii) f(rz, t) = rf(z, t), and
(iii) f(z, t) = 0 implies z8 = |z|8
for all z ∈ C, t ∈ [0, 1], r ∈ [0,∞). Then the map
Y˜ ′ × [0, 1]→ Y˜ ′ , ([u, u¯, v, v¯, 0], t) ↦→ [f(u, t), f(u, t), f(v, t), f(v, t), 0])
defines a deformation retraction of Y˜ ′ onto
K˜ := {[u, u¯, v, v¯, 0] ∈ Vρ2 : Reu8 = −|u|8,Re v8 = −|v|8} .
Now let Y ′ := Y˜ ′/⟨ρ1⟩ and K := K˜/⟨ρ1⟩. The above deformation retractions
are ρ1-invariant. So K is a deformation retract of X4, and Y is the closed double
of Y ′. Note that K and K˜ are 1-dimensional CW-complexes. Furthermore,
Y and Y˜ are connected since K and K˜ are connected.
Now the set
{(u, v) ∈ C2 : |u|8 + |v|8 = 1,Reu8 = −|u|8,Re v8 = −|v|8}
is a 1-dimensional CW-complex with 16 vertices (0-cells) and 64 edges (1-cells).
So K has 4 vertices and 16 edges. Hence χ(K) = 4 − 16 = −12, and so
b1(K) = b0(K)− χ(K) = 13. In particular, π1(Y ′) ∼= π1(K) is a free group of
rank 13.
Given any closed loop in Y ′, there exists a smooth loop homotopic to it.
Furthermore, we may assume that it meets K transversely, which means that it
does not intersect K. Then we may use the earlier deformation retraction back-
wards to push the loop onto ∂Y ′. This shows that the map π1(∂Y ′)→ π1(Y ′)
induced by the inclusion is surjective. Hence
π1(Y ) ∼= π1(Y ′) ∗π1(∂Y ′) π1(Y ′) ∼= π1(Y ′)
by van Kampen’s Theorem [Hat02, Theorem 1.20]. So π1(Y ) is a free group of
rank 13, and hence Y ∼= 13 (S1 × S2) by [Hem76, Exercise 5.3].
149
We should also note the following consequences of the above proof. We
have σ(X4) = σ(X˜4) = 0 since b2(X4) = b2(K) = 0 and b2(X˜4) = b2(K˜) = 0.
Furthermore, Y˜ ∼= 25 (S1 × S2) since K˜ has 8 vertices and 32 edges. Also
b1(∂Y ′) = 2 b1(Y ′) = 26. So ∂Y ′ is a closed orientable surface of genus 13.
Lemma 7.3. The closed orientable surface Z has genus 3. Furthermore,
dimCH0(Z, νCZ) = 4 .
Proof. Note that
Z ∼= {[u, 0, v, 0, w] ∈ Vρ1ρ2 : u8 + v8 + w2 = 0} .
The map
f : Z → CP1 , [u, 0, v, 0, w] ↦→ [u, v]
is a branched double cover with 8 branched points. So χ(Z) = −4 by the
Riemann–Hurwitz Formula [Har77, Corollary 2.4 in Chapter IV]. Hence Z is a
complex curve of genus 3. In particular, c1(νCZ) = −c1(Z) = 4 since C is a
Calabi–Yau manifold.
Furthermore, let f˜ : C → CP3, [z0, . . . , z4] ↦→ [z0, . . . , z3]. Then (df˜)|νCZ
defines an isomorphism νCZ ∼= f ∗νCP3CP1, where νCP3CP1 is the normal bundle
with respect to the embedding CP1 → CP3, [u, v] ↦→ [u, 0, v, 0]. Note that
νCP3CP1 ∼= O(1)⊕O(1), and hence
νCZ ∼= f ∗O(1)⊕ f ∗O(1) .
Note further that ρ˜ : C → C, [z0, . . . , z4] ↦→ [z0, . . . , z3,−z4] defines an in-
volution (dρ˜)|νCZ on H0(Z, f ∗O(1)). The +1 eigenspace is isomorphic to
H0(CP1,O(1)) since ρ˜ : C → C and ρ˜|Z : Z → Z are the deck transformations
with respect to the branched double covers f˜ : C → CP3 and f : Z → CP1,
respectively. On the other hand, any section in the −1 eigenspace must vanish
at all 8 branched points, and hence the −1 eigenspace is 0 since the degree
of f ∗O(1) is equal to c1(f ∗O(1)) = 12c1(νCZ) = 2. So
dimCH0(Z, νCZ) = 2 dimCH0(Z, f ∗O(1)) = 2 dimCH0(CP1,O(1)) = 4 .
150
Let S˜ be the fixed-point set of ρ2 in Σ, and let S := S˜/⟨ρ1⟩.
Lemma 7.4. The closed orientable surface S has genus 13.
Proof. Note that
S ∼= {[u, u¯, v, v¯, τ ] ∈ Vρ2/⟨ρ1⟩ : Reu8+Re v8+ 12τ 2 = Im u8+2 Im v8+ 12τ 2 = 0} .
We have
Reu8 +Re v8 = Im u8 + 2 Im v8 ⇔ Re((1 + i)u8) + Re((1 + 2i)v8) = 0
⇔ Re(λu)8 +Re(µv)8 = 0 ,
where λ, µ ∈ C are such that λ8 = 1 + i and µ8 = 1 + 2i. So S is diffeomorphic
to the closed double of
S ′ := {[u, u¯, v, v¯, 0] ∈ Vρ2/⟨ρ1⟩ : Reu8 +Re v8 ≤ 0,Re(λu)8 +Re(µv)8 = 0} .
Since the map
[u, u¯, v, v¯, 0] ↦→ [e 18πiu, e− 18πiu¯, e 18πiv, e− 18πiv¯, 0]
defines a diffeomorphism of S ′ to
{[u, u¯, v, v¯, 0] ∈ Vρ2/⟨ρ1⟩ : Reu8 +Re v8 ≥ 0,Re(λu)8 +Re(µv)8 = 0} ,
we see that
S ∼= {[u, u¯, v, v¯, 0] ∈ Vρ2/⟨ρ1⟩ : Re(λu)8 +Re(µv)8 = 0} .
This surface is diffeomorphic to ∂Y ′. Hence S is a closed orientable surface of
genus 13.
Define
X˜4 := {[u, u¯, v, v¯, τ ] ∈ Vρ2 : Reu8 +Re v8 + 12τ 2 ≤ 0}
151
and X4 := X˜4/⟨ρ1⟩ as in the proof of Lemma 7.2. Furthermore, let
X˜5 := {[u, u¯, v, v¯, τ ] ∈ Vρ2 \ {p0} : Reu8 +Re v8 + 12τ 2 ≥ 0}
and X5 := X˜5/⟨ρ1⟩.
Note that Vρ2 \ (C ∪{p0}) consists of two connected components, namely the
interiors of X˜4 and X˜5. In the blow-up V˜ \ C˜, they are glued together along an
interval I times the intersection of Vρ2 with Σ, that is, along I × S˜. So
X1 ∼= X4 ∪I×S X5 .
Note that, by construction, we also have a diffeomorphism
X4 ∪Y X5 ∼= R× L(4; 1) .
Hence
χ(X1) = χ(X4) + χ(X5)− χ(S) = χ(L(4; 1)) + χ(Y )− χ(S) = 24
by the inclusion–exclusion formula. Also note that X1 has an orientation-
reversing diffeomorphism (coming from V → V , [z0, . . . , z4] ↦→ [z0, . . . , z3,−z4]),
and hence
σ(X1) = 0 .
The intersection of Vρ1ρ2 with Σ, that is, the set
{[u, 0, v, 0, w] ∈ V : u8 + v8 + w2 = 0, u8 + 2v8 + iw2 = 0} ,
consists of 16 points. So in the blow-up V˜ , the submanifold Vρ1ρ2 \ {p0} ∼=
CP21,1,4 \ {p0} will be blown up at 16 points. Hence
X2 ∪S1×Z (D2 × Z) ∼= (CP21,1,4 \ {p0}) # 16CP2 .
152
Proposition 7.5. The topological type of X is
(CP2 # 17CP2) \ (X13 ⨿X13 ⨿ (D2 ×Σ3)) ,
where Σ3 is a closed orientable surface of genus 3 and X13 is the null-cobordism
of 13 (S1× S2) coming from S1× S2 ∼= ∂(S1×D3). Note that the cross-section
at infinity is diffeomorphic to
13 (S1 × S2)⨿ 13 (S1 × S2)⨿ (S1 ×Σ3) .
Proof. The relation X4 ∪Y X5 ∼= R× L(4; 1) implies
π1(X4) ∗π1(Y ) π1(X5) ∼= π1(L(4; 1)) ∼= Z4
by van Kampen’s Theorem [Hat02, Theorem 1.20]. Since the inclusion Y ↪→ X4
induces an isomorphism π1(Y ) ∼= π1(X4) (see the proof of Lemma 7.2), we
must have π1(X5) ∼= Z4. In fact, the inclusion L(4; 1) ↪→ X5 induces this
isomorphism. Now X1 ∼= X4 ∪I×Σ13 X5 implies
π1(X1) ∼= π1(X4) ∗π1(Σ13) π1(X5)
by van Kampen’s Theorem. Here the map π1(Σ13) → π1(X4) induced by
the inclusion I × Σ13 ↪→ X4 is surjective since Σ13 ∼= ∂Y ′ and the map
π1(∂Y ′)→ π1(Y ) induced by the inclusion ∂Y ′ ↪→ Y ′ ↪→ Y is surjective as
seen at the end of the proof of Lemma 7.2. So the map π1(L(4; 1))→ π1(X1)
induced by the inclusion L(4; 1) ↪→ X5 ↪→ X1 is surjective.
Note that CP21,1,4 \ {p0} is simply-connected [DD85, Corollary 8]. Hence
X2 ∪S1×Z (D2 × Z) ∼= (CP21,1,4 \ {p0}) # 16CP2
is simply-connected. Furthermore, X3 is homotopy equivalent to CP1 (the
exceptional divisor), and hence also X3 is simply-connected.
153
So if we define
X := X ∪Y X13 ∪Y X13 ∪S1×Z (D2 × Z) ,
then π1(X) = 0 by van Kampen’s Theorem since X ∼= X1 ∪L(4;1) X2 ∪L(4;1) X3
and the inclusion Y ↪→ X13 induces an isomorphism π1(Y ) ∼= π1(X13).
We calculate
χ(X) = χ(X1) + 2χ(X13) + χ(CP21,1,4 \ {p0}) + 16 · (χ(CP2)− 2) + χ(X3)
= 24 + 2 · (1− 13) + 2 + 16 · (3− 2) + 2 = 20
by the inclusion–exclusion formula since the Euler characteristic of a closed
orientable 3-manifold is 0, and
σ(X) = σ(X1) + 2σ(X13) + σ(CP21,1,4 \ {p0}) + 16σ(CP2) + σ(X3)
= 0 + 2 · 0 + 1 + 16 · (−1) + (−1) = −16
by Novikov additivity (Lemma 6.11) since b2(X13) = 0. Also note that the
intersection form on X is odd since the intersection form on CP2 is odd. Hence
X is homeomorphic to CP2 # 17CP2 by [FQ90, Theorem 10.1 (2)].
Note that
χ(X) = χ(CP2) + 17 · (χ(CP2)− 2)− 2χ(X13)− χ(D2 ×Σ3)
= 3 + 17 · (3− 2)− 2 · (1− 13)− (−4) = 48
and
σ(X) = σ(CP2) + 17 · σ(CP2)− 2σ(X13)− σ(D2 ×Σ3) = −16 .
Lemma 7.6. We have ∫
X
e(νMX) = 24 .
154
Proof. Recall from above that s := ∂
∂θ
is a parallel normal vector field of Y˜
in S1×C such that s◦ρ = −s, where θ denotes the coordinate on the S1-factor.
Furthermore, νCZ has a non-vanishing section s˜ by [Ste51, Corollary 29.3] since
Z is 2-dimensional and νCZ has rank 4. Extend s˜ to a (non-vanishing) section
of νS1×C(S1 × Z) that is invariant under rotations of S1, which, by abuse of
notation, we will also call s.
When X1, X2, and X3 are glued together along L(4; 1), the normal bundles
are also glued together. Now fix a non-vanishing section of the normal bundle
νS7/Q8L(4; 1) (which always exists by [Ste51, Corollary 29.3] since L(4; 1) is
3-dimensional and νS7/Q8L(4; 1) has rank 4). By abuse of notation, we will also
call this s. Then
∫
X
e(νMX) =
1
2e(π
∗νV˜ /⟨ρ1⟩X1, s) + e(νV˜ /⟨ρ1⟩X2, s) + e(νW/⟨ρ3⟩X3, s) ,
by Theorem 6.14, where π : V˜ → V˜ /⟨ρ1⟩ is the natural projection.
Let Vˆ be the blow-up of V˜ at the singular point p0, and let Xˆ1 be the
fixed-point set of ρˆ2 (the extension of ρ2 to Vˆ ) in Vˆ \ C˜. Then
e(π∗νV˜ /⟨ρ1⟩X1, s) = e(νVˆ Xˆ1, s) = χ(Xˆ1) = χ(X˜1) = 2χ(X1) = 48
since Xˆ1 is a special Lagrangian submanifold of Vˆ and
Xˆ1 ∼= X˜1 ∪L(4;1)⨿L(4;1) (I × L(4; 1)) .
Furthermore, let F4 be the blow-up of CP21,1,4 at the singularity p0. Then
X2 ∪L(4;1)X3 ∪S1×Z (D2×Z) is F4 blown up at 16 points. Denote it by Fˆ4. We
can view Fˆ4 as a submanifold of Vˆ . Then
e(νV˜ /⟨ρ1⟩X2, s) + e(νW/⟨ρ3⟩X3, s) = [Fˆ4] · [Fˆ4]
by Remark 6.27, where [Fˆ4] · [Fˆ4] is the self-intersection number of Fˆ4 inside Vˆ .
155
Note that [F4] · [F4] = 0 inside the blow-up of V at the singularity p0 since the
map
CP21,1,4 × [0, 1]→ CP21,1,1,1,4 , ([u, v, w], t) ↦→ [u, tu, v, tv, w]
extends to the blow-ups at the singularity p0. In fact, this map also extends to
Fˆ4 × [0, 1]→ Vˆ . So [Fˆ4] · [Fˆ4] = 0. Hence∫
X
e(νMX) = 24 .
Proposition 7.7. We have
indλD = −24 ,
where λ > 0 is such that [−λ, 0) contains no eigenvalue of D˜.
Proof. We have
indλD =
1
2χ(X) +
1
2σ(X)−
∫
X
e(νMX) +
1
2(2σ(X4)− σ(X˜4))
+ (b0(Y ) + b1(Y )− b0(Y˜ )− b1(Y˜ ))− dimCH0(Z, νCZ)
= 12 · 48 +
1
2 · (−16)− 24 +
1
2 · (2 · 0− 0) + (1 + 13− 1− 25)− 4
= −24
by Theorem 6.18 and parts of Proposition 6.25 and Proposition 6.26. Note that
the change of sign of σ(X) in the above formula compared to (6.15) comes from
our convention (2.10) of the Spin(7)-structure (complex surfaces of Calabi–Yau
4-folds are Cayley with respect to the opposite orientation).
Remark 7.8. Note that
dim ker(D˜|Y ) = b0(Y˜ ) + b1(Y˜ )− b0(Y )− b1(Y ) = 12 and
dim ker(D˜|S1×Z) = 2 dimCH0(Z, νCZ) = 8 .
In particular, the index of the deformation problem of all asymptotically cyl-
indrical Cayley submanifolds near X with varying asymptotic limit is equal
156
to −24 + 2 · 12 + 8 = 8 by Theorem 6.30, which is positive. In fact, the sub-
manifolds Y and S1×Z have smooth moduli spaces of associative deformations
of the above dimensions so that we can apply Theorem 6.30. This can be seen
as follows.
First note that Y˜ is a special Lagrangian submanifold of C. Now any
deformation of Y as an associative submanifold of (S1 × C)/⟨ρ⟩ lifts to a
deformation of Y˜ as an associative submanifold of S1 × C, which is therefore
necessarily of the form {p} × Y˜ ′ with p ∈ S1, where Y˜ ′ is a special Lagrangian
submanifold of C [Gay14, Proposition 4.6]. Moreover, the deformation must be
invariant under ρ in order to be well-defined in the quotient (S1×C)/⟨ρ⟩. So the
associative deformations of Y correspond precisely to the special Lagrangian
deformations of Y˜ that are invariant under ρ. The moduli space of such
deformations is a smooth manifold of dimension dim ker(D˜|Y ) = 12, which can
be proved similarly to [McL98, Section 3] using spaces of differential forms that
change sign under ρ.
Furthermore, the deformations of S1 × Z as an associative submanifold
of (S1×C)/⟨ρ⟩ correspond to the deformations of Z as a complex curve in C by
[CHNP15, Lemma 5.11] since Z˜ consists of two copies of Z. The moduli space of
such deformations is a smooth manifold of dimension dim ker(D˜|S1×Z) = 8 since
CP1 has a complex 4-dimensional moduli space of deformations as a complex
curve in CP3, which lift to C under the branched double cover C → CP3,
[z0, . . . , z4] ↦→ [z0, . . . , z3].
7.1.3 Cayley submanifolds in Spin(7)-manifolds
constructed from a hypersurface in CP51,1,1,1,4,4
Here we construct two asymptotically cylindrical Cayley submanifolds inside the
asymptotically cylindrical Spin(7)-manifolds constructed in [Kov13, Section 6.3]
and compute the indices. Note that the construction in [Kov13] involves a
choice, namely which asymptotically locally Euclidean (ALE) Spin(7)-manifolds
are used to resolve which singularities (compare also [Joy99, Section 5.2]). This
gives a priori four different ways of resolving two singularities. However, one of
157
the two ALE Spin(7)-manifolds that can be used in the construction must be
used at least once in order to get a manifold with full holonomy Spin(7) (by
[Joy99, Proposition 5.9] and [Kov13, Theorem 3.2]). Furthermore, if both ALE
Spin(7)-manifolds are used, the resulting Spin(7)- and Cayley submanifolds (for
the example using [Kov13, Section 6.3]) are the same by the symmetries involved
(i.e., regardless of which ALE Spin(7)-manifold is used for which singularity).
For the remaining two choices, we will construct Cayley submanifolds that
are topologically distinct (they will have different fundamental groups). But
both Cayley submanifolds will have the same index because the topological
invariants that determine the index will be the same. That is why we will
treat both Cayley submanifolds similarly as far as possible. In particular, the
notation refers to either Cayley submanifold unless stated otherwise.
Let
V := {[z0, . . . , z5] ∈ CP51,1,1,1,4,4 : z80 + z81 + z82 + z83 + z24 + z25 = 0} ,
V ′ := {[z0, . . . , z5] ∈ CP51,1,1,1,4,4 : z4 + z5 = 0} ,
V ′′ := {[z0, . . . , z5] ∈ CP51,1,1,1,4,4 : z4 − z5 = 0} ,
C := V ∩ V ′ , and
Σ := V ∩ V ′ ∩ V ′′ .
Note that V is an orbifold with two singular points p± := [0, 0, 0, 0, i,±1] and
that C is smooth. Let V˜ be the blow-up of V along Σ. Then C lifts to a
submanifold C˜ of V˜ which is isomorphic to C.
Define two antiholomorphic involutions
ρ1 : [z0, . . . , z5] ↦→ [z1,−z0, z3,−z2, z5, z4] and
ρ2 : [z0, . . . , z5] ↦→ [z1, z0, z3, z2, z5, z4] .
Note that ρ1ρ2 = ρ2ρ1. Furthermore, both ρ1 and ρ2 preserve p±, V , V ′, V ′′,
and hence also C and Σ. So they lift to antiholomorphic involutions ρ˜1 and ρ˜2
of V˜ such that ρ˜1ρ˜2 = ρ˜2ρ˜1. Note further that ρ1 fixes only p±.
158
LetM1 := (V˜ \(C˜∪{p±}))/⟨ρ˜1⟩, and defineM2 and f : S7/Q8 → S7/Q8 as in
the last section. Then both M :=M1∪f M2∪f M2 and M ′ :=M1∪idM2∪f M2
admit asymptotically cylindrical Riemannian metrics with holonomy Spin(7)
[Kov13, Theorem 5.9]. Note that the cross-sections at infinity of M and M ′
are diffeomorphic to N := (S1 × C)/⟨ρ⟩, where ρ(eiθ, z) = (e−iθ, ρ1(z)).
Consider the fixed-point set of ρ2 in V ,
Vρ2 := {[u, u, v, v, w, w] : u, v, w ∈ C,Reu8 +Re v8 +Rew2 = 0} .
The action of ρ1 on Vρ2 is given by
ρ1([u, u, v, v, w, w]) = [iu, iu, iv, iv, w, w] .
In particular, ρ1 acts orientation-preserving on Vρ2 (i.e., the quotient Vρ2/⟨ρ1⟩
is orientable). This shows that (ρ˜2)∗(Ω) = Ω as seen in the last section. So by
resolving the singularity in a ρ˜2-equivariant way like in the last section, ρ˜2 yields
an involution of M (resp., M ′) that preserves the Spin(7)-structure. Hence
its fixed-point set is a Cayley submanifold of M (resp., M ′) by Lemma 2.14.
Denote it by X (resp., X ′). We will now deduce the topological types of X
and X ′, and calculate the index of the deformation map.
Let X˜6 be the fixed-point set of ρ˜2 in V˜ \ (C˜ ∪ {p±}), let X˜7 be the fixed-
point set of ρ˜1ρ˜2 in V˜ \ (C˜ ∪ {p±}), and let X6 := X˜6/⟨ρ˜1⟩ and X7 := X˜7/⟨ρ˜1⟩.
Similarly to the proof of Lemma 7.1, we have (recall that there are two
singularities here)
X ∼= X6 ∪L(4;1)⨿L(4;1) X7 ∪L(4;1) X3 ∪L(4;1) X3 and
X ′ ∼= X6 ∪L(4;1)⨿L(4;1) (I × L(4; 1)) ∪L(4;1) (X7 ∪L(4;1) X3) ∪L(4;1) X3 ,
where X3 was defined in the last section and L(4; 1) is a lens space, defined
by considering S3 ⊆ C2 and taking the quotient with respect to the cyclic
group Z4 whose action is induced by multiplication by i.
159
Note that
{[u, u, v, v, w, w] ∈ Vρ2 \ {p±} : Rew ≥ 0} ∼= X˜5 ,
where X˜5 was defined in the last section. So
Vρ2 \ {p±} ∼= X˜5 ∪Y˜ X˜5 ,
where Y˜ was defined in the last section. In the blow-up V˜ , we get (compare
the proof of the relation X1 ∼= X4 ∪I×Σ13 X5 in the last section)
X6 ∼= X5 ∪I×Σ13 X5 .
Note that X6 has an orientation-reversing diffeomorphism (coming from V → V ,
[z0, . . . , z5] ↦→ [z0, . . . , z3, z5, z4]), and hence
σ(X6) = 0 .
Consider also the fixed-point set of ρ1ρ2 in V ,
Vρ1ρ2 := {[u, 0, v, 0, w, z] : u, v, w, z ∈ C, u8 + v8 + w2 + z2 = 0} and
V ′ρ1ρ2 := {[0, u, 0, v, w, z] : u, v, w, z ∈ C, u8 + v8 + w2 + z2 = 0} .
Note that ρ1 interchanges these two components. The intersection of Vρ1ρ2
with Σ, that is, the set
{[u, 0, v, 0, 0, 0] ∈ V : u8 + v8 = 0} ,
consists of 8 points. So in the blow-up V˜ , the submanifold Vρ1ρ2 \ {p±} will be
blown up at 8 points. Hence
X7 ∪S1×Z (D2 × Z) ∼= (Vρ1ρ2 \ {p±}) # 8CP2 .
160
Furthermore, we have h2,0(Vρ1ρ2) = 0 and h1,1(Vρ1ρ2) = 8 by [IF00, Theorem 7.2].
So χ(Vρ1ρ2) = 10 and σ(Vρ1ρ2) = −6. Note also that
Vρ1ρ2 \ ({p±} ∪ {[0, 0, v, 0, w, z] ∈ Vρ1ρ2 : v8 + w2 + z2 = 0})
is simply-connected by [DD85, Lemma 9]. Hence Vρ1ρ2 \ {p±} is simply-
connected since
{[0, 0, v, 0, w, z] ∈ Vρ1ρ2 : v8 + w2 + z2 = 0} → CP1 , [0, 0, v, 0, w, z] ↦→ [w, z]
is an 8-fold branched cover with 2 branched points, and hence
{[0, 0, v, 0, w, z] ∈ Vρ1ρ2 : v8 + w2 + z2 = 0}
is diffeomorphic to S2 (in particular, simply-connected).
Proposition 7.9. The topological type of X is
(13CP2 # 29CP2) \ (X13 ⨿X13 ⨿ (D2 ×Σ3)) ,
where Σ3 is a closed orientable surface of genus 3 and X13 is the null-cobordism
of 13 (S1× S2) coming from S1× S2 ∼= ∂(S1×D3). Note that the cross-section
at infinity is diffeomorphic to
13 (S1 × S2)⨿ 13 (S1 × S2)⨿ (S1 ×Σ3) .
Proof. Recall from above that
X ∼= (X5 ∪I×Σ13 X5) ∪L(4;1)⨿L(4;1) X7 ∪L(4;1) X3 ∪L(4;1) X3 .
In the proof of Proposition 7.5 we have seen that the inclusion L(4; 1) ↪→ X5
induces an isomorphism π(X5) ∼= Z4. Hence the manifold
X := X ∪Y X13 ∪Y X13 ∪S1×Z (D2 × Z)
161
is simply-connected by van Kampen’s Theorem [Hat02, Theorem 1.20] since
X7 ∪S1×Z (D2 × Z) and X3 are simply-connected (see above and the proof of
Proposition 7.5, respectively). Recall further from above that
X7 ∪S1×Z (D2 × Z) ∼= (Vρ1ρ2 \ {p±}) # 8CP2 .
We calculate
χ(X) = 2χ(X5)− χ(Σ13) + 2χ(X13) + χ(Vρ1ρ2 \ {p±})
+ 8 · (χ(CP2)− 2) + 2χ(X3)
= 2 · 12− (−24) + 2 · (−12) + 8 + 8 · (3− 2) + 2 · 2 = 44
by the inclusion–exclusion formula since the Euler characteristic of a closed
orientable 3-manifold is 0, and
σ(X) = σ(X6) + 2σ(X13) + σ(Vρ1ρ2 \ {p±}) + 8σ(CP2) + 2σ(X3)
= 0 + 2 · 0 + (−6) + 8 · (−1) + 2 · (−1) = −16
by Novikov additivity (Lemma 6.11) since b2(X13) = 0. Also note that the
intersection form on X is odd since the intersection form on CP2 is odd. Hence
X is homeomorphic to 13CP2 # 29CP2 by [FQ90, Theorem 10.1 (2)].
Note that
χ(X) = 2 + 13 · (χ(CP2)− 2) + 29 · (χ(CP2)− 2)− 2χ(X13)− χ(D2 ×Σ3)
= 2 + 13 · (3− 2) + 29 · (3− 2)− 2 · (1− 13)− (−4) = 72
and
σ(X) = 13σ(CP2) + 29 · σ(CP2)− 2σ(X13)− σ(D2 ×Σ3) = −16 .
Proposition 7.10. The topological type of X ′ is
((S1 × S3) # 14CP2 # 30CP2) \ (X13 ⨿X13 ⨿ (D2 ×Σ3)) ,
162
where Σ3 is a closed orientable surface of genus 3 and X13 is the null-cobordism
of 13 (S1× S2) coming from S1× S2 ∼= ∂(S1×D3). Note that the cross-section
at infinity is diffeomorphic to
13 (S1 × S2)⨿ 13 (S1 × S2)⨿ (S1 ×Σ3) .
Proof. Recall from above that
X ′ ∼= X6 ∪L(4;1)⨿L(4;1) (I × L(4; 1)) ∪L(4;1) (X7 ∪L(4;1) X3) ∪L(4;1) X3 .
Note that each X5 in the splitting X6 ∼= X5 ∪I×Σ13 X5 approaches both singu-
larities. Recall further that
X5 ∼= {[u, u¯, v, v¯, τ ] ∈ ‘(Vρ2 \ {p0})/⟨ρ1⟩’ : Reu8 +Re v8 + 12τ 2 ≥ 0} ,
where ‘(Vρ2 \ {p0})/⟨ρ1⟩’ refers to the notation in the last section. Consider
X ′5 := {[u, u¯, v, v¯, τ ] ∈ X5 : τ ≥ 0} .
The boundary of X ′5 consists of
{[u, u¯, v, v¯, τ ] ∈ X5 : τ = 0} ∼= {[u, u¯, v, v¯, 0] ∈ X5 : Reu8 +Re v8 ≥ 0} ∼= Y ′
and
{[u, u¯, v, v¯, τ ] ∈ X5 : Reu8 +Re v8 + 12τ 2 = 0, τ ≥ 0}
∼= {[u, u¯, v, v¯, 0] ∈ X5 : Reu8 +Re v8 ≤ 0} ∼= Y ′ ,
where Y ′ was defined in the proof of Lemma 7.2. Both of these parts intersect
in
{[u, u¯, v, v¯, 0] ∈ X5 : Reu8 +Re v8 = 0} ∼= ∂Y ′ ∼= Σ13 .
Hence
X5 ∼= X ′5 ∪Y ′ X ′5 .
163
So in the splitting
Vρ2 \ {p±} ∼= X˜5 ∪Y˜ X˜5 ∼= (X˜ ′5 ∪Y˜ ′ X˜ ′5) ∪Y˜ (X˜ ′5 ∪Y˜ ′ X˜ ′5) ,
we can ‘swap’ two X˜ ′5. In other words, we may assume that one X5 in the
splitting X6 ∼= X5 ∪I×Σ13 X5 approaches p+ and the other approaches p−. So
X6 ∪L(4;1)⨿L(4;1) (I × L(4; 1)) ∼= (X5 ∪L(4;1)⨿L(4;1) (I × L(4; 1))) ∪I×Σ13 X5 .
Recall from the last section that X4 ∪Y X5 ∼= R× L(4; 1). Hence
X4 ∪Y (X5 ∪L(4;1)⨿L(4;1) (I × L(4; 1))) ∼= S1 × L(4; 1) ,
which has fundamental group Z× Z4. Since the inclusion Y ↪→ X4 induces an
isomorphism π1(Y ) ∼= π1(X4) (see the proof of Lemma 7.2), we therefore have
π1(X5 ∪L(4;1)⨿L(4;1) (I × L(4; 1))) ∼= Z× Z4 .
Note that the map π1(Σ13)→ π1(X5) induced by the inclusion I ×Σ13 ↪→ X5
is surjective by similar arguments as in the proof of Lemma 7.2 (noting also
that π1(Σ13) = π1(∂Y ′)→ π1(Y ′) induced by the inclusion is surjective as seen
at the end of the proof of Lemma 7.2). So
π1(X6 ∪L(4;1)⨿L(4;1) (I × L(4; 1))) ∼= Z× Z4
by van Kampen’s Theorem [Hat02, Theorem 1.20], where the Z4-factor is
induced by the inclusion L(4; 1) ↪→ X6 ∪L(4;1)⨿L(4;1) (I × L(4; 1)). Hence if
X
′ := X ′ ∪Y X13 ∪Y X13 ∪S1×Z (D2 × Z) ,
then π1(X) ∼= Z by van Kampen’s Theorem. We have χ(X) = 44 and
σ(X) = −16 as seen in the proof of Proposition 7.9. Furthermore, the intersec-
tion form on X ′ is odd since the intersection form on CP2 is odd. Hence X ′ is
homeomorphic to (S1 × S3) # 14CP2 # 30CP2 by [HT97, Corollary 3 (2)].
164
Note that the map
Vρ1ρ2 × [0, 1]→ V ,
([u, 0, v, 0, w, z], t) ↦→ [(1− t8)1/8u, tu, (1− t8)1/8v, tv, w, z]
extends to the blow-ups, which shows that the self-intersection number of
the blow-up of Vρ1ρ2 in the blow-up of V˜ at the singularities p± is 0. Hence
(compare the proof of Lemma 7.6)
∫
X
e(νMX) = χ(X5 ∪I×Σ13 X5) = 2χ(X5)− χ(Σ13) = 2 · 12− (−24) = 48 .
Lemma 7.11. We have
indλD = −36 ,
where λ > 0 is such that [−λ, 0) contains no eigenvalue of D˜.
Proof. We have
indλD =
1
2χ(X) +
1
2σ(X)−
∫
X
e(νMX) +
1
2(2σ(X4)− σ(X˜4))
+ (b0(Y ) + b1(Y )− b0(Y˜ )− b1(Y˜ ))− dimCH0(Z, νCZ)
= 12 · 72 +
1
2 · (−16)− 48 +
1
2 · (2 · 0− 0) + (1 + 13− 1− 25)− 4
= −36
by Theorem 6.18 and parts of Proposition 6.25 and Proposition 6.26. Note that
the change of sign of σ(X) in the above formula compared to (6.15) comes from
our convention (2.10) of the Spin(7)-structure (complex surfaces of Calabi–Yau
4-folds are Cayley with respect to the opposite orientation).
7.2 Relation to other calibrations
Harvey and Lawson noted in [HL82, Remark 2.12 in Chapter IV] that the
geometry of Cayley submanifolds includes the geometries of other calibrations.
165
In particular, if the holonomy reduces to a proper subgroup of Spin(7), then
Cayley submanifolds can be constructed out of submanifolds that are calibrated
with respect to another calibration. An application of the volume-minimising
property of calibrated submanifolds shows that any deformation of such a
closed Cayley submanifold as a Cayley submanifold must again be of that
form. Here we show that this is also true for asymptotically cylindrical Cayley
submanifolds. We further simplify the index formulae in these cases.
In particular, in the case of the special Lagrangian and the coassociative
calibration, the moduli space of asymptotically cylindrical Cayley deformations
is a smooth manifold (we have the same linearisation up to isomorphism but
better control on the non-linear terms due to Hodge theory).
7.2.1 Special Lagrangian calibration
Let M be an asymptotically cylindrical Calabi–Yau 4-fold with Kähler form ω
and holomorphic volume form Ω, which we assume to be normalised, that is,
ω4 = 32Ω ∧ Ω¯. Then
Φ := −12 ω ∧ ω +ReΩ
defines a Spin(7)-structure on M (see Sections 2.2.5 and 2.2.6). This Spin(7)-
structure is asymptotically cylindrical and torsion-free since ω and Ω are
asymptotically cylindrical and closed.
An orientable 4-dimensional submanifold X of M is called special Lagrangian
if (ReΩ)|X = volX for some orientation of X. This is equivalent to ω|X = 0,
(ImΩ)|X = 0 (see Sections 2.2.5 and 2.2.6). So every special Lagrangian
submanifold is Cayley, but not every Cayley submanifold is special Lagrangian
(for example, complex surfaces are also Cayley).
Proposition 7.12. Let M be an asymptotically cylindrical Calabi–Yau 4-fold,
let X be an asymptotically cylindrical special Lagrangian submanifold of M ,
and let Y be an asymptotically cylindrical local deformation of X with the same
asymptotic limit as X.
166
If Y is a Cayley submanifold ofM , then Y is a special Lagrangian submanifold
of M . So the moduli space of all local deformations of X as an asymptotically
cylindrical Cayley submanifold of M with the same asymptotic limit as X
can be identified with the moduli space of all local deformations of X as an
asymptotically cylindrical special Lagrangian submanifold of M with the same
asymptotic limit as X.
Proof. We have
lim
T→∞
(
vol({y ∈ Y : t ≤ T})− vol({x ∈ X : t ≤ T})
)
= 0
by Proposition 6.15 with φ = Φ. So Proposition 6.15 with φ = ReΩ implies
that Y is special Lagrangian.
Now suppose that M is an asymptotically cylindrical Calabi–Yau 4-fold with
cross-section N , and that X is an asymptotically cylindrical special Lagrangian
submanifold of M with cross-section Y . Then the map v ↦→ Jv defines an
isomorphism TX ∼= νMX, where J is the complex structure onM . Furthermore,
∂
∂t
is mapped to a parallel section s ∈ Γ(νNY ) under this isomorphism since J
is parallel. In particular, e(νMX, s) = e(TX, ∂∂t) = χ(X). So the index in this
case is
indλD = −12χ(X)−
1
2σ(X)−
b0(Y ) + b1(Y )
2 (7.1)
by Proposition 6.24, where λ > 0 is such that [−λ, 0) contains no eigenvalue
of Bev.
Note that ω ∈ Γ(Λ27M) since ∗(ω ∧Φ) = 3ω (which can be checked using the
coordinate expression (2.26)). Moreover, ω|Λ27M |X ∈ E by (2.23) since ω|TX = 0
as X is Lagrangian. Hence E splits as the sum of two vector bundles. One is the
trivial rank 1 vector bundle generated by ω, which we will identify with Λ0X.
The other summand can be identified with Λ2+X via the map TX × TX → E,
(u, v) ↦→ u× Jv. Under the isomorphisms νMX ∼= Λ1X and E ∼= Λ0X ⊕ Λ2+X,
the Dirac operator D is identified with
Ω1(X)→ Ω0(X)⊕ Ω2+(X) , α ↦→ (δα, 12(dα + ∗dα)) . (7.2)
167
The dimension of the kernel of this map is equal to the dimension of the
image of H1cs(X) in H1(X) by [APS75a, Proposition (4·9)], which is equal
to b3(X) + b0(X) − b4(X) − b0(Y ) (note that b4(X) = 0 if X has no closed
connected component).
Salur and Todd [ST10, Theorem 1.1] proved that if M is an asymptotically
cylindrical Calabi–Yau 3-fold and X is an asymptotically cylindrical special
Lagrangian submanifold of M , then the moduli space of all local deformations
of X as an asymptotically cylindrical special Lagrangian submanifold of M
with the same asymptotic limit as X is a smooth manifold whose dimension is
given by the dimension of the image of H1cs(X) in H1(X) (which is equal to
b2(X)+b0(X)−b3(X)−b0(Y ) in dimension 3). The proof of [ST10, Theorem 1.1]
generalises to higher dimensions.
7.2.2 Coassociative calibration
Let M˜ be an asymptotically cylindrical 7-manifold with an asymptotically
cylindrical G2-structure φ˜, let ψ˜ be the Hodge-dual of φ˜ (with respect to the
metric and orientation induced by φ˜), let M := S1 × M˜ , and let θ denote the
coordinate on the S1-factor. Then
Φ := dθ ∧ φ˜+ ψ˜
defines a Spin(7)-structure on M (see Section 2.2.2). The Spin(7)-structure Φ
is asymptotically cylindrical since φ˜ is asymptotically cylindrical. Furthermore,
the Spin(7)-structure Φ is torsion-free if and only if the G2-structure φ˜ is
torsion-free.
An orientable 4-dimensional submanifold X˜ of M˜ is called coassociative if
ψ˜|X˜ = volX˜ for some orientation of X˜. This is equivalent to φ˜|X˜ = 0 [HL82,
Corollary 1.20 in Chapter IV]. So X˜ is a coassociative submanifold of M˜ if and
only if X := {1} × X˜ is a Cayley submanifold of M .
Proposition 7.13. Let M˜ be an asymptotically cylindrical 7-manifold with an
asymptotically cylindrical torsion-free G2-structure, and let X˜ be an asymptot-
168
ically cylindrical coassociative submanifold of M˜ that has no closed connected
component. Define M := S1 × M˜ and X := {1} × X˜. Furthermore, let Y be
an asymptotically cylindrical local deformation of X with the same asymptotic
limit as X.
If Y is a Cayley submanifold ofM , then Y = {1}×Y˜ for some asymptotically
cylindrical coassociative submanifold Y˜ of M˜ . So the moduli space of all local
deformations of X as an asymptotically cylindrical Cayley submanifold of M
with the same asymptotic limit as X can be identified with the moduli space
of all local deformations of X˜ as an asymptotically cylindrical coassociative
submanifold of M˜ with the same asymptotic limit as X˜.
Proof. Let π : M = S1 × M˜ → M˜ be the projection. If Z is an asymptotically
cylindrical local deformation of X, then the volume of ZT := {z ∈ Z : t ≤ T}
is greater than or equal to the volume of π(ZT ), and equality holds if and only
if ZT is a submanifold of {p} × M˜ for some p ∈ S1. We have
lim
T→∞
(
vol({y ∈ Y : t ≤ T})− vol({x ∈ X : t ≤ T})
)
= 0 and
lim
T→∞
(
vol({y ∈ {1} × π(Y ) : t ≤ T})− vol({x ∈ X : t ≤ T})
)
≥ 0
by Proposition 6.15 with φ = Φ, and hence
lim
T→∞
(
vol({y ∈ Y : t ≤ T})− vol({y ∈ {1} × π(Y ) : t ≤ T})
)
≤ 0 .
So Y = {p} × Y˜ for some p ∈ S1 and some submanifold Y˜ of M˜ since
vol(YT )− vol(π(YT )) is increasing as T →∞. We have p = 1 since Y and X
have the same asymptotic limit. Furthermore, Y˜ is an asymptotically cylindrical
coassociative submanifold since Y is an asymptotically cylindrical Cayley
submanifold.
Now suppose that M˜ is an asymptotically cylindrical 7-manifold with cross-
section N˜ and with an asymptotically cylindrical torsion-free G2-structure,
and that X˜ is an asymptotically cylindrical coassociative submanifold of M˜
with cross-section Y˜ . Define M := S1 × M˜ , N := S1 × N˜ , X := {1} × X˜, and
169
Y := {1} × Y˜ . Let θ denote the coordinate on the S1-factor. Then ∂
∂θ
is a
non-trivial parallel section of νMX. So e(νMX, ∂∂θ |Y ) = 0, and hence the index
in this case is
indλD =
1
2χ(X)−
1
2σ(X)−
b0(Y ) + b1(Y )
2 (7.3)
by Proposition 6.24, where λ > 0 is such that [−λ, 0) contains no eigenvalue
of Bev.
The maps s ↦→ (s ⌟ ( ∂
∂θ
⌟ Φ), g(s, ∂
∂θ
)) and v ↦→ ∂
∂θ
× v define isomorphisms
νMX ∼= Λ2−X ⊕ Λ0X and TX ∼= E of vector bundles, respectively. Under the
isomorphisms νMX ∼= Λ2−X ⊕ Λ4X and E ∼= Λ3X, the Dirac operator D is
identified with
Ω2−(X)⊕ Ω4(X)→ Ω3(X) , (α, β) ↦→ dα + δβ . (7.4)
The dimension of the kernel of this map is equal to the dimension of the negative
subspace of the image of H2cs(X) in H2(X) by [APS75a, Proposition (4·9) and
Corollary (4·11)] if X has no closed connected component.
Joyce and Salur [JS05, Theorem 1.1] proved that the moduli space of all local
deformations of X˜ as an asymptotically cylindrical coassociative submanifold
of M˜ with the same asymptotic limit as X˜ is a smooth manifold whose dimension
is given by the dimension of the negative subspace of the image of H2cs(X)
in H2(X). Note that they get the dimension of the positive subspace since
they use a different convention for the G2-structure (not our convention (2.7)).
Examples of asymptotically cylindrical coassociative submanifolds inside
asymptotically cylindrical Riemannian 7-manifolds with holonomy G2 were
constructed by Kovalev and Nordström in [KN10, Section 5.3].
7.2.3 Complex surfaces
Recall from Section 7.2.1 that ifM is an asymptotically cylindrical Calabi–Yau 4-fold
with Kähler form ω and holomorphic volume formΩ, then Φ := −12ω ∧ ω +ReΩ
defines an asymptotically cylindrical torsion-free Spin(7)-structure on M .
170
Furthermore, every complex surface in M is Cayley, but not every Cayley
submanifold is a complex surface (for example, special Lagrangian submanifolds
are also Cayley).
Proposition 7.14. Let M be an asymptotically cylindrical Calabi–Yau 4-fold,
let X be an asymptotically cylindrical complex surface in M , and let Y be an
asymptotically cylindrical local deformation of X with the same asymptotic
limit as X.
If Y is a Cayley submanifold of M , then Y is a complex surface in M . So
the moduli space of all local deformations of X as an asymptotically cylindrical
Cayley submanifold of M with the same asymptotic limit as X can be identified
with the moduli space of all local deformations of X as an asymptotically
cylindrical complex surface in M with the same asymptotic limit as X.
Proof. We have
lim
T→∞
(
vol({y ∈ Y : t ≤ T})− vol({x ∈ X : t ≤ T})
)
= 0
by Proposition 6.15 with φ = Φ. So Proposition 6.15 with φ = −12ω∧ω implies
that Y is a complex surface.
Now suppose that M is an asymptotically cylindrical Calabi–Yau 4-fold with
cross-section N , and that X is an asymptotically cylindrical complex surface
in M with cross-section Y . Assume that N ∼= S1 × C for some Calabi–Yau
3-fold C. Then also Y ∼= S1 × Z for some complex curve Z in C. Write
M and X for the compactifications as in Remark 6.27. So M ∼= M \ C and
X ∼= X \ Z. Then χ(X) = χ(X) − χ(Z) by the inclusion–exclusion formula
and σ(X) = σ(X) by Novikov additivity (Lemma 6.11) since σ(D2 × Z) = 0.
So the index in this case is
indλD =
1
2χ(X) +
1
2σ(X)− [X] · [X]−
1
2χ(Z)− dimCH
0(Z, νCZ) (7.5)
by Proposition 6.26 and Remark 6.27, where λ > 0 is such that [−λ, 0) contains
no eigenvalue of D˜. Note that here σ(X) refers to the orientation that is
171
induced by the complex structure (compared to the orientation as a Cayley
submanifold with respect to our convention (2.10) of the Spin(7)-structure,
where complex surfaces of Calabi–Yau 4-folds are Cayley with respect to the
opposite orientation).
7.2.4 Associative calibration
Recall from Section 7.2.2 that if M˜ is an asymptotically cylindrical 7-manifold
with an asymptotically cylindrical G2-structure φ˜, ψ˜ is the Hodge-dual of φ˜,
M := S1×M˜ , and θ denotes the coordinate on the S1-factor, then Φ := dθ∧φ˜+ψ˜
defines an asymptotically cylindrical Spin(7)-structure on M . Furthermore, the
Spin(7)-structure Φ is torsion-free if and only the G2-structure φ˜ is torsion-free.
An orientable 3-dimensional submanifold X˜ of M˜ is called associative if
φ˜|X˜ = volX˜ for some orientation of X˜. So X˜ is an associative submanifold
of M˜ if and only if X := S1 × X˜ is a Cayley submanifold of M .
Proposition 7.15. Let M˜ be an asymptotically cylindrical 7-manifold with an
asymptotically cylindrical torsion-free G2-structure, and let X˜ be an asymp-
totically cylindrical associative submanifold of M˜ . Define M := S1 × M˜ and
X := S1 × X˜. Furthermore, let Y be an asymptotically cylindrical local de-
formation of X with the same asymptotic limit as X.
If Y is a Cayley submanifold of M , then Y = S1× Y˜ for some asymptotically
cylindrical associative submanifold Y˜ of M˜ . So the moduli space of all local
deformations of X as an asymptotically cylindrical Cayley submanifold of M
with the same asymptotic limit as X can be identified with the moduli space of all
local deformations of X˜ as an asymptotically cylindrical associative submanifold
of M˜ with the same asymptotic limit as X˜.
Proof. For p ∈ S1, let Yp := Y ∩ ({p} × M˜). Note that Yp is an asymptotically
cylindrical submanifold of M˜ for all p ∈ S1 as Y is a local deformation of X.
Then
vol({y ∈ Y : t ≤ T}) ≥
∫
S1
vol({y ∈ Yeiθ : t ≤ T}) dθ ,
172
and equality holds if and only if Y = S1 × Y1. Furthermore,
lim
T→∞
(
vol({y ∈ Yp : t ≤ T})− vol({x ∈ X˜ : t ≤ T})
)
≥ 0
by Proposition 6.15 with φ = φ˜. Here we have equality if and only if Yp is
associative. But
lim
T→∞
(
vol({y ∈ Y : t ≤ T})− vol({x ∈ X : t ≤ T})
)
= 0
by Proposition 6.15 with φ = Φ. Hence we get equality in the above inequalities
(noting that the difference in the first inequality is increasing as T →∞). So
Y = S1 × Y1, and Y1 is an asymptotically cylindrical associative submanifold
of M˜ .
Now suppose that M˜ is an asymptotically cylindrical 7-manifold with cross-
section N˜ and with an asymptotically cylindrical torsion-free G2-structure,
and that X˜ is an asymptotically cylindrical associative submanifold of M˜ with
cross-section Y˜ . Then N˜ is a Calabi–Yau 3-fold and Y˜ is a complex curve in N˜ .
Define M := S1 × M˜ , N := S1 × N˜ , X := S1 × X˜, and Y := S1 × Y˜ . We
have χ(X) = χ(S1) · χ(X˜) = 0. Furthermore, X ∼= S1 × X˜ has an orientation-
reversing diffeomorphism, and hence σ(X) = 0. Moreover, νM˜X˜ has a non-
vanishing section s by [Ste51, Corollary 29.3] since X˜ is 3-dimensional and
νM˜X˜ has rank 4. Extend s to a section of νMX that is invariant under rotations
of S1. Then s is a non-vanishing section of νMX, and hence e(νMX, s) = 0.
Therefore,
indλD = − dimCH0(Y˜ , νN˜ Y˜ ) (7.6)
by Proposition 6.26, where λ > 0 is such that [−λ, 0) contains no eigenvalue
of D˜.
173

Concluding remarks
In this thesis, we have investigated the deformations of compact Cayley sub-
manifolds with boundary and of asymptotically cylindrical Cayley submanifolds.
For other calibrations, two further classes of submanifolds to which McLean’s
results were extended are asymptotically conical submanifolds and submanifolds
with conical singularities. See, for example, Marshall [Mar02], Joyce [Joy03],
Pacini [Pac04], and Lotay [Lot07, Lot09, Lot11]. Now one could also investigate
the deformations of asymptotically conical Cayley submanifolds or of Cayley
submanifolds with conical singularities.
Asymptotically conical submanifolds can be modelled as submanifolds with
cylindrical ends whose metrics are asymptotic to conical metrics, that is,
metrics of the form g = dr2 + r2gY for r > 0 as r →∞. Elliptic operators on
such manifolds can be dealt with via the Fredholm theory on asymptotically
cylindrical manifolds (if g = dr2 + r2gY is conical, then e−2tg = dt2 + gY
is cylindrical, where t = log r). In fact, the index calculation should be
similar, except that there will be a correction term depending on the spectrum
of D˜, where D˜ is the operator of Dirac type that arises as the linearisation
of the deformation map for associative deformations of Y . Furthermore,
a similar genericity statement should hold, that is, we expect that for a
generic asymptotically conical Spin(7)-structure, asymptotically conical Cayley
submanifolds will form a smooth moduli space.
Cayley submanifolds with conical singularities can also be modelled as
submanifolds with cylindrical ends whose metrics are asymptotic to conical
metrics, but this time the asymptotic behaviour occurs as r → 0. So assuming
that we demand that the singular points and tangent cones are fixed (although
generalisations are also sensible), the index calculation should again be similar.
175
For the genericity statement, there is one additional constraint: We regard these
as submanifolds in a smooth Spin(7)-manifold. So the problem is the extension
of the infinitesimal deformation of the Spin(7)-structure to the singular point.
However, in the proof of Theorem 1.6, note that the image of the operator D
can be characterised as the L2-orthogonal complement of the kernel of the
formal adjoint D∗ with respect to the negative weight. Since this kernel is
stable under small variations of the weight, we see that we may complement the
image of D by sections with compact support. With a similar argument in the
case of a Cayley submanifold with conical singularities, one then could extend
that section up to the singular point (which is only singular for the Cayley
submanifold but not for the ambient Spin(7)-manifold). So we expect that
for a generic Spin(7)-structure that agrees with the initial Spin(7)-structure
at all singular points (including its first derivative), asymptotically conical
Cayley submanifolds with the same singular points and tangent cones will form
a smooth moduli space.
For example, the complete Spin(7)-manifold constructed by Bryant and
Salamon in [BS89] (which we discussed in Section 5.3) is asymptotically conical.
This manifold is (topologically) the negative spin bundle over S4. On the one
hand, the fibres of this bundle are asymptotically conical Cayley submanifolds.
Moreover, if Σ is a minimal surface in S4, then there are certain rank 2
subbundles over Σ such that the total spaces are Cayley submanifolds [KMO05,
Theorem 4.8]. These Cayley submanifolds are also asymptotically conical. Note
that for every genus g, there is an embedding of the closed orientable surface
of genus g as a minimal submanifold of S3 [Law70, Theorem 2]. So we have
an infinite family of (topologically) different asymptotically conical Cayley
submanifolds inside this Spin(7)-manifold.
176
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