Index theory for Callias-type operators

by Pengshuai Shi

B.S. and M.S. in Mathematics, Zhejiang University

A dissertation submitted to

The Faculty ofthe College of Science ofNortheastern University

in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy

March 30, 2018

Dissertation directed by

Maxim BravermanProfessor of Mathematics

1

Acknowledgments

First and foremost I would like to express my sincere gratitude to my advisor Professor

Maxim Braverman, who led me to the field of index theory and guided me with his knowledge

and encouragement throughout my PhD study. His mathematical insight inspired me alot

in our meetings and discussions.

I am grateful to Professor Robert McOwen and Professor Petar Topalov for their support

both mathematically and in my applications.

I would like to thank Professor Ivan Loseu, Professor Alexandru Suciu, Professor Gordana

Todorov, Professor Valerio Toledano Laredo, Professor Jonathan Weitsman, Professor Ryan

Kinser, Professor Chris Kottke, Professor Gideon Maschler, Dr. Simone Cecchini and others

for all I learned from them.

Thanks to the department staff for their assistance. Thanks to my friends including my

office mates and all whom I worked with for their help.

Last but not least, I wish to thank my family especially my parents for all the uncondi-

tional love and support over the years.

Pengshuai Shi

Northeastern University

March 2018

2

Abstract of Dissertation

We address several index problems for Callias-type operators — a certain class of per-

turbed Dirac operators, on non-compact manifolds. We first show that the index of Callias-

type operators is preserved under cobordism, which generalizes a well-known result for the

index of elliptic operators on closed manifolds. We then concentrate on boundary value

problems, with emphasis on the case where the boundary is non-compact. We obtain index

theorems for (strongly) Callias-type operators under Atiyah–Patodi–Singer boundary con-

ditions. An interesting boundary invariant shows up in the theorem which we call relative

eta invariant and study its properties. We also investigate the relationship between Cauchy

data spaces and the Atiyah–Patodi–Singer index of strongly Callias-type operators.

3

Table of Contents

Acknowledgments 2

Abstract of Dissertation 3

Table of Contents 4

Introduction 6

0.1 The object — Callias-type operators . . . . . . . . . . . . . . . . . . . . . . . . 7

0.2 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Chapter 1 Cobordism Invariance of the Index of Callias-Type Operators 16

1.1 The outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2 Index of the operator Ba,δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3 The model operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.4 Proof of Theorem 1.1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.5 The gluing formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.6 Relative index theorem for Callias-type operators . . . . . . . . . . . . . . . . . 39

Chapter 2 The Index of Callias-Type Operators with Atiyah–Patodi–Singer

Boundary Conditions 44

2.1 Manifolds with compact boundary . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3 Callias-type operators with APS boundary conditions . . . . . . . . . . . . . . . 56

2.4 Proof of Theorem 2.3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Chapter 3 Boundary Value Problems for Strongly Callias-Type Operators 69

3.1 Operators on manifolds with non-compact boundary . . . . . . . . . . . . . . . 69

3.2 Domains of strongly Callias-type operators . . . . . . . . . . . . . . . . . . . . . 73

4

3.3 Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.4 Index theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Chapter 4 The Atiyah–Patodi–Singer Index on Manifolds with Non-Compact

Boundary: Odd-Dimensional Case 104

4.1 The outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.2 Reduction to an essentially cylindrical manifold . . . . . . . . . . . . . . . . . . 108

4.3 The index of operators on essentially cylindrical manifolds . . . . . . . . . . . . 113

4.4 The relative η-invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.5 The spectral flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Chapter 5 The Atiyah–Patodi–Singer Index on Manifolds with Non-Compact

Boundary: Even-Dimensional Case 129

5.1 The index of operators on essentially cylindrical manifolds . . . . . . . . . . . . 130

5.2 The relative η-invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.3 The spectral flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Chapter 6 Cauchy Data Spaces and Atiyah–Patodi–Singer Index on Non-

Compact Manifolds 147

6.1 Maximal Cauchy data spaces and index formulas . . . . . . . . . . . . . . . . . 148

6.2 Cauchy data spaces and boundary value problems . . . . . . . . . . . . . . . . . 156

Bibliography 168

5

Introduction

The Atiyah–Singer index theorem (on closed manifolds) [7, 8] is one of the great mathematical

achievements of the twentieth century, building a bridge between analysis (the analytical

index which describes the solutions of a system of differential equations) and topology (the

topological index which is determined by topological data). Since the theory was established,

people have been interested in generalizing it to various situations. Among the numerous

generalizations, there are two directions that are closely related to the work of this thesis.

The first one is the study of the index of a Dirac-type operator with potential on a

complete odd-dimensional manifold, which was initiated by Callias [35] and further studied

by many authors, cf. for example, [21, 32, 3, 57, 34]. A celebrated Callias index theorem

discovered by these authors in different forms states that the index of a Callias-type operator

can be computed as an index of a certain operator induced by the potential on a compact

hypersurface. Several generalizations and applications of the Callias index theorem were

obtained recently in [50, 36, 66, 51, 28]. In Chapter 1, we show the cobordism invariance of

the Callias index.

The other one is the so-called Atiyah–Patodi–Singer (or APS) index theorem on (com-

pact) manifolds with boundary investigated by Aityah, Patodi and Singer [5]. It expresses

the index of a first-order elliptic operator under APS boundary condition as the sum of the

cohomological term that appeared in the classical Atiyah–Singer index theorem and a bound-

ary term. It was later realized that this non-local APS boundary condition is a rather typical

representative of elliptic boundary conditions. Recently, Bar and Ballmann [11, 12] provided

a thorough and comprehensive description of boundary value problems for first-order elliptic

operators on (not necessarily compact) manifolds with compact boundary, making one ready

6

for the study of the APS index of Callias-type operators. In Chapter 2, we give an index

formula about it.

The study of Callias-type operators on manifolds with non-compact boundary was initi-

ated by Fox and Haskell [39, 40]. Under rather strong conditions on the manifold and the

operator they showed the Fredholmness and proved a version of the APS index theorem

in this situation. We want to address this problem in the general situation. Therefore we

need to first develop a theory of boundary value problems on manifolds with non-compact

boundary. This can be seen as a generalization of Bar and Ballmann’s work and is presented

in Chapter 3. On the basis of it, we derive a formula for the index of strongly Callias-

type operators under APS boundary conditions, in Chapters 4 (odd-dimensional case) and 5

(even-dimensional case). We found an interesting boundary invariant in the formula which

behaves like the difference of two individual η-invariants. As a result, we call it the relative

η-invariant and illustrate its properties. At last, we study the relationship between Cauchy

data spaces and the APS index of strongly Callias-type operators, in Chapter 6.

Now we introduce the object studied in the thesis and formulate the main results.

0.1 The object — Callias-type operators

0.1.1 Dirac operators

Let M be a complete Riemannian manifold (with or without boundary) and let E → M

be a Hermitian vector bundle over M . We use the Riemannian metric of M to identify the

tangent and the cotangent bundles, T ∗M ' TM .

Definition 0.1.1 ([52, Definition II.5.2]). The bundle E is called a Dirac bundle over M if

the following data is given

(i) a Clifford multiplication c : TM ' T ∗M → End(E), such that c(ξ)2 = −|ξ|2 and

c(ξ)∗ = −c(ξ) for every ξ ∈ T ∗M ;

7

(ii) a Hermitian connection ∇E on E which is compatible with the Clifford multiplication

in the sense that

∇E(c(ξ)u

)= c(∇LCξ)u + c(ξ)∇Eu, u ∈ C∞(M,E).

Here ∇LC denotes the Levi-Civita connection on T ∗M .

If E is a Dirac bundle we consider the Dirac operator D : C∞(M,E)→ C∞(M,E) defined

by

D =n∑j=1

c(ej)∇Eej, (0.1.1)

where e1, . . . , en is an orthonormal basis of TM ' T ∗M . One easily checks that D is formally

self-adjoint, D∗ = D.

0.1.2 Callias-type operators

Let D : C∞(M,E)→ C∞(M,E) be a Dirac operator. Suppose Φ ∈ End(E) is a self-adjoint

bundle map (called the Callias potential). Then

D := D + iΦ

is a Dirac-type operator on E with formal adjoint given by

D∗ = D − iΦ.

So

D∗D = D2 + Φ2 + i[D,Φ],

DD∗ = D2 + Φ2 − i[D,Φ],

(0.1.2)

where

[D,Φ] := DΦ − ΦD

is the commutator of the operators D and Φ.

8

Definition 0.1.2. Let D = D + iΦ be as above.

(1) We call D a Callias-type operator if

(i) [D,Φ] is a zeroth order differential operator, i.e. a bundle map;

(ii) there exist a compact subset K ⊂M and a constant c > 0 such that

Φ2(x) − |[D,Φ](x)| ≥ c

for all x ∈ M \ K. Here |[D,Φ](x)| denotes the operator norm of the linear map

[D,Φ](x) : Ex → Ex. In this case, the compact set K is called an essential support of

D.

(2) We call D a strongly Callias-type operator if it satisfies (i) and

(ii′) for any R > 0, there exists a compact subset KR ⊂M such that

Φ2(x) −∣∣[D,Φ](x)

∣∣ ≥ R (0.1.3)

for all x ∈ M \KR. In this case, the compact set KR is called an R-essential support

of D. For any R > 0, an R-essential support can serve as an essential support of D.

Remark 0.1.3. Strongly Callias-type operators are a stronger version of the Callias-type

operators in the sense that one requires the Callias potential to grow to infinity at the

infinite ends of the manifold. Note that D is a (strongly) Callias-type operator if and only

if D∗ is.

Remark 0.1.4. Condition (i) of Definition 0.1.2 is equivalent to the condition that Φ com-

mutes with the Clifford multiplication[c(ξ),Φ

]= 0, for all ξ ∈ T ∗M. (0.1.4)

Example 0.1.5. A natural choice of a Callias potential that satisfies condition (i) is Φ =

f ∈ C∞(M,R), a real-valued function on M . Then condition (ii) is the restriction that

f 2 − |df | ≥ c

outside a compact set. Similarly, one can derive the restriction from condition (ii′).

9

0.1.3 Graded Callias-type operators

It is often convenient to consider Z2-graded operators in index theory 1. So we introduce the

notion of graded Callias-type operators.

Let D : C∞(M,E) → C∞(M,E) be a Dirac operator and Ψ ∈ End(E) be a self-adjoint

bundle map (which is also called a Callias potential). Then

D := D + Ψ (0.1.5)

is a formally self-adjoint Dirac-type operator on E and

D2 = D2 + Ψ2 + [D,Ψ]+,

where [D,Ψ]+ := D ◦Ψ + Ψ ◦D is the anticommutator of the operators D and Ψ.

Suppose now that E = E+ ⊕ E− is a Z2-graded Dirac bundle such that the Clifford

multiplication c(ξ) is odd and the Clifford connection is even with respect to this grading.

Then

D :=

0 D−

D+ 0

is the Z2-graded Dirac operator, where D± : C∞(M,E±) → C∞(M,E∓) are formally

adjoint to each other. Assume that the Callias potential Ψ has odd grading degree, i.e.,

Ψ =

0 Ψ−

Ψ+ 0

,

where Ψ± ∈ Hom(E±, E∓) are adjoint to each other. Then we have

D = D + Ψ =

0 D− + Ψ−

D+ + Ψ+ 0

=:

0 D−

D+ 0

, (0.1.6)

where D+ and D− are formal adjoint of each other.

Definition 0.1.6. (1) We call D (or D+,D−) a graded Callias-type operator if

1For example, on even-dimensional spin manifolds.

10

(i) [D,Ψ]+ is a zeroth order differential operator, i.e. a bundle map;

(ii) there exist a compact subset K ⊂M and a constant c > 0 such that

Ψ2(x) −∣∣[D,Ψ]+(x)

∣∣ ≥ c

for all x ∈M \K. In this case, the compact set K is called an essential support of D

(or D+,D−).

(2) We call D (or D+,D−) a graded strongly Callias-type operator if it satisfies (i) and

(ii′) for any R > 0, there exists a compact subset KR ⊂M such that

Ψ2(x) −∣∣[D,Ψ]+(x)

∣∣ ≥ R (0.1.7)

for all x ∈ M \KR. In this case, the compact set KR is called an R-essential support

of D (or D+,D−). For any R > 0, an R-essential support can serve as an essential

support of D (or D+,D−).

Sometimes we will omit the term “graded” if it is clear from context.

Remark 0.1.7. Condition (i) of Definition 0.1.6 is equivalent to the condition that Ψ anti-

commutes with the Clifford multiplication:[c(ξ),Ψ

]+

= 0, for all ξ ∈ T ∗M .

Remark 0.1.8. When M is an oriented even-dimensional manifold, there is a natural grading

of E induced by the Hodge ∗-operator. We will consider this situation in Chapter 5.

Remark 0.1.9. Definition 0.1.6 is more general than Definition 0.1.2. Suppose there is a

skew-adjoint isomorphism γ : E± → E∓, γ∗ = −γ, which anticommutes with multiplication

c(ξ) for all ξ ∈ T ∗M , satisfies γ2 = −1, and is flat with respect to the connection ∇E, i.e.

[∇E, γ] = 0. Then ξ 7→ γ ◦ c(ξ) defines a Clifford multiplication of T ∗M on E+ and the

corresponding Dirac operator is D+ = γ ◦D+. Suppose also that γ commutes with Ψ. Then

Φ+ = −iγ ◦Ψ+ is a self-adjoint endomorphism of E+. In this situation,

D+ + iΦ+ = γ ◦ D+ : C∞(M,E+) → C∞(M,E+)

is a strongly Callias-type operator in the sense of Definition 0.1.2.

11

0.2 The main results

A well-known feature of the index on closed manifolds is the so-called cobordism invariance

(cf. [56, Chapter XVII]), which roughly says that if an elliptic operator P on a closed

manifold M can be extended to an elliptic operator P on a compact manifold M such that

the boundary of M is M , then the index of P is equal to 0. In Chapter 1, we generalize this

result to Callias-type operators, which can be stated as the following.

Theorem A. Let D be a graded Callias-type operator on a complete non-compact manifold

M . Suppose that D can be extended to a Callias-type operator D on a complete non-compact

manifold M such that ∂M = M . Then indD+ = 0.

Since we are on non-compact manifolds now, we need to take extra effort in some steps of

the proof due to the non-compact setting. As applications of Theorem A, we also prove a

gluing formula and a relative index theorem.

Chapter 2 concerns the APS index problem for Callias-type operators on non-compact

manifolds with compact boundary. Our main theorem is:

Theorem B. Let M be an odd-dimensional complete manifold with compact boundary and

let D = D+ iΦ be an ungraded Callias-type operator on M . Let A be the restriction of D to

the boundary ∂M (the tangential operator) with inward as positive direction. We denote by

DAPS the operator D imposed with Atiyah–Patodi–Singer boundary condition. Then

indDAPS =1

2(ind ∂+

+ − ind ∂+−) − 1

2(dim kerA + η(A)), (0.2.1)

where ∂± are the graded Dirac operators induced by the Callias potential Φ on a compact

hypersurface outside the essential support of D and η(A) is the η-invariant of A as in the

APS index theorem.

This result can be regarded as a generalization of both the APS index theorem (where M

is compact, so the first summand on the right-hand-side of (0.2.1) vanishes just like the

12

cohomological term does in odd-dimensional case) and the Callias index theorem (where

∂M = ∅, so the second summand on the right-hand-side of (0.2.1) vanishes). The proof

uses a relative index theorem (with boundary) which is indicated in [11] and the APS index

theorem of [5]. In particular, it provides a new proof of the usual Callias index theorem.

In Chapter 3, we discuss the boundary value problems on manifolds with non-compact

boundary, following the framework of Bar and Ballmann [11]. To get spectral decomposition

on the boundary, we want the restriction of the operator to the boundary to have discrete

spectrum. As a result, we consider the class of strongly Callias-type operators.

Let D be a strongly Callias-type operator on a complete manifold M and A be the

restriction of D to the boundary ∂M . We use the spectral decomposition of A to give a

delicate definition of Sobolev spaces on ∂M . We can actually prove that they share the

properties like Sobolev spaces on a compact domain. It turns out that they determine the

boundary value problems of D, which is our first main result.

Theorem C. The boundary value problems of D are characterized by closed subspaces of

the hybrid Sobolev space H(A) (cf. (3.2.13)) on the boundary. An elliptic boundary value

problem is a boundary value problem such that both the boundary condition and its adjoint

boundary condition lie in the H1/2-Sobolev spaces.

Like in the compact case, (generalized) APS boundary value problems are still elliptic. Hence

we can study its index as a result of the following theorem.

Theorem D. Let DB be a strongly Callias-type operator with an elliptic boundary condition

B. Then DB is a Fredholm operator.

Theorem D enables us to explore the index of strongly Callias-type operators with APS

boundary conditions, which is implemented in Chapters 4 and 5. Basically we can apply a

splitting theorem to reduce the index to a model manifold called an essentially cylindrical

manifold. We then solve the problem there.

13

Theorem E. Let DAPS be an APS boundary value problem for a strongly Callias-type op-

erator on an essentially cylindrical manifold M whose boundary is the disjoint union of two

non-compact manifolds N0 and N1. Then

indDAPS −∫M

αAS(D)

depends only on the restrictions A0 and −A1 of D to the boundary, where αAPS(D) denotes

the local Atiyah–Singer index density of D.

If two operators A0 and −A1 can be realized as the restriction of a strongly Callias-type

operator to the boundary of an essentially cylindrical manifold, we call them almost compact

cobordant. It follows from the theorem that we can define the relative η-invariant as

η(A1,A0) := 2(

indDAPS −∫M

αAS(D))

+ dim kerA0 + dim kerA1.

It satisfies

• Antisymmetry: η(A0,A1) = −η(A1,A0);

• Cocycle condition: η(A2,A0) = η(A2,A1) + η(A1,A0).

The proof techniques of Theorem E are fairly different for the odd-dimensional case (Chapter

4, where αAS(D) vanishes) and the even-dimensional case (Chapter 5). We also get a formula

for the relative η-invariant in terms of the spectral flow.

In Chapter 6, we relate the APS index of strongly Callias-type operators to their Cauchy

data spaces. We manage to prove that

Theorem F. Suppose D is a strongly Callias-type operator on a Dirac bundle E over M

whose restriction to the boundary is A. Let L2[0,∞)(A) be the subspace of L2(∂M,E|∂M) which

is generated by the eigensections of A corresponding to non-negative eigenvalues and let C

be the L2-Cauchy data space of D. Let

Π+(A) : L2(∂M,E|∂M)� L2[0,∞)(A) and P (D) : L2(∂M,E|∂M)� C.

14

be the orthogonal projections. Set T : C → L2[0,∞)(A) to be the restriction of Π+(A) to the

range of P (D). Then T is a Fredholm operator and indT = indDAPS.

In addition, we can realize the counterpart of the H1/2-Cauchy data space as an elliptic

boundary condition.

Chapter 1 is based on the joint work [29] with Maxim Braverman. Chapter 2 is based on

the author’s work [62]. Chapters 3, 4 and 5 are based on the joint work [31, 30] with Maxim

Braverman. Chapter 6 is based on the author’s work [63].

15

Chapter 1

Cobordism Invariance of the Index of

Callias-Type Operators

In this chapter we define a class of cobordisms between Callias-type operators and show that

the Callias-type index is preserved by this class of cobordisms. The proof of this theorem is

similar to the proof of the cobordism invariance of the index on a compact manifold, given

in [24], but a more careful analysis is needed. We also present several applications of this

result.

1.1 The outline

1.1.1 The index of a Z2-graded Callias-type operator

Let (M, g) be a complete Riemannian manifold without boundary. Suppose M is endowed

with a Hermitian vector bundle E. We denote by C∞0 (M,E) the space of smooth sections

of E with compact support, and by L2(M,E) the completion of C∞0 (M,E) with respect to

the norm ‖ · ‖ induced by the L2-inner product

(s1, s2) =

∫M

〈s1(x), s2(x)〉Ex dvol(x), (1.1.1)

where 〈·, ·〉Ex denotes the fiberwise inner product and dvol(x) is the canonical volume form

induced by the metric g.

16

Let D : C∞0 (M,E)→ C∞0 (M,E) be a first-order formally self-adjoint elliptic differential

operator (not necessarily of Dirac type) and let Ψ ∈ End(E) be a self-adjoint bundle map.

Suppose that E = E+ ⊕ E− is a Z2-graded vector bundle and that D and Ψ are odd with

respect to this grading.

Definition 1.1.1. We say that D + Ψ is a (generalized) Callias-type operator if it satisfies

conditions (i) and (ii) of Definition 0.1.6.

In this chapter, we will further assume that D satisfies the following assumption.

Assumption 1.1.2. There exists a constant k > 0 such that

0 < |σ(D)(x, ξ)| ≤ k‖ξ‖, for all x ∈M, ξ ∈ T ∗xM \ {0}, (1.1.2)

where ‖ξ‖ denotes the length of ξ defined by the metric g, σ(D)(x, ξ) : E±x → E∓x is the

leading symbol of D.

An interesting class of examples of operators satisfying (1.1.2) is given by Dirac-type

operators. In this case the operators defined in Definition 1.1.1 is just the usual Callias-type

operators in the sense of Definition 0.1.6.

By [43, Theorem 1.17], (1.1.2) implies that D and D + Ψ are essentially self-adjoint

operators with initial domain C∞0 (M,E). We view D + Ψ as an unbounded operator on

L2(M,E). By a slight abuse of notation we also denote by D+ Ψ the closure of D+ Ψ. Let

‖ · ‖ denote the norm on L2(M,E) induced by (1.1.1).

Remark 1.1.3. It is easy to see from (0.1.5) that a Callias-type operator D + Ψ satisfying

Assumption 1.1.2 is invertible at infinity, i.e.,

‖(D + Ψ)s‖ ≥√c‖s‖, for all s ∈ L2(M,E), supp(s) ∩K = ∅. (1.1.3)

It follows from [2, Theorem 2.1] and Remark 1.1.3 that

Lemma 1.1.4. If Assumption 1.1.2 is satisfied, then a Callias-type operator D+ Ψ is Fred-

holm.

17

Thus ker(D+ Ψ) = ker(D+ + Ψ+)⊕ ker(D−+ Ψ−) ⊂ L2(M,E) is finite-dimensional, and

the index,

ind(D+ + Ψ+) := dim ker(D+ + Ψ+) − dim ker(D− + Ψ−) (1.1.4)

is well-defined. To simplify the notation, we use ind(D + Ψ) to denote the index in this

chapter.

1.1.2 Cobordism of Callias-type operators

We now introduce a class of non-compact cobordisms similar to those considered in [42, 45,

23, 25]. One of the main results of this chapter is that the index (1.1.4) is preserved by this

class of cobordisms.

Definition 1.1.5. Suppose (M1, E1, D1 + Ψ1) and (M2, E2, D2 + Ψ2) are two triples which

are described as above. (W,F, D + Ψ) is a cobordism between them if

(i) W is a complete manifold with boundary ∂W and there is an open neighborhood U of

∂W and a metric-preserving diffeomorphism

φ : (M1 × (−ε, 0]) t (M2 × [0, ε)) → U. (1.1.5)

In particular, ∂W is diffeomorphic to the disjoint union M1 tM2.

(ii) F is a vector bundle (may not be graded) over W , whose restriction to U is isomorphic

to the lift of E1 and E2 over (M1 × (−ε, 0]) t (M2 × [0, ε));

(iii) D+ Ψ : C∞0 (W,F )→ C∞0 (W,F ) is a Callias-type operator with D satisfying Assump-

tion 1.1.2, and takes the form

D + Ψ = Di + γ∂t + Ψi (1.1.6)

on U , where t is the normal coordinate and γ|E±i = ±√−1, i = 1, 2.

18

If there exists a cobordism between (M1, E1, D1 + Ψ1) and (M2, E2, D2 + Ψ2) then the

operators D1 + Ψ1 and D2 + Ψ2 are called cobordant.

Remark 1.1.6. If M2 is the empty manifold, then (W,F, D + Ψ) is a null-cobordism of

(M1, E1, D1 + Ψ1). In this case the operator D1 + Ψ1 is called null-cobordant.

Remark 1.1.7. Let Eop2 denote the vector bundle E2 with opposite grading, namely Eop±

2 =

E∓2 . Consider the vector bundle E over M = M1 t M2 induced by E1 and Eop2 . Let

D + Ψ : C∞0 (M,E±) → C∞0 (M,E∓) be the operator such that D|Mi= Di, Ψ|Mi

= Ψi,

i = 1, 2. Then (W,F, D + Ψ) makes D + Ψ null-cobordant.

1.1.3 Cobordism invariance of the index

We now formulate the main result of this chapter:

Theorem 1.1.8. Let D1 + Ψ1 and D2 + Ψ2 be cobordant Callias-type operators. Then

ind(D1 + Ψ1) = ind(D2 + Ψ2).

By Remark 1.1.7, this is equivalent to the following

Theorem 1.1.9. The index of a null-cobordant Callias-type operator D+Ψ is equal to zero.

1.1.4 An outline of the proof of Theorem 1.1.9

Sections 1.2-1.4 deal with the proof of Theorem 1.1.9. We use the method of [24, 22] with

necessary modifications.

Suppose (W,F, D+Ψ) is a null-cobordism of (M,E,D+Ψ). In Section 1.2, we denote by

W the manifold obtained from W by attaching a semi-infinite cylinder M× [0,∞). Then for

small enough number δ > 0 we construct a family of Fredholm operators Ba,δ on W whose

index is independent of a ∈ R.

19

An easy computation, cf. Lemma 1.2.4, shows that for a � 0 the operator B2a,δ > 0.

Hence, its index is equal to 0. Hence,

ind Ba,δ = 0, for all a ∈ R. (1.1.7)

In Sections 1.3 and 1.4 we study the operator Ba,δ for a � 0. It turns out that the

sections in the kernel of this operator are concentrated on the cylinder M × [0,∞) near

the hypersurface M × {a}. Then we construct an operator Bmodδ on the cylinder M × R,

whose restriction to a neighborhood of M × {0} is very close to the restriction of Ba,δ to a

neighborhood of M × {a}. In a certain sense, Bmodδ is the limit of Ba,δ as a→∞. We refer

to Bmodδ as the model operator for Ba,δ. In Lemma 1.3.1 we compute the kernel of

ind Bmodδ = ind(D + Ψ). (1.1.8)

Finally, in Proposition 1.4.2 we show that

ind Bmodδ = ind Ba,δ. (1.1.9)

Theorem 1.1.9 follows immediately from (1.1.7), (1.1.8), and (1.1.9).

1.1.5 The gluing formula

As a first application of Theorem 1.1.8 we prove the gluing formula, cf. Section 1.5.

Suppose that (M,E,D + Ψ) is as in Subsection 1.1.1 and that Σ is a hypersurface in

M . Under certain conditions (cf. Assumption 1.5.1), if one cuts M along Σ and converts

it to a complete manifold without boundary by rescaling the metric, one gets a new triple

(MΣ, EΣ, DΣ + ΨΣ), with DΣ + ΨΣ being a Callias-type and, hence, a Fredholm operator.

Then the gluing formula asserts that

Theorem 1.1.10. The operators D + Ψ and DΣ + ΨΣ are cobordant. In particular,

ind(D + Ψ) = ind(DΣ + ΨΣ).

20

If M is partitioned into two relatively open submanifolds M1 and M2 by Σ, namely,

M = M1 ∪ Σ ∪M2, then the complete metric on MΣ induces complete Riemannian metrics

on M1 and M2. Let Ei, Di,Ψi denote the restrictions of the graded vector bundle EΣ and

operators DΣ,ΨΣ to Mi (i = 1, 2). The above theorem implies the additivity of the index

(cf. Corollary 1.5.5).

1.1.6 The relative index theorem

Section 1.6 is occupied with the second application of Theorem 1.1.8, which is a new proof

of the well-known relative index theorem for Callias-type operators.

Consider two triples (Mj, Ej, Dj + Ψj) as before (j = 1, 2). Suppose M ′j ∪Σj

M ′′j are

partitions of Mj into relatively open submanifolds, where Σj are compact hypersurfaces.

Suppose there exist tubular neighborhoods U(Σj) of Σj. We assume isomorphisms of

structures between Σ1 and Σ2, U(Σ1) and U(Σ2), E1|U(Σ1) and E2|U(Σ2). We also assume

that Ψj are invertible on U(Σj), and that D1,Ψ1 coincide with D2,Ψ2 on U(Σ1) ' U(Σ2)

(cf. Assumption 1.6.1). Then we can cut Mj along Σj and use the isomorphism map to glue

the pieces together interchanging M ′′1 and M ′′

2 . In this way we obtain the manifolds

M3 := M ′1 ∪Σ M

′′2 , M4 := M ′

2 ∪Σ M′′1 .

Similarly, we can do this cut-and-glue procedure to Ej to get new vector bundles E3 over

M3, E4 over M4. After restricting Dj,Ψj to each piece, we obtain Callias-type operators

D3 + Ψ3 on M3, D4 + Ψ4 on M4, both having well-defined indexes.

For simplicity, we denote the Callias-type operators Dj + Ψj, j = 1, 2, 3, 4 by Dj. Then

the relative index theorem can be stated as

Theorem 1.1.11. indD1 + indD2 = indD3 + indD4.

Our proof of the theorem involves the gluing formula. However, one has to do deformations

to Dj, j = 1, 2 first in order to have (Mj, Ej,Dj) along with Σj satisfy the hypothesis of the

21

gluing formula (cf. Subsections 1.6.3 and 1.6.4).

1.1.7 Callias-type index theorem

Using the relative index theorem, Anghel proved an important Callias-type index theorem

in [3]. Since we give a new proof of the relative index theorem here, we also obtain a new

proof of the Callias-type index theorem.

1.2 Index of the operator Ba,δ

In this section, we construct a family of operators Ba,δ on W := W ∪(∂W × [0,∞)

), such

that the index ind Ba,δ = 0. Later in Section 1.4, we show that ind Ba,δ = ind(D + Ψ) for

a� 0.

1.2.1 Construction of Ba,δ

Consider two anti-commuting actions (“left” and “right” action) of the Clifford algebra of R

on the exterior algebra ∧•C = ∧0C⊕ ∧1C given by

cL(t)ω = t ∧ ω − ιtω, cR(t)ω = t ∧ ω + ιtω. (1.2.1)

We define W := W ∪ (M × [0,∞)) as in Subsection 1.1.4 and extend the vector bundle

F and the operators D, Ψ to W in the natural way. Set F := F ⊗ ∧•C and consider the

operator

B :=√−1 (D + Ψ)⊗ cL(1) : C∞0 (W , F )→ C∞0 (W , F ).

Let f : R→ [0,∞) be a smooth function with f(t) = t for t ≥ 1, and f(t) = 0 for t ≤ 1/2.

Consider the map p : W → R such that p(y, t) = f(t) for (y, t) ∈ M × (0,∞) and p(x) = 0

for x ∈ W . For any a ∈ R and δ > 0, define the operator

Ba,δ := B − 1⊗ δ · cR (p(x)− a). (1.2.2)

22

Note that as a first order differential operator on the complete manifold W , the leading

symbol of Ba,δ is equal to σ(D). Hence it satisfies (1.1.2). We conclude that Ba,δ is essentially

self-adjoint by [43, Theorem 1.17].

Lemma 1.2.1. Let Πi : F → F ⊗ ∧iC (i = 0, 1) be the projections. Then

B2a,δ = (D + Ψ)2 ⊗ 1 − δ ·R + δ2 |p(x)− a|2, (1.2.3)

where R is a uniformly bounded bundle map whose restriction to W vanishes, and

R|M×(1,∞) =√−1γ (Π1 − Π0), where γ|F± = ±

√−1. (1.2.4)

Proof. Note first that p(x)− a ≡ −a on W . Thus, since cR(−a) anti-commutes with B, we

have B2a,δ|W = B2|W + δ2a2 = (D + Ψ)2 ⊗ 1|W + δ2a2, which is (1.2.3).

Restricting Ba,δ to the cylinder M × (0,∞), we obtain

Ba,δ|M×(0,∞) =√−1(D + Ψ)⊗ cL(1) +

√−1γ ⊗ cL(1)∂t − 1⊗ δ(f(t)− a) · cR(1).

Since cL and cR anti-commute, we get

B2a,δ|M×(0,∞) = (D + Ψ)2 ⊗ 1−

√−1δf ′γ ⊗ cL(1)cR(1) + δ2|t− a|2.

Since cL(1)cR(1) = Π1 − Π0, (1.2.3) and (1.2.4) follow with R = f ′√−1γ(Π1 − Π0).

1.2.2 Fredholmness of Ba,δ

Lemma 1.2.2. There exists a small enough δ, such that Ba,δ is a Fredholm operator for

every a ∈ R.

Proof. By [2, Theorem 2.1], it is enough to show that the operator Ba,δ is invertible at

infinity (cf. (1.1.3)). Since Ba,δ is self-adjoint, (1.1.3) is equivalent to the fact that there

exists a constant c > 0 and a compact K b W such that

(B2a,δs, s) ≥ c ‖s‖2, for all s ∈ L2(W , F ), supp(s) ∩ K = ∅, (1.2.5)

23

where (·, ·) denotes the inner product on L2(W , F ). Note that if we denote the bundle map

Q := (Ψ2 + [D, Ψ]+)⊗ 1 − δ ·R + δ2 |p(x)− a|2,

then (1.2.3) can also be written as

B2a,δ = D2 ⊗ 1 + Q.

Since D2 is a non-negative operator on W , (1.2.5) can be reduced to

|Q(x)| ≥ c, for all x ∈ W \ K. (1.2.6)

Since both D + Ψ and D + Ψ are Callias-type operators, there exist compact subsets

K bM , KW b W and positive constants c, cW > 0, such that

|(Ψ2 + [D,Ψ]+)(y)| ≥ c, for all y ∈M \K, (1.2.7)

and

|(Ψ2 + [D, Ψ]+)(x)| ≥ cW , for all x ∈ W \KW . (1.2.8)

Now consider |(Ψ2 + [D, Ψ]+)(y, t)| for (y, t) ∈ M × [0,∞). Note that Ψ is independent

of t, and anti-commutes with γ∂t. So

[D, Ψ]+ = (D + γ∂t)Ψ + Ψ(D + γ∂t) = DΨ + ΨD = [D, Ψ]+.

Thus

|(Ψ2 + [D, Ψ]+)(y, t)| = |(Ψ2 + [D,Ψ]+)(y)|,

which does not depend on t. From (1.2.7), we get

|(Ψ2 + [D, Ψ]+)(y, t)| ≥ c, for all (y, t) ∈ (M × [0,∞)) \ (K × [0,∞)). (1.2.9)

Furthermore, since K is compact, Ψ2 + [D, Ψ]+ is bounded from below on M × [0,∞).

Set Wr := W ∪ (M × [0, r]), Kr := KW ∪ (K × [0, r]), r > 0 and d1 := min{c, cW}. By

(1.2.8) and (1.2.9),

|(Ψ2 + [D, Ψ]+)(x)| ≥ d1, for all x ∈ Wr \ Kr.

24

Since R is uniformly bounded on W , we can choose δ small enough such that

δ · supx∈W|R(x)| ≤ d1/2.

So

|Q(x)| ≥ d1

2, for all x ∈ Wr \ Kr. (1.2.10)

Since Ψ2 + [D, Ψ]+ has a uniform lower bound on M × [0,∞), and |p(y, t) − a|2 grows

quadratically as t→∞, there exist r = r(a, δ) and d2 > 0, such that

|Q(y, t)| ≥ d2, for all (y, t) ∈M × [r,∞). (1.2.11)

Set K := Kr(a,δ) and c := min{d1/2, d2}. Combining (1.2.10) and (1.2.11) yields (1.2.6).

Therefore the lemma is proved.

1.2.3 Index of Ba,δ

From now on we fix δ which satisfies Lemma 1.2.2. Define a grading on the vector bundle

F = F ⊗ ∧•C by

F+ := F ⊗ ∧0C, F− := F ⊗ ∧1C, (1.2.12)

and denote by B±a,δ := Ba,δ|L2(W ,F±) the restrictions. We consider the index

ind Ba,δ := dim ker B+a,δ − dim ker B−a,δ.

Lemma 1.2.3. ind Ba,δ is independent of a.

Proof. Since for every a, b ∈ R the operator Bb,δ −Ba,δ = 1 ⊗ δ · cR(b − a) is bounded and

depends continuously on b − a ∈ R, the lemma follows from the stability of the index of a

Fredholm operator.

Lemma 1.2.4. ind Ba,δ = 0 for all a ∈ R.

Proof. By Lemma 1.2.3, it suffices to prove this result for a particular value of a. If a

is a negative number such that a2 > supx∈W |R(x)|/δ, then B2a,δ > 0 by (1.2.3), so that

ker Ba,δ = 0 = ind Ba,δ.

25

1.3 The model operator

When a is large, all the sections s ∈ ker Ba,δ are concentrated on the cylinder M × [0,∞)

near M × {a}. Thus index of Ba,δ is related to the index of a certain operator on M × R,

whose restriction to a neighborhood of M ×{a} in W is an approximation of the restriction

of Ba,δ to the neighborhood of M ×{a} in M ×R. We call this operator the model operator

for Ba,δ and denote it by Bmodδ . In this section we construct the model operator and show

that ind Bmodδ = ind(D+Ψ). In the next section we show that its index is equal to the index

of Ba,δ.

1.3.1 The operator Bmodδ

Consider the lift of the bundle E = E+ ⊕ E− to the cylinder M × R, which will still be

denoted by E = E+ ⊕ E−.

Consider the vector bundle Fmod = (E+ ⊕ E−)⊗ ∧•C over M × R and the operator

Bmodδ : L2(M × R, Fmod) → L2(M × R, Fmod)

defined by

Bmodδ :=

√−1 (D + Ψ)⊗ cL(1) +

√−1 γ ⊗ cL(1)∂t − 1⊗ δ · cR(t), (1.3.1)

where t is the coordinate along the axis of the cylinder, γ|E± = ±√−1, and δ is fixed with

the same value as in Subsection 1.2.3. The operator Bmodδ satisfies Assumption 1.1.2 as well

and, hence, is self-adjoint. Like in Lemma 1.2.1, we have

(Bmodδ )2 = (D + γ∂t + Ψ)2 ⊗ 1 −

√−1 δγ (Π1 − Π0) + δ2t2.

Then by the same argument as in the proof of Lemma 1.2.2, Bmodδ is a Fredholm operator.

Clearly, the restrictions of Fmod and F to the cylinder M × (1,∞) are isomorphic. We

give Fmod grading similar to (1.2.12),

Fmod+ := E ⊗ ∧0C, Fmod

− := E ⊗ ∧1C.

26

Set

ind Bmodδ := dim ker(Bmod

δ )+ − dim ker(Bmodδ )−,

where (Bmodδ )± := Bmod

δ |L2(W ,Fmod± ).

Lemma 1.3.1. The space ker Bmodδ is isomorphic (as a graded space) to ker(D + Ψ). In

particular,

ind Bmodδ = ind(D + Ψ). (1.3.2)

Proof. The space L2(M × R, E± ⊗ ∧•C) decomposes into a tensor product

L2(M × R, E± ⊗ ∧•C) = L2(M,E±)⊗ L2(R,∧•C).

From (1.3.1) it follows that with respect to this decomposition we have

(Bmodδ )2|L2(M×R,E±⊗∧•C) = (D + Ψ)2 ⊗ 1 + 1⊗ (−∂tt ± δ (Π1 − Π0) + δ2t2).

Notice that both summands on the right hand side are non-negative.

The space ker (−∂tt + δ(Π1−Π0) + δ2t2) ⊂ L2(R,∧•C) is one-dimensional and is spanned

by

α+(t) = e−δt2/2 ∈ L2(R,∧0C).

Similarly, the space ker (−∂tt + δ(Π0 − Π1) + δ2t2) is one-dimensional and is spanned by

α−(t) = e−δt2/2ds ∈ L2(R,∧1C).

It follows that

ker (Bmodδ )2|L2(M×R,E±⊗∧•C) ' {σ ⊗ α±(t) : σ ∈ ker(D + Ψ)2|L2(M,E±)}.

27

1.3.2 The operator Bmoda,δ

Let

Ta : M × R → M × R, Ta(x, t) = (x, t+ a)

be the translation and consider the pull-back map

T ∗a : L2(M × R, Fmod) → L2(M × R, Fmod).

Set

Bmoda,δ := T ∗−a ◦Bmod

δ ◦ T ∗a =√−1(D + Ψ)⊗ cL(1) +

√−1γ ⊗ cL(1)∂t − 1⊗ δ · cR(t− a).

Then

dim ker(Bmoda,δ )± = dim ker(Bmod

δ )± (1.3.3)

for all a ∈ R.

1.4 Proof of Theorem 1.1.9

In this section, we finish the proof of the cobordism invariance of the Callias-type index by

showing that ind Ba,δ = ind Bmodδ . Since δ is fixed throughout the section, we omit it from

the notation, and write Ba,Bmod and Bmod

a for Ba,δ,Bmodδ and Bmod

a,δ , respectively.

1.4.1 The spectral counting function

For a self-adjoint operator P and a real number λ, we denote by N(λ, P ) the number of

eigenvalues of P not exceeding λ (counting with multiplicities). If the intersection of the

continuum spectrum of P with the set (−∞, λ] is not empty, then we set N(λ, P ) =∞.

Let B±a denote the restrictions of Ba to the spaces L2(W , F±) and let Bmod± , Bmod

a,± denote

the restrictions of Bmod, Bmoda to the spaces L2(M × R, Fmod

± ).

28

Since the operator Bmod is self-adjoint, by von Neumann’s theorem (cf. [58, Theorem

X.25]), the operators (Bmod)2± = Bmod

∓ Bmod± = (Bmod

± )∗Bmod± are also self-adjoint. Since

the operators (Bmod)2± are Fredholm, they have smallest non-zero elements of the spectra,

denoted by λ±.

Lemma 1.4.1. λ+ = λ−.

Proof. Since (Bmod)2+ = Bmod

− Bmod+ , (Bmod)2

− = Bmod+ Bmod

− , by [46, Theorem 1.1], their

spectra satisfy

σ((Bmod)2+) \ {0} = σ((Bmod)2

−) \ {0}.

In particular, λ+ = λ−.

From now on, we set

λ := λ+ = λ−.

Proposition 1.4.2. For any 0 < ε < λ, there exists A = A(ε, δ, p) > 0, such that

N(λ− ε, (B2a)±) = dim ker(Bmod)2

±, for all a > A, (1.4.1)

where δ > 0, p : W → R are as in Subsection 1.2.1. In particular,

ind Bmod = N(λ− ε, (B2a)

+) − N(λ− ε, (B2a)−). (1.4.2)

Before proving this proposition we show how it implies Theorem 1.1.9.

1.4.2 Proof of Theorem 1.1.9

By Proposition 1.4.2, N(λ− ε, (B2a)±) <∞. Let

V ±ε,a ⊂ L2(W , F±)

denote the vector spaces spanned by the eigenvectors of the operators (B2a)± with eigenvalues

within (0, λ− ε]. Then dimV ±ε,a <∞ and the restrictions of the operators B±a to V ±ε,a define

bijections

B±a : V ±ε,a −→ V ∓ε,a.

29

Hence,

dimV +ε,a = dimV −ε,a.

Thus

N(λ− ε, (B2a)

+)−N(λ− ε, (B2a)−) = (dim ker B+

a + dimV +ε,a) − (dim ker B−a + dimV −ε,a)

= dim ker B+a − dim ker B−a = ind Ba.

From Proposition 1.4.2 we now obtain

ind Ba = ind Bmod

and Theorem 1.1.9 follows from Lemma 1.2.4 and 1.3.1. �

The rest of this section is occupied with the proof of Proposition 1.4.2.

1.4.3 Estimate from above on N(λ− ε, (B2a)±)

First we show that

N(λ− ε, (B2a)±) ≤ dim ker Bmod

± . (1.4.3)

This is done through the following techniques.

1.4.4 The IMS localization

Let j, j : R→ [0, 1] be smooth functions such that j2 + j2 ≡ 1 and j(t) = 1 for t ≥ 3, while

j(t) = 0 for t ≤ 2. Set ja(t) = j(a−1/2t), ja(t) = j(a−1/2t). Now we view them as functions

on the cylinder M × [0,∞) (whose points are written as (y, t)). Similarly, we still use the

same notations ja(x) = j(a−1/2p(x)), ja(x) = j(a−1/2p(x)) to denote the functions on W ,

where p(x) is defined in Subsection 1.2.1 .

We use the following verison of the IMS localization, cf. [64, §3], [24, Lemma 4.5]1

1The abbreviation IMS is formed by the initials of the surnames of R. Ismagilov, J. Morgan, I. Sigal andB. Simon.

30

Lemma 1.4.3. The following operator identity holds:

B2a = jaB

2aja + jaB

2aja +

1

2[ja, [ja,B

2a]] +

1

2[ja, [ja,B

2a]]. (1.4.4)

Now we estimate each summand on the right-hand side of (1.4.4).

Lemma 1.4.4. There exists A = A(δ, p) > 0 such that

jaB2aja ≥

δ2a2

8j2a

for all a > A.

Proof. Note that if x ∈ supp ja, then p(x) ≤ 3a1/2. Hence for a > 36, we have

j2a |p(x)− a|2 ≥ a2

4j2a.

Set A = max{36, 4δ1/2 supx∈W ‖R(x)‖1/2} and let a > A. By Lemma 1.2.1,

jaB2aja ≥ j2

aδ2 |p(x)− a|2 − ja (δ ·R)ja ≥

δ2a2

8j2a.

1.4.5

Let Πa : L2(M×R, Fmod)→ ker Bmoda be the orthogonal projection and Π±a be the restrictions

of Πa to the spaces L2(M ×R, Fmod± ). Then Π±a are finite rank operators and their ranks are

dim ker Bmoda,± , which are equal to dim ker Bmod

± by (1.3.3). Since (Bmoda )2

± are nonnegative

operators, it’s clear that

(Bmoda )2

± + λΠ±a ≥ λ. (1.4.5)

Observe that supp ja in M×R is a subset of M×[0,∞). It’s a subset of W = W∪(M×[0,∞))

as well. So we can consider jaΠaja and jaBmoda ja as operators on W . Then jaB

2aja =

ja(Bmoda )2ja. Hence, (1.4.5) implies the following.

Lemma 1.4.5. ja(B2a)±ja + λjaΠ

±a ja ≥ λj2

a, rank jaΠ±a ja ≤ dim ker Bmod

± .

31

The next lemma estimates the last two summands on the right-hand side of (1.4.4).

Lemma 1.4.6. Let C = 2 max{

max{|j′(t)|2, |j′(t)|2} : t ∈ R}

. Then

‖[ja, [ja,B2a]]‖ ≤ Ca−1, ‖[ja, [ja,B2

a]]‖ ≤ Ca−1 for all a > 0. (1.4.6)

Proof. By Lemma 1.2.1, we get

‖[ja, [ja,B2a]]‖ = 2 |j′a(t)|2 = 2a−1 |j′(a−1/2t)|2,

‖[ja, [ja,B2a]]‖ = 2 |j′a(t)|2 = 2a−1 |j′(a−1/2t)|2.

Then (1.4.6) follows immediately.

Since λ is fixed, combining Lemma 1.4.3, 1.4.4, 1.4.5 and 1.4.6, we obtain

Corollary 1.4.7. For any ε > 0, there exists A = A(ε, δ, p) > 0 such that, for all a > A,

(B2a)± + λjaΠ

±a ja ≥ λ − ε, rank jaΠ

±a ja ≤ dim ker Bmod

± . (1.4.7)

The estimate (1.4.3) now follows from Corollary 1.4.7 and the following result, [59, p. 270]:

Lemma 1.4.8. Assume that P,Q are self-adjoint operators on a Hilbert space such that

rankQ ≤ k and there exists µ > 0 such that 〈(P + Q)u, u〉 ≥ µ〈u, u〉 for any u ∈ Dom(P ).

Then N(µ− ε, P ) ≤ k for any ε > 0.

1.4.6 Estimate from below on N(λ− ε, (B2a)±)

Now it remains to prove that

N(λ− ε, (B2a)±) ≥ dim ker Bmod

± = dim ker Bmoda,± . (1.4.8)

By (1.4.3), N(λ − ε, (B2a)±) are finite for a large enough. Under this circumstance, let

V ±ε,a ⊂ L2(W , F±) denote the vector spaces spanned by the eigenvectors of the operators

(B2a)± for eigenvalues within [0, λ − ε]. Let Θ±ε,a : L2(W , F±) → V ±ε,a be the orthogonal

32

projections. Then rank Θ±ε,a = N(λ − ε, (B2a)±). As in Subsection 1.4.5, we can consider

jaΘ±ε,aja as operators on L2(M × R, Fmod

± ). Then the same argument as in the proof of

Corollary 1.4.7 works here and we have

Lemma 1.4.9. For any ε > 0, there exists A = A(ε, δ) > 0 such that, for all a > A,

(Bmoda )2

± + λjaΘ±a ja ≥ λ − ε, rank jaΘ

±a ja ≤ dimN(λ− ε, (B2

a)±). (1.4.9)

Similarly, the estimate (1.4.8) follows from Lemma 1.4.9 and 1.4.8.

Now the proof of Proposition 1.4.2 and, hence, of Theorem 1.1.9 is complete. �

1.5 The gluing formula

Our first application of Theorem 1.1.8 is the gluing formula. If we cut a complete manifold

along a hypersurface Σ, we obtain a manifold with boundary. By rescaling the metric near

the boundary, we may convert it to a complete manifold without boundary. In this section,

we show that the index of a Callias-type operator is invariant under this type of surgery. In

particular, if M is partitioned into two pieces M1 and M2 by Σ, we see that the index on M

is equal to the sum of the indexes on M1 and M2. In other words, the index is additive.

1.5.1 The surgery

Let (M,E,D + Ψ) be as in Subsection 1.1.1 with dimM = n, where

D : C∞0 (M,E±) → C∞0 (M,E∓)

satisfies Assumption 1.1.2 and D+Ψ is a Callias-type operator. Suppose Σ ⊂M is a smooth

hypersurface. For simplicity, we assume that Σ is compact.

Throughout this section we make the following assumption.

Assumption 1.5.1. There exist a compact set K b M and two constants c1, c2 > 0 such

that

33

(i) for all x ∈M \K,

|(Ψ2 + [D,Ψ]+)(x)| ≥ c1, |Ψ2(x)| ≥ c2;

(ii) Σ ⊂M \K, which indicates that K is still a compact subset of MΣ := M \ Σ.

We denote by EΣ the restriction of the graded vector bundle E to MΣ. Let g denote the

Riemannian metric on M . By a rescaling of g near Σ, one can obtain a complete Riemannian

metric on MΣ and a Callias-type operator DΣ + ΨΣ on MΣ. It follows from the cobordism

invariance of the index (cf. Theorem 1.1.8) that the index of DΣ + ΨΣ is independent of the

choice of a rescaling.

1.5.2 A rescaling of the metric

We now present one of the possible constructions of a complete metric on MΣ.

Let τ : M → [−1, 1] be a smooth function, such that τ−1(0) = Σ and τ is regular at Σ.

Set α(x) = (τ(x))2. Define the metric gΣ

on MΣ by

gΣ

:=1

α(x)2g. (1.5.1)

This makes (MΣ, gΣ) a complete Riemannian manifold.

Let dvolg(x) and dvolgΣ

(x) denote the canonical volume forms on (M, g) and (MΣ, gΣ),

respectively. It’s easy to see that dvolgΣ

(x) = 1α(x)n

dvolg(x). So the L2-inner product on

L2(MΣ, EΣ) becomes

(s1, s2)Σ =

∫MΣ

〈s1(x), s2(x)〉(EΣ)x

1

α(x)ndvolg(x). (1.5.2)

1.5.3 The Callias-type operator on (MΣ, gΣ)

In order to get a natural Callias-type operator acting on C∞0 (MΣ, EΣ) we set

ΨΣ := Ψ|MΣ,

34

and

DΣ(x)(s) := α(x)n+1

2 D(x)(α(x)−n−1

2 s), for all x ∈MΣ, s ∈ C∞0 (MΣ, EΣ). (1.5.3)

It’s easy to check that

σ(DΣ)(x, ξ) = α(x)σ(D)(x, ξ)

So DΣ also satisfies Assumption 1.1.2. Thus DΣ and DΣ+ΨΣ : C∞0 (MΣ, E±Σ )→ C∞0 (MΣ, E

∓Σ )

are still Z2-graded first-order elliptic operators, which are essentially self-adjoint with respect

to the L2-inner product defined by (1.5.2).

Remark 1.5.2. If E is a Dirac bundle with respect to g, and D is the Dirac operator, then

EΣ also has a Clifford structure with respect to gΣ, and DΣ defined by (1.5.3) is precisely

the associated Dirac operator.

Lemma 1.5.3. DΣ + ΨΣ is a Callias-type operator, and, hence, is Fredholm.

Proof. Since [D,Ψ]+ is a bundle map, a direct computation gives that

[DΣ,ΨΣ]+(s) = DΣΨΣ(s) + ΨΣDΣ(s)

= αn+1

2 D(α−n−1

2 Ψ(s)) + Ψ(αn+1

2 D(α−n−1

2 s))

= αn+1

2 D(Ψ(α−n−1

2 s)) + αn+1

2 Ψ(D(α−n−1

2 s))

= αn+1

2 [D,Ψ]+(α−n−1

2 s) = α[D,Ψ]+(s).

So [DΣ,ΨΣ]+ is a bundle map as well. Then

Ψ2Σ + [DΣ,ΨΣ]+ = (Ψ2 + α [D,Ψ]+)|MΣ

= ((1− α)Ψ2 + α (Ψ2 + [D,Ψ]+))|MΣ.

Note that α(x) ∈ [0, 1], by Assumption 1.5.1,

|(Ψ2Σ + [DΣ,ΨΣ]+)(x)| ≥ c, for all x ∈MΣ \K,

where c := min{c1, c2}. Thus DΣ + ΨΣ is a Callias-type operator and, hence, is Fredholm

by Lemma 1.1.4.

It follows from the above lemma that the index ind(DΣ + ΨΣ) is well defined.

35

1.5.4 The gluing formula

Under the above setting, there are two well-defined indexes ind(D + Ψ) and ind(DΣ + ΨΣ).

Theorem 1.5.4. The operators D + Ψ and DΣ + ΨΣ are cobordant in the sense of Defini-

tion 1.1.5. In particular,

ind(D + Ψ) = ind(DΣ + ΨΣ).

We refer to Theorem 1.5.4 as a gluing formula, meaning that M is obtained from MΣ by

gluing along Σ.

Proof. The goal is to find a triple (W,F, D + Ψ), such that it is the cobordism between

(M,E,D + Ψ) and (MΣ, EΣ, DΣ + ΨΣ) and then apply Theorem 1.1.8.

Consider

W :={

(x, t) ∈M × [0,∞) : t ≤ 1

α(x)+ 1}.

Then W is a non-compact manifold whose boundary is diffeomorphic to the disjoint union of

M 'M ×{0} and MΣ = M \Σ ' {(x, 1α(x)

+ 1)}. Essentially, W is the required cobordism.

However, to be precise, we need to define a complete Riemannian metric gW on W , such

that condition (i) of Definition 1.1.5 is fulfilled.

Let β : W → [0, 1] be a smooth function such that β(x, t) = 1 for 0 ≤ t ≤ 1/2, β(x, t) > 0

for 1/2 < t < 1/α(x) + 1/2 and β(x, t) = α(x) for 1/α(x) + 1/2 ≤ t ≤ 1/α(x) + 1. Define

the metric gW on W by

gW ((ξ1, η1), (ξ2, η2)) :=1

β(x, t)2g(ξ1, ξ2) + η1η2, (1.5.4)

where (ξ1, η1), (ξ2, η2) ∈ TxM ⊕ R ' T(x,t)W . Then gW is a complete metric.

Consider the neighborhood

U = U1 t U2 := {(x, t) : 0 ≤ t < 1/3} t{

(x, t) :1

α(x)+

2

3< t ≤ 1

α(x)+ 1}

of ∂W 'M tMΣ. We define a diffeomorphism

φ : (M × [0, 1/3)) t (MΣ × (−1/3, 0]) → U

36

by the formulas

φ(x, t) := (x, t), x ∈M, 0 ≤ t < 1/3,

φ(x, t) :=(x,

1

α(x)+ 1 + t

), x ∈MΣ, −1/3 < t ≤ 0.

We claim that the diffeomorphism φ is metric-preserving. Clearly, the restriction of φ to

M × [0, 1/3) is metric-preserving. So we only need to show that the restriction of φ to

MΣ × (−1/3, 0] is metric preserving. Here MΣ × (−1/3, 0] is endowed with the product of

the metric gΣ

(cf. (1.5.1)) and the standard metric on the interval (−1/3, 0]. Thus

gMΣ×(−1/3,0]((ξ1, η1), (ξ2, η2)) =1

α(x)2g(ξ1, ξ2) + η1η2, (1.5.5)

for ξ1, ξ2 ∈ TxMΣ, η1, η2 ∈ R. The restriction of φ to MΣ×(−1/3, 0] is basically a translation

in t direction. Hence,

φ∗(ξ, η) = (ξ, η), ξ ∈ TxMΣ, η ∈ R.

Since the restriction of β to U2 = φ(MΣ× (−1/3, 0]

)is equal to α, we conclude from (1.5.5)

and (1.5.4) that φ is metric-preserving. The claim is proved.

Let π : M × [0,∞) → M be the projection. Then the pull-back π∗E is a vector bundle

over M × [0,∞). Define

F := π∗E|W .

So F is a vector bundle over W , whose restriction to the first part of U is isomorphic to the

lift of E over M × [0, 1/3) and whose restriction to the second part of U is isomorphic to the

lift of EΣ over MΣ× (−1/3, 0]. Hence condition (ii) of Definition 1.1.5 is fulfilled. Note that

here we can give F a natural grading which is compatible with that on E and EΣ:

F+ := π∗E+|W , F− := π∗E−|W .

We still use D and Ψ to denote the lifts of D and Ψ to M × [0,∞). Now we define

D, Ψ : C∞0 (W,F ) → C∞0 (W,F )

37

by

D(s) := (βn+1

2 D|W )(β−n−1

2 s) + γ∂t(s), for all s ∈ C∞0 (W,F ),

Ψ := Ψ|W ,(1.5.6)

where γ|F± = ±√−1. Then σ(D) = β(σ(D|W )) + σ(γ∂t). Since β lies in [0, 1], D satisfies

Assumption 1.1.2. Moreover, D + Ψ takes the form D + γ∂t + Ψ on one end M × [0, 1/3)

and the form DΣ + γ∂t + ΨΣ on the other end MΣ × (−1/3, 0]. So D + Ψ has exactly the

form required in condition (iii) of Definition 1.1.5.

It remains to verify that D + Ψ is a Callias-type operator. Note that γ∂t anti-commutes

with Ψ, by the same computation as in the proof of Lemma 1.5.3, we have

[D, Ψ]+ = β [D,Ψ]+|W

is a bundle map. And

Ψ2 + [D, Ψ]+ = ((1− β) Ψ2 + β (Ψ2 + [D,Ψ]+))|W .

By Assumption 1.5.1,

K :={

(x, t) ∈M × [0,∞) : x ∈ K, t ≤ 1

α(x)+ 1}

is a compact subset of W , and

|(Ψ2 + [D,Ψ]+)(x, t)| ≥ c1, |Ψ2(x, t)| ≥ c2, for all (x, t) ∈ W \ K.

Again by β ⊂ [0, 1], we get

|(Ψ2 + [D, Ψ]+)(x, t)| ≥ c, for all (x, t) ∈ W \ K,

where c is the same as in the proof of Lemma 1.5.3. Thus D + Ψ is a Callias-type operator,

and condition (iii) of Definition 1.1.5 is also fulfilled.

Therefore, (W,F, D+ Ψ) is a cobordism between (M,E,D) and (MΣ, EΣ, DΣ + ΨΣ), and

by Theorem 1.1.8, ind(D + Ψ) = ind(DΣ + ΨΣ).

38

1.5.5 The additivity of the index

Suppose that M is partitioned into two relatively open submanifolds M1 and M2 by Σ, so

that MΣ = M1 tM2. The metric gΣ

induces complete Riemannian metrics gM1, g

M2on M1

and M2, respectively. Let Ei, Di,Ψi denote the restrictions of the graded vector bundle EΣ

and operators DΣ,ΨΣ to Mi (i = 1, 2). Then Theorem 1.5.4 implies the following corollary.

Corollary 1.5.5. ind(D + Ψ) = ind(D1 + Ψ1) + ind(D2 + Ψ2).

Thus we see that the index is “additive”.

1.6 Relative index theorem for Callias-type operators

As a second application of Theorem 1.1.8, and also as an application of Corollary 1.5.5, we

give a new proof of the relative index theorem for Callias-type operators. There are several

different forms of relative index theorem. In this chapter we follow the approach of [34].

1.6.1 Setting

Let (Mj, Ej, Dj + Ψj), j = 1, 2 be two triples of complete Riemannian manifold endowed

with a Z2-graded Hermitian vector bundle and with the associated Callias-type operator

acting on the compactly supported smooth sections of the bundle. Suppose they satisfy

Assumption 1.5.1.(i) of Subsection 1.5.1. In particular, the indexes ind(Dj + Ψj), j = 1, 2

are well-defined.

Suppose M ′j ∪Σj

M ′′j are partitions of Mj into relatively open submanifolds, where Σj are

compact hypersurfaces. We make the following assumption.

Assumption 1.6.1. There exist tubular neighborhoods U(Σ1), U(Σ2) of Σ1 and Σ2 such

that:

39

(i) there is a commutative diagram of isometric diffeomorphisms

ψ : E1|U(Σ1) → E2|U(Σ2)

↓ ↓

φ : U(Σ1) → U(Σ2)

↑ ↑

φ|Σ1 : Σ1 → Σ2

(ii) Ψj are invertible bundle maps on U(Σj), j = 1, 2.

(iii) D1 and D2, Ψ1 and Ψ2 coincide on the neighborhoods, i.e.,

ψ ◦D1 = D2 ◦ ψ, ψ ◦Ψ1 = Ψ2 ◦ ψ.

We cut Mj along Σj and use the map φ to glue the pieces together interchanging M ′′1 and

M ′′2 . In this way we obtain the manifolds

M3 := M ′1 ∪Σ M ′′

2 , M4 := M ′2 ∪Σ M ′′

1 ,

where Σ ∼= Σ1∼= Σ2. We use the map ψ to cut the bundles E1, E2 at Σ1, Σ2 and glue the

pieces together interchanging E1|M ′′1 and E2|M ′′2 . With this procedure we obtain Z2-graded

Hermitian vector bundles E3 → M3 and E4 → M4. At last, we define D3 and D4, Ψ3 and

Ψ4 by

D3 =

D1 on M ′1

D2 on M ′′2

, D4 =

D2 on M ′2

D1 on M ′′1

;

Ψ3 =

Ψ1 on M ′1

Ψ2 on M ′′2

, Ψ4 =

Ψ2 on M ′2

Ψ1 on M ′′1

.

Then by Assumption 1.6.1.(iii), Dj + Ψj : C∞0 (Mj, Ej) → C∞0 (Mj, Ej), j = 3, 4 are also

Z2-graded essentially self-adjoint Callias-type operators. So again we have two well-defined

indexes ind(D3 + Ψ3) and ind(D4 + Ψ4).

40

1.6.2 Relative index theorem

As in Subsection 1.1.6, we set Dj := Dj + Ψj, j = 1, 2, 3, 4. The we have the following

version of the relative index theorem

Theorem 1.6.2. indD1 + indD2 = indD3 + indD4.

The idea of the proof is to use Corollary 1.5.5 to write indDj as the sum of the indexes

on two pieces. However, as one might notice, in our setting, Σ1 and Σ2 might not satisfy

condition (ii) of Assumption 1.5.1. So Corollary 1.5.5 cannot be applied directly. In the

next subsection, we construct deformations of the operators D1 and D2 which preserve the

indexes such that the deformed operators satisfy Assumption 1.5.1.(ii).

1.6.3 Deformations of the operators D1 and D2

Let Uj, j = 1, 2 denote the neighborhoods U(Σj) of Σj in Subsection 1.6.1. Since Σj are

compact hypersurfaces, we can find their relatively compact neighborhoods Vj,Wj satisfying

V2 = φ(V1), W2 = φ(W1) and

Vj ⊂ Vj ⊂ Wj ⊂ Wj ⊂ Uj.

Fix smooth functions fj : Mj → [0, 1] such that fj ≡ 1 on Vj and fj ≡ 0 outside of Wj.

Notice that fj have compact supports.

For each t ∈ [0,∞) define

Ψj,t := (1 + tfj)Ψj,

and set

Dj,t := Dj + tfjΨj = Dj + (1 + tfj)Ψj = Dj + Ψj,t.

Lemma 1.6.3. For j = 1, 2, we have

(i) For every t ≥ 0, the operator Dj,t = Dj + Ψj,t : C∞0 (Mj, E±j ) → C∞0 (Mj, E

∓j ) is of

Callias-type, and, hence, is Fredholm.

41

(ii) The exists a constant b > 0 and a compact subset Kj,b bMj, such that Σj ⊂Mj \Kj,b

and for every t ≥ b, the essential support of Dj,t is contained in Kj,b.

Notice, that Lemma 1.6.3.(ii) implies that for large t, condition (ii) of Assumption 1.5.1

is satisfied for the operators Dj,t.

Proof. (i) Direct computation yields

[Dj,Ψj,t]+ = (1 + tfj)[Dj,Ψj]+ +√−1 tσ(Dj)(dfj)Ψj,

Ψ2j,t + [Dj,Ψj,t]+ = (1 + tfj)

2Ψ2j + (1 + tfj)[Dj,Ψj]+ +

√−1 tσ(Dj)(dfj)Ψj

= Ψ2j + [Dj,Ψj]+ + (t2f 2

j + 2tfj)Ψ2j + tfj[Dj,Ψj]+ +

√−1 tσ(Dj)(dfj)Ψj.

Since both [Dj,Ψj]+ and σ(Dj)(dfj)Ψj are bundle maps, so are [Dj,Ψj,t]+. Suppose Kj bMj

are the essential supports of Dj. The supports of tfj and dfj both lie in the compact sets

Wj, so Kj ∪Wj is still compact and can serve as the essential supports of Dj,t. Therefore,

Dj,t are Callias-type operators and, hence, are Fredholm.

(ii) Since Kj are the essential supports of Dj, there exist constants cj > 0, such that

|(Ψ2j + [Dj,Ψj]+)(xj)| ≥ cj, for all xj ∈Mj \Kj.

Since Vj are compact sets, Ψ2j + [Dj,Ψj]+ have finite lower bounds and Ψ2

j have positive

lower bounds on them. Note that on Vj, t2f 2j ≡ t2 and dfj ≡ 0. One can find b large enough

such that for any t ≥ b,

|(Ψ2j,t + [Dj,Ψj,t]+)(xj)| ≥ cj, for all xj ∈ Vj.

Now we set Kj,b := Kj \ Vj. It’s easy to see that they are still compact sets and are essential

supports of Dj,t for t ≥ b. Clearly, Σj 6⊂ Kj,b. So we are done.

From this lemma, we see that after the deformations of the operators, Σj satisfy Assump-

tion 1.5.1.(ii). It remains to prove the following.

42

Lemma 1.6.4. Let b be the positive constant as in last lemma. Then for j = 1, 2,

indDj,b = indDj. (1.6.1)

Proof. Using a similar argument as in the proof of Lemma 1.2.3, for any t, t′ ∈ [0,∞),

Dj,t −Dj,t′ = (t− t′)fjΨj are bounded operators depending continuously on t− t′ ∈ R. By

the stability of the index of a Fredholm operator, indDj,t are independent of t. Then the

lemma follows from setting t = b and t = 0.

1.6.4 Proof of Theorem 1.6.2

Applying the construction of Subsection 1.6.1 to the operators D1,b and D2,b we obtain

operators D3,b and D4,b on M3 and M4 respectively. By Lemma 1.6.4 the indexes of these

operators are equal to the indexes of D3 and D4 respectively. It follows that it is enough to

prove Theorem 1.6.2 for operators Dj,b, j = 1, . . . , 4. In other words it is enough to prove the

theorem for the case when Σ satisfies Assumption 1.5.1.(ii). Then we can apply Corollary

1.5.5.

From now on we assume that Σ satisfies Assumption 1.5.1.(ii) for operators D1 and D2.

As in Section 1.5, we can define operators DΣj ,j, j = 1, 2. Let D′j,D′′j be the restrictions

of DΣj ,j to M ′j,M

′′j . By Corollary 1.5.5,

indD1 = indD′1 + indD′′1 ,

indD2 = indD′2 + indD′′2 .

Similarly, we also have

indD3 = indD′1 + indD′′2 ,

indD4 = indD′2 + indD′′1 .

Combining these four equations, we get

indD1 + indD2 = indD3 + indD4

and complete the proof. �

43

Chapter 2

The Index of Callias-Type Operators

with Atiyah–Patodi–Singer Boundary

Conditions

Constantine Callias in [35], considered a class of perturbed Dirac operators on an odd-

dimensional Euclidean space which are Fredholm and found a beautiful formula for the

index of such operators. This result was soon generalized to Riemannian manifolds by many

authors, [21, 32, 3, 57, 34]. A nice character of the Callias index theorem is that it reduces

a non-compact index to a compact one. Recently, many new properties, generalizations and

applications of Callias-type index were found, cf. for example, [50, 36, 66, 51, 28, 29].

In this chapter we extend the Callias-type index theory to manifolds with compact bound-

ary. The study of the index theory on compact manifolds with boundary was initiated in

[4]. In the seminal paper [5], Atiyah, Patodi and Singer computed the index of a first order

elliptic operator with a non-local boundary condition. This so-called Atiyah–Patodi–Singer

(APS) boundary condition is defined using the spectrum of a self-adjoint operator associ-

ated to the restriction of the original operator to the boundary. The Atiyah–Patodi–Singer

index theorem inspired an intensive study of boundary value problems for first-order elliptic

operators, especially Dirac-type operators (see [20] for compact manifolds). Recently, Bar

and Ballmann in [11] gave a thorough description of boundary value problems for first-order

44

elliptic operators on (not necessarily compact) manifolds with compact boundary. They

obtained the Fredholm property for Callias-type operators with APS boundary conditions,

making it possible to study the index problem on non-compact manifolds with boundary.

The results in [11] were also partially generalized to Spinc manifolds of bounded geometry

with non-compact boundary in [44].

In this chapter we combine the results of [5], [11] and [35] and compute the index of

Callias-type operators with APS boundary conditions. We show that this index is equal to a

combination of indexes of the induced operators on a compact hypersurface and a boundary

term which appears in APS index theorem. Thus our result generalizes the Callias index

theorem to manifolds with boundary. We point out that our proof technique leads to a new

proof of the classical (boundaryless) Callias index theorem.

This chapter is organized as follows. In Section 2.1, we introduce the basic setting for

manifolds with compact boundary. In Section 2.2, we discuss some results from [11] about

boundary value problems of Dirac-type operators with the focus on APS boundary condition.

Also, we recall the splitting theorem and relative index theorem which will play their roles in

proving the main theorem. Then in Section 2.3, we study the above-mentioned APS-Callias

index problem and give our main result in Theorem 2.3.5, followed by some consequences.

The theorem is proved in Section 2.4.

2.1 Manifolds with compact boundary

We introduce the basic notations that will be used later.

2.1.1 Setting

Let M be a Riemannian manifold with compact boundary ∂M . We assume the manifold

is complete in the sense of metric spaces and call it a complete Riemannian manifold. We

denote by dV the volume element on M and by dS the volume element on ∂M . The interior

45

of M is denoted by M . For a vector bundle E over M , C∞(M,E) is the space of smooth

sections of E, C∞c (M,E) is the space of smooth sections of E with compact support, and

C∞cc (M,E) is the space of smooth sections of E with compact support in M . Note that

C∞cc (M,E) ⊂ C∞c (M,E) ⊂ C∞(M,E).

When M is compact, C∞c (M,E) = C∞(M,E); when ∂M = ∅, C∞cc (M,E) = C∞c (M,E).

We denote by L2(M,E) the Hilbert space of square-integrable sections of E, which is the

completion of C∞c (M,E) with respect to the norm induced by the L2-inner product

(u1, u2) :=

∫M

〈u1, u2〉 dV,

where 〈·, ·〉 denotes the fiberwise inner product.

Let E,F be two Hermitian vector bundles over M and D : C∞(M,E)→ C∞(M,F ) be a

first-order differential operator. The formal adjoint of D, denoted by D∗, is defined by∫M

〈Du, v〉 dV =

∫M

〈u,D∗v〉 dV,

for all u ∈ C∞cc (M,E) and v ∈ C∞(M,F ). If E = F and D = D∗, then D is called formally

self-adjoint.

2.1.2 Minimal and maximal extensions

Suppose Dcc := D|C∞cc (M,E), and view it as an unbounded operator from L2(M,E) to

L2(M,F ). The minimal extension Dmin of D is the operator whose graph is the closure

of that of Dcc. The maximal extension Dmax of D is defined to be Dmax =((D∗)cc

)ad,

where “ad” means adjoint of the operator in the sense of functional analysis. Both Dmin and

Dmax are closed operators. Their domains, domDmin and domDmax, become Hilbert spaces

equipped with the graph norm, which is the norm associated with the inner product

(u1, u2)D :=

∫M

(〈u1, u2〉+ 〈Du1, Du2〉) dV.

46

2.1.3 Green’s formula

Let τ ∈ TM |∂M be the unit inward normal vector field along ∂M . Using the Riemannian

metric, τ can be identified with its associated one-form. We have the following formula (cf.

[20, Proposition 3.4]).

Proposition 2.1.1 (Green’s formula). Let D be as above. Then for all u ∈ C∞c (M,E) and

v ∈ C∞c (M,F ), ∫M

〈Du, v〉 dV =

∫M

〈u,D∗v〉 dV −∫∂M

〈σD(τ)u, v〉 dS, (2.1.1)

where σD denotes the principal symbol of the operator D.

Remark 2.1.2. By [11, Theorem 6.7], the formula (2.1.1) can be generalized to the case where

u ∈ domDmax and v ∈ dom(D∗)max.

2.1.4 Sobolev spaces

Let ∇E be a Hermitian connection on E. For any u ∈ C∞(M,E), the covariant derivative

∇Eu ∈ C∞(M,T ∗M ⊗ E). For k ∈ Z+, we define the kth Sobolev space

Hk(M,E) := {u ∈ L2(M,E) : ∇Eu, (∇E)2u, . . . , (∇E)ku ∈ L2(M)},

where the covariant derivatives are understood in distributional sense. It is a Hilbert space

with Hk-norm

‖u‖2Hk(M) := ‖u‖2

L2(M) + ‖∇Eu‖2L2(M) + · · · + ‖(∇E)ku‖2

L2(M).

Note that when M is compact, Hk(M,E) does not depend on the choices of ∇E and Rie-

mannian metric, but when M is non-compact, it does.

We say u ∈ L2loc(M,E) if the restrictions of u to compact subsets of M have finite L2-norm.

For k ∈ Z+, we say u ∈ Hkloc(M,E), the kth local Sobolev space, if u,∇Eu, (∇E)2u, . . . , (∇E)ku

all lie in L2loc(M,E). This Sobolev space is independent of the preceding choices.

47

Similarly, we fix a Hermitian connection on F and define the spaces L2(M,F ), L2loc(M,F ),

Hk(M,F ), and Hkloc(M,F ).

2.2 Preliminary results

In this section, we summarize some results on boundary value problems on complete mani-

folds with compact boundary. We mostly follow [11, 12].

2.2.1 Adapted operators to Dirac-type operators

Let E be a Dirac bundle over M with Clifford multiplication denoted by c(·). We say that

D : C∞(M,E)→ C∞(M,E) is a Dirac-type operator if the principal symbol of D is c(·). In

local coordinates, D can be written as

D =n∑j=1

c(ej)∇Eej

+ V (2.2.1)

at x ∈ M , where e1, . . . , en is an orthonormal basis of TxM (using Riemannian metric to

identify TM and T ∗M), ∇E is a Hermitian connection on E and V ∈ End(E) is the potential.

When V = 0, D is merely a Dirac operator as in Subsection 0.1.1.

The formal adjoint D∗ of a Dirac-type operator D is also of Dirac type. Note that for

x ∈ ∂M , one can identify T ∗x∂M with the space {ξ ∈ T ∗xM : 〈ξ, τ(x)〉 = 0}.

Definition 2.2.1. A formally self-adjoint first-order differential operator A : C∞(∂M,E)→

C∞(∂M,E) is called an adapted operator to D if the principal symbol of A is given by

σA(ξ) = σD(τ(x))−1 ◦ σD(ξ).

Remark 2.2.2. Adapted operators always exist and are also of Dirac type. They are unique

up to addition of a Hermitian bundle map of E (cf. [12, Section 3]).

If A is adapted to D, then

A] = c(τ) ◦ (−A) ◦ c(τ)−1 (2.2.2)

48

is an adapted operator to D∗. Moreover, if D is formally self-adjoint, we can find an adapted

operator A to D such that

A ◦ c(τ) = − c(τ) ◦ A, (2.2.3)

and, hence, A] = A.

By definition, A is an essentially self-adjoint elliptic operator on the closed manifold ∂M .

Hence A has discrete spectrum consisting of real eigenvalues {λj}j∈Z, each of which has finite

multiplicity. In particular, the corresponding eigenspaces Vj are finite-dimensional. Thus we

have decomposition of L2(∂M,E) into a direct sum of eigenspaces of A:

L2(∂M,E) =⊕

λj∈spec(A)

Vj. (2.2.4)

For any s ∈ R, the positive operator (id +A2)s/2 is defined by functional calculus. Then the

Hs-norm on C∞(∂M,E) is defined by

‖u‖2Hs(∂M,E) := ‖(id +A2)s/2u‖2

L2(∂M,E).

The Sobolev space Hs(∂M,E) is the completion of C∞(∂M,E) with respect to this norm.

Remark 2.2.3. When s ∈ Z+, this definition of Sobolev spaces coincides with that of Sub-

section 2.1.4 via covariant derivatives.

For I ⊂ R, let

PI : L2(∂M,E) →⊕λj∈I

Vj (2.2.5)

be the orthogonal spectral projection. It’s easy to see that

PI(Hs(∂M,E)) ⊂ Hs(∂M,E)

for all s ∈ R. Set HsI (A) := PI(H

s(∂M,E)). For a ∈ R, we define the hybrid Sobolev space

H(A) := H1/2(−∞,a)(A) ⊕ H

−1/2[a,∞)(A) (2.2.6)

with H-norm

‖u‖2H(A)

:= ‖P(−∞,a)u‖2H1/2(∂M,E) + ‖P[a,∞)u‖2

H−1/2(∂M,E).

The space H(A) is independent of the choice of a (cf. [11, p. 27]).

49

2.2.2 Boundary value problems

Let D be a Dirac-type operator. If ∂M = ∅, then D has a unique extension, i.e., Dmin =

Dmax. (When D is formally self-adjoint, this is called essentially self-adjointness, cf. [37],

[43, Theorem 1.17].) But when ∂M 6= ∅, the minimal and maximal extensions may not be

equal. Those closed extensions lying between Dmin and Dmax give rise to boundary value

problems.

One of the main results of [11] is the following.

Theorem 2.2.4. For any closed subspace B ⊂ H(A), denote by DB the extension of D with

domain

domDB = {u ∈ domDmax : u|∂M ∈ B}.

Then DB is a closed extension of D between Dmin and Dmax, and any closed extension of D

between Dmin and Dmax is of this form.

Remark 2.2.5. We recall the trace theorem which says that the trace map ·|∂M : C∞c (M,E)→

C∞(∂M,E) extends to a bounded linear map

T : Hkloc(M,E) → Hk−1/2(∂M,E)

for all k ≥ 1.

Due to this theorem, one can define boundary conditions in the following way.

Definition 2.2.6. A boundary condition for D is a closed subspace of H(A). We use the

notation DB from Theorem 2.2.4 to denote the operator D with boundary condition B.

Regarding DB as an unbounded operator on L2(M,E), its adjoint operator is D∗Bad , where

the boundary condition is

Bad = {v ∈ H(A]) : (σD(τ)u, v) = 0, for all u ∈ B},

and A] is an adapted operator to D∗.

50

2.2.3 Elliptic boundary conditions

Notice that for general boundary conditions, domDB 6⊂ H1loc(M,E).

Definition 2.2.7. A boundary condition B is said to be elliptic if domDB ⊂ H1loc(M,E)

and domD∗Bad ⊂ H1

loc(M,E).

Remark 2.2.8. This definition is equivalent to saying that B ⊂ H1/2(∂M,E) and its adjoint

boundary condition Bad ⊂ H1/2(∂M,E) (cf. [11, Theorem 1.7]). There is another equivalent

but technical way to define elliptic boundary conditions, see [11, Definition 7.5] or [12,

Definition 4.7]. From [11, 12], B is an elliptic boundary condition if and only if Bad is.

The definition of elliptic boundary condition can be generalized as follows.

Definition 2.2.9. A boundary condition B is said to be

(i) m-regular, where m ∈ Z+, if

Dmaxu ∈ Hkloc(M,E) =⇒ u ∈ Hk+1

loc (M,E),

D∗maxv ∈ Hkloc(M,E) =⇒ v ∈ Hk+1

loc (M,E)

for all u ∈ domDB, v ∈ domD∗Bad , and k = 0, 1, . . . ,m− 1.

(ii) ∞-regular if it is m-regular for all m ∈ Z+.

Remark 2.2.10. By this definition, an elliptic boundary condition is 1-regular.

It is clear that if B is an ∞-regular boundary condition, then

kerDB ⊂ C∞(M,E), kerD∗Bad ⊂ C∞(M,E).

2.2.4 The Atiyah–Patodi–Singer boundary condition

A typical example of elliptic boundary condition, which is called Atiyah–Patodi–Singer

boundary condition (or APS boundary condition), is introduced in [5].

51

Let D : C∞(M,E) → C∞(M,E) be a Dirac-type operator. Assume the Riemannian

metric and the Dirac bundle E (with the associated Clifford multiplication and Clifford

connection) have product structure near the boundary ∂M . So D can be written as

D = c(τ)(∂t + A+R

)(2.2.7)

in a tubular neighborhood of ∂M , where t is the normal coordinate, A is an adapted operator

to D, and R is a zeroth order operator on ∂M . Then

D∗ = c(τ)(∂t + A] +R]

),

where A] is as in (2.2.2). When D = D∗, one can choose R = R] = 0 so that A = A].

Let P(−∞,0) be the spectral projection as in (2.2.5) and set

H1/2(−∞,0)(A) = P(−∞,0)(H

1/2(∂M,E)).

Definition 2.2.11. The Atiyah–Patodi–Singer boundary condition is

BAPS := H1/2(−∞,0)(A). (2.2.8)

This is a closed subspace of H(A) (recall that the space H(A) is defined in (2.2.6)). The

adjoint boundary condition is given by

BadAPS = c(τ)H

1/2[0,∞)(A) = H

1/2(−∞,0](A

]). (2.2.9)

By [11, Proposition 7.24 and Example 7.27], we have that

Proposition 2.2.12. The APS boundary condition (2.2.8) is an ∞-regular boundary con-

dition.

2.2.5 Invertibility at infinity

If the manifold M is non-compact without boundary, in general, an elliptic operator on it is

not Fredholm. Similarly, for non-compact manifold M with compact boundary, an elliptic

52

boundary condition does not guarantee that the operator is Fredholm. We now define a class

of operators on non-compact manifolds which are Fredholm.

Definition 2.2.13. We say that an operatorD is invertible at infinity (or coercive at infinity)

if there exist a constant C > 0 and a compact subset K bM such that

‖Du‖L2(M) ≥ C ‖u‖L2(M),

for all u ∈ C∞c (M,E) with supp(u) ∩K = ∅.

Remark 2.2.14. (i) By definition, if M is compact, then D is invertible at infinity.

(ii) Boundary conditions have nothing to do with invertibility at infinity since the compact

set K can always be chosen such that a neighborhood of ∂M is contained in K.

An important class of examples for operators which are invertible at infinity is the so-

called Callias-type operators that will be discussed in next section.

2.2.6 Fredholmness

Recall that for ∂M = ∅, a first-order essentially self-adjoint elliptic operator which is in-

vertible at infinity is Fredholm (cf. [2, Theorem 2.1]). For ∂M 6= ∅, we have the following

analogous result ([11, Theorem 8.5, Corollary 8.6]).

Proposition 2.2.15. Assume that DB : domDB → L2(M,E) is a Dirac-type operator with

elliptic boundary condition.

(i) If D is invertible at infinity, then DB has finite-dimensional kernel and closed range.

(ii) If D and D∗ are invertible at infinity, then DB is a Fredholm operator.

Remark 2.2.16. Since for an elliptic boundary condition B, domDB ⊂ H1loc(M,E), the proof

is essentially the same as that for the case without boundary (involving Rellich embedding

theorem). And it is easy to see that (ii) is an immediate consequence of (i).

53

Under the hypothesis of Proposition 2.2.15.(ii), we define the index of D subject to the

boundary condition B as the integer

indDB := dim kerDB − dim kerD∗Bad ∈ Z.

2.2.7 The splitting theorem

We recall the splitting theorem of [11] which can be thought of as a more general version

of [57, Proposition 2.3]. Let D : C∞(M,E) → C∞(M,E) be a Dirac-type operator on M .

Let N be a closed and two-sided hypersurface in M which does not intersect the compact

boundary ∂M . Cut M along N to obtain a manifold M ′, whose boundary ∂M ′ consists of

disjoint union of ∂M and two copies N1 and N2 of N . One can pull back E and D from

M to M ′ to define the bundle E ′ and operator D′. Then D′ : C∞(M ′, E ′)→ C∞(M ′, E ′) is

still a Dirac-type operator. Assume that there is a unit inward normal vector field τ along

N1 and choose an adapted operator A to D′ along N1. Then −A is an adapted operator to

D′ along N2.

Theorem 2.2.17 ([11, Theorem 8.17]). Let M,D,M ′, D′ be as above.

(i) D and D∗ are invertible at infinity if and only if D′ and (D′)∗ are invertible at infinity.

(ii) Let B be an elliptic boundary condition on ∂M . Fix a ∈ R and let B1 = H1/2(−∞,a)(A)

and B2 = H1/2[a,∞)(A) be boundary conditions along N1 and N2, respectively. Then the

operators DB and D′B⊕B1⊕B2are Fredholm operators and

indDB = indD′B⊕B1⊕B2.

2.2.8 Relative index theorem

Let Mj, j = 1, 2 be two complete manifolds with compact boundary and Dj,Bj: domDj,Bj

→

L2(Mj, Ej) be two Dirac-type operators with elliptic boundary conditions. Suppose M ′j ∪Nj

54

M ′′j are partitions of Mj into relatively open submanifolds, where Nj are closed hypersurfaces

of Mj that do not intersect the boundaries. We assume that Nj have tubular neighborhoods

which are diffeomorphic to each other and the structures of Ej (resp. Dj) on the neighbor-

hoods are isomorphic.

Cut Mj along Nj and glue the pieces together interchanging M ′′1 and M ′′

2 . In this way we

obtain the manifolds

M3 := M ′1 ∪N M ′′

2 , M4 := M ′2 ∪N M ′′

1 ,

where N ∼= N1∼= N2. Then we get operators D3,B3 and D4,B4 on M3 and M4, respectively.

The following relative index theorem, which generalizes [11, Theorem 8.19], is a direct con-

sequence of Theorem 2.2.17. (One can see [34, Theorem 1.2] for a boundaryless version.)

Theorem 2.2.18. If Dj and D∗j , j = 1, 2, 3, 4 are all invertible at infinity, then Dj,Bjare

all Fredholm operators, and

indD1,B1 + indD2,B2 = indD3,B3 + indD4,B4 .

Proof. Clearly the hypersurfaces Nj satisfy the hypothesis of Theorem 2.2.17. As in last

subsection, choose boundary conditions B′Njand B′′Nj

along Nj on M ′j and M ′′

j , respectively.

Since Dj and D∗j are invertible at infinity, from Theorem 2.2.17,

indDj,Bj= indD′j,B′j⊕B′Nj

+ indD′′j,B′′j ⊕B′′Nj

, j = 1, 2,

where B′j and B′′j are the restrictions of the boundary condition Bj to M ′j and M ′′

j , respec-

tively. By the construction of M3 and M4,

indD3,B3 = indD′1,B′1⊕B′N1

+ indD′′2,B′′2⊕B′′N2

,

indD4,B4 = indD′2,B′2⊕B′N2

+ indD′′1,B′′1⊕B′′N1

.

Adding together, the theorem is proved.

55

2.3 Callias-type operators with APS boundary condi-

tions

2.3.1 Invertibility at infinity of Callias-type operators

Let M be a complete odd-dimensional Riemannian manifold with boundary ∂M . Suppose

that E is a Dirac bundle over M . Let D := D + iΦ be an ungraded Callias-type operator

on E as in Definition 0.1.2.

Proposition 2.3.1. Callias-type operators are invertible at infinity in the sense of Definition

2.2.13.

Proof. Since ∂M is compact, we can always assume that the essential support K contains a

neighborhood of ∂M . Thus for all u ∈ C∞c (M,E) with supp(u) ∩ K = ∅, u ∈ C∞cc (M,E).

Then by Proposition 2.1.1, (0.1.2), and Definition 0.1.2,

‖Du‖2L2(M) = (Du,Du)L2(M) = (D∗Du, u)L2(M)

= (D2u, u)L2(M) + ((Φ2 + i [D,Φ])u, u)L2(M)

≥ ‖Du‖2L2(M) + c ‖u‖2

L2(M)

≥ c ‖u‖2L2(M).

Therefore ‖Du‖L2(M) ≥√c‖u‖L2(M) and D is invertible at infinity.

Remark 2.3.2. When ∂M = ∅, D has a unique closed extension to L2(M,E), and it is a

Fredholm operator. Thus one can define its L2-index,

indD := dim{u ∈ L2(M,E) : Du = 0} − dim{u ∈ L2(M,E) : D∗u = 0}.

A seminal result says that this index is equal to the index of a Dirac-type operator (the

operator ∂++ of (2.3.1)) on a compact hypersurface outside of the essential support. This was

first proved by Callias in [35] for Euclidean space (see also [21]) and was later generalized

56

to manifolds in [3, 57, 34], etc. In [29] and [28], the relationship between such result and

cobordism invariance of the index was being discussed for usual and von Neumann algebra

cases, respectively.

Remark 2.3.3. If ∂M 6= ∅, then in general, D is not Fredholm. By Proposition 2.2.15, we

need an elliptic boundary condition in order to have a well-defined index and study it.

2.3.2 The APS boundary condition for Callias-type operators

We impose the APS boundary condition as discussed in Subsection 2.2.4 that enables us to

define the index for Callias-type operators.

As in Subsection 2.2.4, we assume the product structure (2.2.7) for D near ∂M . We also

assume that Φ does not depend on t near ∂M . Then near ∂M ,

D = c(τ)(∂t + A− ic(τ)Φ

)= c(τ)

(∂t +A

),

where A := A− ic(τ)Φ is still formally self-adjoint and thus is an adapted operator to D.

Replacing D and A in Subsection 2.2.4 byD andA, we define the APS boundary condition

BAPS as in (2.2.8) for the Callias-type operator D. It is an elliptic boundary condition. Com-

bining Proposition 2.2.15, Remark 0.1.3 and Proposition 2.3.1, we obtain the Fredholmness

for the operator DBAPS.

Proposition 2.3.4. The operator DBAPS: domDBAPS

→ L2(M,E) is Fredholm, thus has an

index

indDBAPS= dim kerDBAPS

− dim kerD∗BadAPS∈ Z.

2.3.3 The APS-Callias index theorem

We now formulate the main result of this chapter — a Callias-type index theorem for oper-

ators with APS boundary conditions.

57

By Definition 0.1.2, the Callias potential Φ is nonsingular outside of the essential support

K. Then over M \K, there is a bundle decomposition

E|M\K = E+ ⊕ E−,

where E± are the positive/negative eigenspaces of Φ. Since Definition 0.1.2.(i) implies that

Φ commutes with Clifford multiplication, E± are also Dirac bundles.

Let L b M be a compact subset of M containing the essential support K such that

(K \ ∂M) ⊂ L. Suppose that ∂L = ∂M tN , where N is a closed hypersurface partitioning

M . Denote

EN := E|N , EN± := E±|N .

The restriction of the Clifford multiplication on E± defines a Clifford multiplication cN(·)

on EN±. Let ∇EN be the restriction of the connection ∇E on E. In general, ∇EN does not

preserve the decomposition EN = EN+ ⊕ EN−. However, if we define

∇EN± := ProjEN±◦ ∇EN ,

where ProjEN±are the projections onto T ∗N⊗EN±. One can check that these are Hermitian

connections on EN± (cf. [1, Lemma 2.7]). Then EN± are Dirac bundles over N , and we define

the (formally self-adjoint) Dirac operators on EN± by

∂± :=n−1∑j=1

cN(ej)∇EN±ej

at x ∈ N , where e1, . . . , en−1 is an orthonormal basis of TxN . They can be seen as adapted

operators associated to D± := D|E± .

Let τN be a unit inward (with respect to L) normal vector field on N and set

ν := ic(τN).

Since ν2 = id, ν induces a grading on EN±

E±N± = {u ∈ EN± : νu = ±u},

58

It’s easy to see from (2.2.3) that ∂± anti-commute with ν. We denote by ∂±± the restrictions

of ∂± to E±N±. Then

∂±± : C∞(N,E±N±) → C∞(N,E∓N±). (2.3.1)

As mentioned in Remark 2.3.2, when ∂M = ∅, the classical Callias index theorem asserts

that

indD = ind ∂++ .

The following theorem generalizes this result to the case of manifolds with boundary.

Theorem 2.3.5. Let D = D + iΦ : C∞(M,E) → C∞(M,E) be a Callias-type operator on

an odd-dimensional complete manifold M with compact boundary ∂M . Let BAPS be the APS

boundary condition described in Subsection 2.3.2. Then

indDBAPS=

1

2(ind ∂+

+ − ind ∂+−)− η(A), (2.3.2)

where ∂+± : C∞(N,E+

N±)→ C∞(N,E−N±) are the Dirac-type operators on the closed manifold

N ,

η(A) :=1

2(dim kerA + η(0;A)), (2.3.3)

and the η-function η(s;A) is defined by

η(s;A) :=∑

λ∈spec(A)\{0}

sign(λ)|λ|−s.

Remark 2.3.6. Since ∂M is a closed manifold, η(s;A) converges absolutely for Re(s) large.

Then η(0;A) can be defined using meromorphic continuation of η(s;A) and we call it η-

invariant for A on ∂M . Note that ∂M is an even-dimensional manifold. In general, the

η-invariant on even-dimensional manifolds is much simpler than on odd-dimensional ones.

We refer the reader to [41] for details.

Theorem 2.3.5 will be proved in the next section. The main idea of the proof is as follows.

Recall that we have chosen a compact subset L of M containing the essential support of D

59

with boundary ∂L = ∂M tN . First use Theorems 2.2.18 and 2.2.17 to transfer the index we

want to find to an index on L with APS boundary condition. Then by APS index formula

[5, Theorem 3.10] and dimension reason, we get

indDBAPS= − η(AN) − η(A).

Then the proof is completed by a careful study of the η-invariant η(0;AN).

2.3.4 Connection between Theorem 2.3.5 and the usual Callias

index theorem

Consider the special case when ∂M = ∅. Clearly, η(A) vanishes. Since N = ∂L now, by

cobordism invariance of the index (see for example [56, Chapter XVII] or [24]),

0 = ind ∂+ = ind ∂++ + ind ∂+

− .

Hence ind ∂++ = − ind ∂+

− , and (2.3.2) becomes

indD = ind ∂++ ,

which is exactly the usual Callias index theorem. Therefore, our Theorem 2.3.5 can be seen

as a generalization of the Callias index theorem to manifolds with boundary. In particular,

we give a new proof of the Callias index theorem for manifolds without boundary.

2.3.5 An asymmetry result

One can see from (2.2.8) and (2.2.9) that the APS boundary condition BAPS involves spec-

tral projection onto (−∞, 0), while its adjoint boundary condition BadAPS involves spectral

projection onto a slightly different interval (−∞, 0]. This shows that Atiyah–Patodi–Singer

boundary condition is not symmetric. When the manifold M is compact, this asymmetry

can be expressed in terms of the kernel of the adapted operator (cf. [5, pp. 58-60]). For

60

our Callias-type operator on non-compact manifold, a similar result still holds. To avoid

confusion of notations, we use D + iΦ for D and D − iΦ for D∗.

Corollary 2.3.7. Under the same hypothesis as in Theorem 2.3.5,

ind(D + iΦ)BAPS+ ind(D − iΦ)BAPS

= − dim kerA.

Proof. Recall that D + iΦ and D − iΦ can be written as

D + iΦ = c(τ)(∂t +A

),

D − iΦ = c(τ)(∂t +A]

),

where the adapted operators A and A satisfy

A] ◦ c(τ) = −c(τ) ◦ A. (2.3.4)

Apply Theorem 2.3.5 to D+ iΦ and D− iΦ. Notice that ∂++ and ∂+

− are interchanged for

these two Callias-type operators, so we have

ind(D + iΦ)BAPS=

1

2(ind ∂+

+ − ind ∂+−) − η(A),

ind(D − iΦ)BAPS=

1

2(ind ∂+

− − ind ∂++) − η(A]).

Add them up and it suffices to show that

η(A) + η(A]) = dim kerA.

By (2.3.4), the map c(τ) sends eigensections of A associated with eigenvalue λj to eigen-

sections of A] associated with eigenvalue −λj bijectively and vice versa. In particular, it

induces an isomorphism between the kernel of A and that of A]. So

η(0;A) + η(0;A]) = 0 and dim kerA + dim kerA] = 2 dim kerA.

Now the corollary follows from (2.3.3).

Notice, that if ∂M = ∅ then Corollary 2.3.7 implies a well known result (cf. for example,

[28, (2.10)])

ind(D + iΦ) = − ind(D − iΦ).

61

2.4 Proof of Theorem 2.3.5

We prove Theorem 2.3.5 following the idea sketched in Subsection 2.3.3. To simplify nota-

tions, we will write indD for indDBAPSin this section.

2.4.1 Deformation of structures near N

Remember we assumed that (K\∂M) ⊂ L, so there exists a relatively compact neighborhood

U(N) of N which does not intersect with the essential support K. Out first step is to do

deformation on U(N). The following lemma is from [28, Section 6].

Lemma 2.4.1. Set Nδ := N × (−δ, δ) for any δ > 0. One can deform all the structures in

the neighborhood U(N) of N so that the following conditions are satisfied:

(i) U(N) is isometric to N2ε;

(ii) (see also [28, Lemma 5.3]) the restrictions of the Dirac bundles E|Nε and E±|Nε are

isomorphic to the pull backs of EN and EN± to Nε respectively along with connections;

(iii) (see also [28, Lemma 5.4]) Φ|Nε is a constant multiple of its unitarization Φ0 :=

Φ(Φ2)−1/2, i.e., Φ|E± = ±h on Nε, where h > 0 is a constant;

(iv) the potential V from (2.2.1) of the Dirac-type operator D vanishes on Nε;

(v) D is always a Callias-type operator throughout the deformation, and the essential sup-

port of the Callias-type operator associated to the new structures is still contained in

L \ (N × (−ε, 0]).

Remark 2.4.2. As a result of (i) and (ii), we can write

D|Nε = c(τN)(∂t + ∂),

D±|Nε = c(τN)(∂t + ∂±),

62

where t is the normal coordinate pointing inward L and ∂, ∂± are as in Subsection 2.3.3.

Furthermore, by (iii), our Callias-type operator has the form

D|Nε = c(τN)(∂t +AN), (2.4.1)

where AN = ∂ − ic(τN)Φ|N does not depend on t and

AN |EN± = ∂± ∓ ic(τN)h. (2.4.2)

It’s also easy to see from (iii) that

Φ2 = h2 and [D,Φ]|E± = [D,±h] = 0 (2.4.3)

on Nε.

Remark 2.4.3. Below in Lemma 2.4.9 we use the freedom to choose h in (iii) to be arbitrarily

large.

Proposition 2.4.4. The deformation in Lemma 2.4.1 preserves the index of the Callias-type

operator D = D + iΦ under APS boundary condition.

Proof. Let W be the closure of U(N). It is a compact subset of M which does not intersect

the boundary ∂M . This indicates that D keeps unchanged near the boundary and one

can impose the same APS boundary condition. Since the deformation only occurs on the

compact set W and is continuous, the domain of D remains the same under APS boundary

condition. Therefore, throughout the deformation, D is always a bounded operator from this

fixed domain to L2(M,E) which is Fredholm. Now by the stability of the Fredholm index

(cf. [52, Proposition III.7.1]), the index of D is preserved.

Remark 2.4.5. One can also show Proposition 2.4.4 by using relative index theorem and the

fact that the dimension is odd.

Proposition 2.4.4 ensures that we can make the following assumption.

Assumption 2.4.6. We assume that conditions (i)-(v) of Lemma 2.4.1 are satisfied for our

problem henceforth.

63

2.4.2 The index on manifold with a cylindrical end

Note that N gives a partition M = L ∪N (M \ L). Consider M1 = N × (−∞,∞) with the

partition

M1 = (N × (−∞, 0]) ∪N (N × (0,∞)).

Lift the Dirac bundle EN , Dirac operator ∂ and restriction of bundle map Φ|N from N to M1.

By Assumption 2.4.6, there are isomorphisms between structures of M near N and those of

M1 near N ×{0}. One can do the “cut-and-glue” procedure as described in Subsection 2.2.8

to form

M = L ∪N (N × (0,∞)), M2 = (N × (−∞, 0]) ∪N (M \ L). (2.4.4)

We obtain Callias-type operators D,D1, D,D2 acting on E,E1, E, E2 over corresponding

manifold. They satisfy Theorem 2.2.18. Therefore

indD + indD1 = ind D + indD2.

Notice that D1 and D2 are Callias-type operators with empty essential supports on manifolds

without boundary. Therefore, D1, D2 and their adjoints are invertible operators. So indD1 =

indD2 = 0 and we get

Lemma 2.4.7. indD = ind D.

Remark 2.4.8. Now the problem is moved to M , a manifold with a cylindrical end. We point

out that conditions (ii)-(iv) of Lemma 2.4.1 continue holding on the cylindrical end.

2.4.3 Applying the splitting theorem

We have the partition of M as in (2.4.4) and D is of form (2.4.1) near N . Cut M along N .

Define boundary condition on L along N to be the APS boundary condition H1/2(−∞,0)(AN),

and the boundary condition on N × [0,∞) along N to be H1/2[0,∞)(AN). Denote by D1 and D2

64

the restrictions of D to L and N × [0,∞), respectively. Let ind D1 be the index of D1 with

APS boundary condition and ind D2 be the index of D2 with boundary condition H1/2[0,∞)(AN).

Then by Theorem 2.2.17,

ind D = ind D1 + ind D2.

Lemma 2.4.9. ind D = ind D1.

Proof. We need to prove that ind D2 = 0. Remember that D2 = D + iΦ satisfies conditions

of Lemma 2.4.1 on N × [0,∞). For any u ∈ C∞c (N × [0,∞), E) satisfying u|N ∈ H1/2[0,∞)(AN),

by Proposition 2.1.1, (2.4.1), (2.4.3) and Remark 2.4.8,

‖D2u‖2L2 = (D2u, D2u)L2

= (D∗2D2u, u)L2 −∫N

〈c(τN)D2u, u〉 dS

= (D2u, u)L2 + ((Φ2 + i [D, Φ])u, u)L2 +

∫N

(〈∂tu, u〉+ 〈ANu, u〉) dS

≥ (D2u, u)L2 + h2 ‖u‖2L2 +

∫N

〈∂tu, u〉 dS.

By Assumption 2.4.6 and Remark 2.4.8, the potential V for D vanishes on N × [0,∞).

The Weitzenbock identity (or general Bochner identity, cf. [52, Proposition II.8.2]) for Dirac

operator gives that

D2 = ∇∗∇ + R on N × [0,∞),

where the bundle map term R is the curvature transformation associated with the Dirac

bundle E|N×[0,∞). Since this bundle is the lift of EN from the compact base N , R is bounded

on N × [0,∞). As mentioned in Remark 2.4.3, one can choose h large enough so that h2/2

is greater than the upper bound of the norm |R|. Applying Proposition 2.1.1 to ∇, we have

(∇∗∇u, u)L2 − ‖∇u‖2L2 =

∫N

〈σ∇∗(τN)∇u, u〉 dS

= −∫N

〈∇u, σ∇(τN)u〉 dS = −∫N

〈∇u, τN ⊗ u〉 dS

= −∫N

〈∇τNu, u〉 dS = −∫N

〈∂tu, u〉 dS.

65

Thus

‖D2u‖2L2 ≥ (∇∗∇u, u)L2 +

∫N

〈∂tu, u〉 dS + h2 ‖u‖2L2 + (Ru, u)L2

≥ ‖∇u‖2L2 +

h2

2‖u‖2

L2 ≥h2

2‖u‖2

L2 .

Therefore, D2 is invertible on the domain determined by the boundary condition H1/2[0,∞)(AN)

and ker D2 = {0}. Similarly, ker(D2)ad = {0}. Hence ind D2 = 0 and ind D = ind D1.

Standard Atiyah–Patodi–Singer index formula ([5, Theorem 3.10]) applies to ind D1 giving

that

ind D1 =

∫L

AS − η(AN) − η(A), (2.4.5)

where AS is the interior Atiyah–Singer integrand. Since the dimension of L is odd, this

integral vanishes. Combining Lemmas 2.4.7, 2.4.9 and (2.4.5), we finally obtain

indD = − η(AN) − η(A). (2.4.6)

2.4.4 The η-invariant of the perturbed Dirac operator on N

In the last subsection, we have expressed the index of DBAPSin terms of η(A) and η(AN) as

in (2.4.6), where

AN = AN+ ⊕ AN− = (∂+ − νh) ⊕ (∂− + νh)

under the splitting EN = EN+ ⊕ EN−, and ν = ic(τN) (cf. (2.4.2)). In this subsection, we

shall show how η(AN) can be written as the difference of two indexes as in the right-hand

side of (2.3.2).

Recall that

η(AN) =1

2(dim kerAN + η(0;AN)). (2.4.7)

AN and ∂ can be viewed as adapted operators to D and D on N , respectively. Using the

fact that ∂ anti-commutes with ν, we have

A2N = ∂2 + h2. (2.4.8)

66

Since h > 0 is a constant, AN is an invertible operator, and hence

dim kerAN = 0. (2.4.9)

As for η(0;AN), we have the following lemma.

Lemma 2.4.10. η(0;AN) = − ind ∂++ + ind ∂+

− .

Proof. Notice that AN is a perturbation of ∂ by a bundle map ν which anti-commutes with

it. Restricting to EN+, we write AN according to the grading EN+ = E+N+ ⊕ E

−N+ induced

by ν (see Subsection 2.3.3),

AN+ =

−h ∂−+

∂++ h

.The spectrum of AN+ consists of eigenvalues with finite multiplicity. By (2.4.8), the

eigenvalues ofAN have absolute value of at least h. Suppose that u = u+⊕u− ∈ C∞(N,EN+)

is an eigenvector of AN+ with eigenvalue λ. Then

λ

u+

u−

= AN+

u+

u−

=

−h ∂−+

∂++ h

u+

u−

=

∂−+u− − hu+

∂++u

+ + hu−

,which gives ∂−+u

− = (λ+ h)u+

∂++u

+ = (λ− h)u−. (2.4.10)

Then

AN+

(λ+ h)u+

−(λ− h)u−

=

−(λ− h)∂−+u− − h(λ+ h)u+

(λ+ h)∂++u

+ − h(λ− h)u−

= −λ

(λ+ h)u+

−(λ− h)u−

.Note that the map u+⊕u− 7→ (λ+h)u+⊕(−(λ−h)u−) is invertible when |λ| > h. Therefore,

for such λ, this map induces an isomorphism between the eigenspaces of AN+ corresponding

to eigenvalues λ and −λ. This means that the spectrum of AN+ lying in (−∞,−h) is

symmetric to that lying in (h,∞), hence

η(0;AN+) = dim ker(AN+ − h) − dim ker(AN+ + h).

67

If u+ ⊕ u− ∈ ker(AN+ − h), by letting λ = h in (2.4.10), we get ∂−+u− = 2hu+

∂++u

+ = 0.

Applying ∂++ to the first equation yields (∂+

+∂−+)u− = 0. Thus u− ∈ ker(∂+

+∂−+). Since ∂+ is

formally self-adjoint, ker(∂++∂−+) = ker ∂−+ . So u− ∈ ker ∂−+ and u+ = 0. Therefore

ker(AN+ − h) = {0⊕ u− : u− ∈ ker ∂−+}.

Hence dim ker(AN+ − h) = dim ker ∂−+ . Similarly, dim ker(AN+ + h) = dim ker ∂++ . Then

η(0;AN+) = dim ker ∂−+ − dim ker ∂++ = − ind ∂+

+ .

The discussion on EN− is exactly the same as what we just did on EN+. One gets

η(0;AN−) = ind ∂+− .

As a direct sum of AN+ and AN−, by the additivity of the η-invariant, finally we obtain

η(0;AN) = − ind ∂++ + ind ∂+

− .

Now (2.3.2) follows simply from (2.4.6), (2.4.7), (2.4.9) and Lemma 2.4.10. We complete

the proof of Theorem 2.3.5.

68

Chapter 3

Boundary Value Problems for

Strongly Callias-Type Operators

In this chapter, we introduce the boundary value problems for strongly Callias-type operators

on manifolds with non-compact boundary. The theory generalizes results of [11] (where the

boundary is compact, see also Section 2.2) and will be the fondation of the next three

chapters. We use ungraded operator D = D+ iΦ for the presentation. But all the results of

this chapter hold as well for graded strongly Callias-type operators D + Ψ (just replace D

by D+ = D+ + Ψ+ and D∗ by D− = D− + Ψ−).

3.1 Operators on manifolds with non-compact bound-

ary

In this section we discuss different domains for operators on manifolds with boundary.

3.1.1 Setting and notations

Let M be a complete Riemannian manifold with (possibly non-compact) boundary ∂M . We

denote the Riemannian metric on M by gM and its restriction to the boundary by g∂M .

Then (∂M, g∂M) is also a complete Riemannian manifold. We denote by dV the volume

69

form on M and by dS the volume form on ∂M . The interior of M is denoted by M . For a

vector bundle E over M , C∞(M,E) is the space of smooth sections of E, C∞c (M,E) is the

space of smooth sections of E with compact support, and C∞cc (M,E) is the space of smooth

sections of E with compact support in M . Note that

C∞cc (M,E) ⊂ C∞c (M,E) ⊂ C∞(M,E).

We denote by L2(M,E) the Hilbert space of square-integrable sections of E, which is the

completion of C∞c (M,E) with respect to the norm induced by the L2-inner product

(u1, u2)L2(M,E) :=

∫M

〈u1, u2〉 dV,

where 〈·, ·〉 denotes the fiberwise inner product. Similarly, we have spaces C∞(∂M,E∂M),

C∞c (∂M,E∂M) and L2(∂M,E∂M) on the boundary ∂M , where E∂M denotes the restriction of

the bundle E to ∂M . If u ∈ C∞(M,E), we denote by u∂M ∈ C∞(∂M,E∂M) the restriction

of u to ∂M . For general sections on the boundary ∂M , we use bold letters u,v, · · · to denote

them.

Let E,F be two Hermitian vector bundles over M and D : C∞c (M,E)→ C∞c (M,F ) be a

first-order differential operator. The formal adjoint of D, denoted by D∗, is defined by∫M

〈Du, v〉dV =

∫M

〈u,D∗v〉 dV, (3.1.1)

for all u, v ∈ C∞cc (M,E). If E = F and D = D∗, then D is called formally self-adjoint.

3.1.2 Minimal and maximal extensions

We set Dcc := D|C∞cc (M,E) and view it as an unbounded operator from L2(M,E) to L2(M,F ).

The minimal extension Dmin of D is the operator whose graph is the closure of that of Dcc.

The maximal extension Dmax of D is defined to be Dmax =((D∗)cc

)ad, where the superscript

“ad” denotes the adjoint of the operator in the sense of functional analysis. Both Dmin and

70

Dmax are closed operators. Their domains, domDmin and domDmax, become Hilbert spaces

equipped with the graph norm ‖ · ‖D, which is the norm associated with the inner product

(u1, u2)D :=

∫M

(〈u1, u2〉 + 〈Du1, Du2〉

)dV.

It’s easy to see from the following Green’s formula that C∞c (M,E) ⊂ domDmax.

3.1.3 Green’s formula

Let τ ∈ TM |∂M be the unit inward normal vector field along ∂M . Using the Riemannian

metric, τ can be identified with its associated one-form. We have the following formula (cf.

[20, Proposition 3.4]).

Proposition 3.1.1 (Green’s formula). Let D be as above. Then for all u ∈ C∞c (M,E) and

v ∈ C∞c (M,F ),∫M

〈Du, v〉 dV =

∫M

〈u,D∗v〉 dV −∫∂M

〈σD(τ)u∂M , v∂M〉 dS, (3.1.2)

where σD denotes the principal symbol of the operator D.

Remark 3.1.2. A more general version of formula (3.1.2) will be presented in Theorem 3.2.20

below.

3.1.4 Sobolev spaces

Let ∇E be a Hermitian connection on E. For any u ∈ C∞(M,E), the covariant derivative

∇Eu ∈ C∞(M,T ∗M ⊗E). Applying the covariant derivative multiple times we get (∇E)k ∈

C∞(M,T ∗M⊗k ⊗ E) for k ∈ Z+. We define kth Sobolev space by

Hk(M,E) :={u ∈ L2(M,E) : (∇E)ju ∈ L2(M,T ∗M⊗j ⊗ E) for all j = 1, . . . , k

},

where the covariant derivatives are understood in distributional sense. It is a Hilbert space

with Hk-norm

‖u‖2Hk(M,E) := ‖u‖2

L2(M,E) + ‖∇Eu‖2L2(M,T ∗M⊗E) + · · · + ‖(∇E)ku‖2

L2(M,T ∗M⊗k⊗E).

71

Note that when M is compact, Hk(M,E) does not depend on the choices of ∇E and the

Riemannian metric, but when M is non-compact, it does.

We say u ∈ L2loc(M,E) if the restrictions of u to compact subsets of M have finite L2-

norms. For k ∈ Z+, we say u ∈ Hkloc(M,E), the kth local Sobolev space, if u,∇Eu, (∇E)2u, . . . , (∇E)ku

all lie in L2loc. This Sobolev space is independent of the preceding choices.

Similarly, we fix a Hermitian connection on F and define the spaces L2(M,F ), L2loc(M,F ),

Hk(M,F ), and Hkloc(M,F ). Again, definitions of these spaces apply without change to ∂M .

3.1.5 Completeness

We recall the following definition of completeness and a lemma from [11].

Definition 3.1.3. We call D a complete operator if the subspace of compactly supported

sections in domDmax is dense in domDmax with respect to the graph norm of D.

Lemma 3.1.4 ([11, Lemma 3.1]). Let f : M → R be a Lipschitz function with compact

support and u ∈ domDmax. Then fu ∈ domDmax and

Dmax(fu) = σD(df)u + fDmaxu.

The next theorem, again from [11], is still true here with minor changes of the proof.

Theorem 3.1.5. Let D : C∞(M,E) → C∞(M,F ) be a differential operator of first order.

Suppose that there exists a constant C > 0 such that

|σD(ξ)| ≤ C |ξ|

for all x ∈M and ξ ∈ T ∗xM . Then D and D∗ are complete.

Sketch of the proof. Fix a base point x0 ∈ ∂M and let r : M → R be the distance function

from x0, r(x) = dist(x, x0). Then r is a Lipschitz function with Lipschitz constant 1. Now

the proof is exactly the same as that of [11, Theorem 3.3].

72

Example 3.1.6. If D is a Dirac-type operator (cf. Subsection 2.2.1), then σD(ξ) = σD∗(ξ) =

c(ξ) is the Clifford multiplication. So one can choose C = 1 in Theorem 3.1.5 and therefore

D and D∗ are complete.

3.2 Domains of strongly Callias-type operators

In this section we introduce our main object of study – strongly Callias-type operators. The

main property of these operators is the discreteness of their spectra. We discuss natural

domains for a strongly Callias-type operator on a manifold with non-compact boundary. We

also introduce a scale of Sobolev spaces defined by a strongly Callias-type operator.

3.2.1 A product structure

We say that the Riemannian metric gM is product near the boundary if there exists a neigh-

borhood U ⊂M of the boundary which is isometric to the cylinder

Zr := [0, r)× ∂M ⊂ M. (3.2.1)

In the following we identify U with Zr and denote by t the coordinate along the axis of Zr.

Then the inward unit normal vector to the boundary is given by τ = dt.

Further, we assume that the Clifford multiplication c : T ∗M → End(E) and the connec-

tion ∇E also have product structure on Zr. In this situation we say that the Dirac bundle

E is product on Zr. We say that the Dirac bundle E is product near the boundary if there

exists r > 0, a neighborhood U of ∂M and an isometry U ' Zr such that E is product on

Zr. In this situation the restriction of the Dirac operator to Zr takes the form

D = c(τ)(∂t + A

), (3.2.2)

where, by (0.1.1) (with τ = en),

A = −n−1∑j=1

c(τ)c(ej)∇Eej.

73

The operator A is formally self-adjoint A∗ = A and anticommutes with c(τ)

A ◦ c(τ) = − c(τ) ◦ A. (3.2.3)

Let D = D+ iΦ : C∞(M,E)→ C∞(M,E) be a strongly Callias-type operator. Then the

restriction of D to Zr is given by

D = c(τ)(∂t + A− ic(τ)Φ

)= c(τ)

(∂t +A

), (3.2.4)

where

A := A − ic(τ)Φ : C∞(∂M,E∂M) → C∞(∂M,E∂M). (3.2.5)

Definition 3.2.1. We say that a Callias-type operator D is product near the boundary if the

Dirac bundle E is product near the boundary and the restriction of the Callias potential Φ

to Zr does not depend on t. The operator A of (3.2.5) is called the restriction of D to the

boundary.

Remark 3.2.2. One can easily see that the restriction of D to the boundary is an adapted

operator to D in the sense of Definition 2.2.1.

3.2.2 The restriction of the adjoint to the boundary

Recall that Φ is a self-adjoint bundle map, which, by Remark 0.1.4, commutes with the

Clifford multiplication. It follows from (3.2.4), that

D∗ = c(τ)(∂t +A]

)= c(τ)

(∂t + A+ ic(τ)Φ

), (3.2.6)

where

A] := A + ic(τ)Φ. (3.2.7)

Thus, D∗ is product near the boundary.

From (0.1.4) and (3.2.3), we obtain

A] = − c(τ) ◦ A ◦ c(τ)−1. (3.2.8)

74

3.2.3 Self-adjoint strongly Callias-type operators

Notice that A is a formally self-adjoint Dirac-type operator on ∂M and thus is an essentially

self-adjoint elliptic operator by [43, Theorem 1.17]. Since c(τ) anticommutes with A, we

have

A2 = A2 + ic(τ)[A,Φ] + Φ2. (3.2.9)

It follows from Definition 0.1.2 and (3.2.5) that [A,Φ] is also a bundle map with the same

norm as [D,Φ]. Thus the last two terms on the right hand side of (3.2.9) grow to infinity at

the infinite ends of ∂M . By [65, Lemma 6.3], the spectrum of A is discrete. In fact, A is a

self-adjoint strongly Callias-type operator.

3.2.4 Sobolev spaces on the boundary

The operator id +A2 is positive. Hence, for any s ∈ R, its powers (id +A2)s/2 can be defined

using functional calculus.

Definition 3.2.3. Set

C∞A (∂M,E∂M) :={

u ∈ C∞(∂M,E∂M) :∥∥(id +A2)s/2u

∥∥2

L2(∂M,E∂M )< +∞ for all s ∈ R

}.

For all s ∈ R we define the Sobolev HsA-norm on C∞A (∂M,E∂M) by

‖u‖2HsA(∂M,E∂M ) :=

∥∥(id +A2)s/2u∥∥2

L2(∂M,E∂M ). (3.2.10)

The Sobolev space HsA(∂M,E∂M) is defined to be the completion of C∞A (∂M,E∂M) with

respect to this norm.

Remark 3.2.4. In general,

C∞c (∂M,E∂M) ⊂ C∞A (∂M,E∂M) ⊂ C∞(∂M,E∂M).

When ∂M is compact, the above spaces are all equal and the space C∞A (∂M,E∂M) is inde-

pendent of A. However, if ∂M is not compact, these spaces are different and C∞A (∂M,E∂M)

75

does depend on the operator A. Consequently, if ∂M is not compact, the Sobolev spaces

HsA(∂M,E∂M) depend on A.

Remark 3.2.5. Alternatively one could define the s-Sobolev space to be the completion of

C∞c (∂M,E∂M) with respect to the HsA-norm. In general, this leads to a different scale of

Sobolev spaces, cf. [47, §3.1] for more details. We prefer our definition, since the space

Hfin(A), defined below in (3.2.12), which plays an important role in our discussion, is a

subspace of C∞A (∂M,E∂M) but is not a subspace of C∞c (∂M,E∂M).

The rest of this section follows rather closely the exposition in Sections 5 and 6 of [11]

with some changes needed to accommodate the non-compactness of the boundary.

3.2.5 Eigenvalues and eigensections of A

Let

−∞← · · · ≤ λ−2 ≤ λ−1 ≤ λ0 ≤ λ1 ≤ λ2 ≤ · · · → +∞ (3.2.11)

be the spectrum ofA with each eigenvalue being repeated according to its (finite) multiplicity.

Fix a corresponding L2-orthonormal basis {uj}j∈Z of eigensections of A. By definition, each

element in C∞A (∂M,E∂M) is L2-integrable and thus can be written as u =∑∞

j=−∞ ajuj.

Then

‖u‖2HsA(∂M,E∂M ) =

∞∑j=−∞

|aj|2 (1 + λ2j)s.

On the other hand, let

Hfin(A) :={

u =∑j

ajuj : aj = 0 for all but finitely many j}

(3.2.12)

be the space of finitely generated sections. Then Hfin(A) ⊂ C∞A (∂M,E∂M) and for any

s ∈ R, Hfin(A) is dense in HsA(∂M,E∂M). We obtain an alternative description of the

Sobolev spaces

HsA(∂M,E∂M) =

{u =

∑j

ajuj :∑j

|aj|2(1 + λ2j)s < +∞

}.

76

Remark 3.2.6. The following properties follow from our definition and preceding discussion.

(i) H0A(∂M,E∂M) = L2(∂M,E∂M).

(ii) If s < t, then ‖u‖HsA(∂M,E∂M ) ≤ ‖u‖Ht

A(∂M,E∂M ). And we shall show shortly in Theorem

3.2.7 that there is still a Rellich embedding theorem, i.e., the induced embedding

H tA(∂M,E∂M) ↪→ Hs

A(∂M,E∂M) is compact.

(iii)⋂s∈RH

sA(∂M,E∂M) = C∞A (∂M,E∂M).

(iv) For all s ∈ R, the pairing

HsA(∂M,E∂M) × H−sA (∂M,E∂M) → C,

(∑j

ajuj,∑j

bjuj

)7→∑j

ajbj

is perfect. Therefore, HsA(∂M,E∂M) and H−sA (∂M,E∂M) are pairwise dual.

We have the following version of the Rellich Embedding Theorem:

Theorem 3.2.7. For s < t, the embedding H tA(∂M,E∂M) ↪→ Hs

A(∂M,E∂M) mentioned in

Remark 3.2.6.(ii) is compact.

To prove the theorem, we use the following result, cf. for example, [14, Proposition 2.1].

Proposition 3.2.8. A closed bounded subset K in a Banach space X is compact if and only

if for every ε > 0, there exists a finite dimensional subspace Yε of X such that every element

x ∈ K is within distance ε from Yε.

Proof of Theorem 3.2.7. Let B be the unit ball in H tA(∂M,E∂M). We use Proposition 3.2.8

to show that the closure B of B in HsA(∂M,E∂M) is compact in Hs

A(∂M,E∂M).

For simplicity, suppose that λ0 is an eigenvalue of A with smallest absolute value and for

n > 0, set Λn := min{λ2n, λ

2−n}. Then {Λn} is an increasing sequence by (3.2.11). For every

ε > 0, there exists an integer N > 0, such that (1 + Λn)s−t < ε2/4 for all n ≥ N .

Consider the finite-dimensional space

Yε := span {uj : −N ≤ j ≤ N} ⊂ HsA(∂M,E∂M).

77

We claim that every element u ∈ B is within distance ε from Yε. Indeed, choose u =∑j ajuj ∈ B, such that the Hs

A-distance between u and u is less than ε/2. Then u′ :=∑Nj=−N ajuj belongs to Yε and the Hs

A-distance

‖u− u′‖2HsA(∂M,E∂M ) =

∑|j|>N

|aj|2 (1 + λ2j)s ≤

∑|j|>N

|aj|2 (1 + λ2j)t · (1 + ΛN)s−t

;≤ ‖u‖2HtA(∂M,E∂M ) · (1 + ΛN)s−t ≤ (1 + ΛN)s−t <

ε2

4.

Hence u is within distance ε/2 of Yε, and therefore u is within distance ε of Yε. The theorem

then follows from Proposition 3.2.8.

3.2.6 The hybrid Soblev spaces

For I ⊂ R, let

PAI :∑j

ajuj 7→∑λj∈I

ajuj

be the spectral projection. It’s easy to see that

HsI (A) := PAI (Hs

A(∂M,E∂M)) ⊂ HsA(∂M,E∂M)

for all s ∈ R.

Definition 3.2.9. For a ∈ R, we define the hybrid Sobolev space

H(A) := H1/2(−∞,a](A) ⊕ H

−1/2(a,∞)(A) ⊂ H

−1/2A (∂M,E∂M) (3.2.13)

with H-norm

‖u‖2H(A)

:=∥∥PA(−∞,a]u

∥∥2

H1/2A (∂M,E∂M )

+∥∥PA(a,∞)u

∥∥2

H−1/2A (∂M,E∂M )

.

The space H(A) is independent of the choice of a. Indeed, for a1 < a2, the dif-

ference between the corresponding H-norms only occurs on the finite dimensional space

PA[a1,a2](L2(∂M,E∂M)). Thus the norms defined using different values of a are equivalent.

78

Similarly, we define

H(A) := H−1/2(−∞,a](A) ⊕ H

1/2(a,∞)(A)

with H-norm

‖u‖2H(A)

:= ‖PA(−∞,a]u‖2

H−1/2A (∂M,E∂M )

+ ‖PA(a,∞)u‖2

H1/2A (∂M,E∂M )

.

Then

H(A) = H(−A).

The pairing of Remark 3.2.6.(iv) induces a perfect pairing

H(A) × H(A) → C.

3.2.7 The hybrid space of the dual operator

Recall, that the restriction A] of D∗ to the boundary can be computed by (3.2.8). Thus the

isomorphism c(τ) : E∂M → E∂M sends each eigensection uj of A associated to eigenvalue

λj to an eigensection of A] associated to eigenvalue −λj. We conclude that the set of

eigenvalues of A] is {−λj}j∈Z with associated L2-orthonormal eigensections {c(τ)uj}j∈Z.

For u =∑

j ajuj ∈ HsA(∂M,E∂M), we have

‖c(τ)u‖2HsA] (∂M,E∂M ) =

∑j

|aj|2(1 + (−λj)2

)s= ‖u‖2

HsA(∂M,E∂M ).

So c(τ) induces an isometry between Sobolev spaces HsA(∂M,E∂M) and Hs

A](∂M,E∂M) for

any s ∈ R. Furthermore, it restricts to an isomorphism between Hs(−∞,a](A) and Hs

[−a,∞)(A]).

Therefore we conclude that

Lemma 3.2.10. Over ∂M , the isomorphism c(τ) : E∂M → E∂M induces an isomorphism

H(A)→ H(A]). In particular, the sesquilinear form

β : H(A) × H(A]) → C, β(u,v) := −(u,−c(τ)v

)= −

(c(τ)u,v

),

is a perfect pairing of topological vector spaces.

79

3.2.8 Sections in a neighborhood of the boundary

Recall from (3.2.1) that we identify a neighborhood of ∂M with the product Zr = [0, r)×∂M .

The L2-sections over Zr can be written as

u(t, x) =∑j

aj(t) uj(x)

in terms of the L2-orthonormal basis {uj} on ∂M . We fix a smooth cut-off function χ : R→

R with

χ(t) =

1 for t ≤ r/3

0 for t ≥ 2r/3.

(3.2.14)

Recall that Hfin(A) is dense in H(A) and H(A). For u ∈ Hfin(A), we define a smooth

section E u over Zr by

(E u)(t) := χ(t) · exp(−t|A|) u. (3.2.15)

Thus, if u(x) =∑

j ajuj(x), then

(E u)(t, x) = χ(t)∑j

aj · exp(−t|λj|) · uj(x). (3.2.16)

It’s easy to see that E u is an L2-section over Zr. So we get a linear map

E : Hfin(A) → C∞(Zr, E) ∩ L2(Zr, E)

which we call the extension map.

As in Subsection 3.1.2 we denote by ‖ · ‖D the graph norm of D.

Lemma 3.2.11. For all u ∈ Hfin(A), the extended section E u over Zr belongs to domDmax.

And there exists a constant C = C(χ,A) > 0 such that

∥∥E u∥∥D ≤ C ‖u‖H(A) and

∥∥c(τ)E u∥∥D∗ ≤ C ‖u‖H(A).

Proof. For the first claim, we only need to show that D(E u) is an L2-section over Zr. Since

D(E u) = D(EPA(−∞,0]u) + D(EPA(0,∞)u),

80

it suffices to consider each summand separately. Recall that D = c(τ)(∂t + A) on Zr. By

(3.2.15), we have

D(EPA(0,∞)u) = c(τ)χ′ exp(−tA)PA(0,∞)u,

which is clearly an L2-section over Zr. On the other hand,

D(EPA(−∞,0]u) = c(τ)(2χA+ χ′

)exp(tA)PA(−∞,0]u,

which is again an L2-section over Zr. Therefore E u ∈ domDmax.

The proof of the first inequality is exactly the same as that of [11, Lemma 5.5]. For the

second one, just notice thatA] is the restriction to the boundary of D∗ and, by Lemma 3.2.10,

c(τ) : H(A])→ H(A) is an isomorphism of Hilbert spaces.

The following lemma is an analogue of [11, Lemma 6.2] with exactly the same proof.

Lemma 3.2.12. There is a constant C > 0 such that for all u ∈ C∞c (Zr, E),

‖u∂M‖H(A) ≤ C ‖u‖D.

3.2.9 A natural domain for boundary value problems

For closed manifolds the ellipticity of D implies that dom(Dmax) ⊂ H1loc(M,E). However,

if ∂M 6= ∅, then near the boundary the sections in dom(Dmax) can behave badly. That is

why, if one wants to talk about boundary value of sections, one needs to consider a smaller

domain for D.

Definition 3.2.13. We define the norm

‖u‖2H1D(Zr,E) := ‖u‖2

L2(Zr,E) + ‖∂tu‖2L2(Zr,E) + ‖Au‖2

L2(Zr,E). (3.2.17)

and denote by H1D(Zr, E) the completion of C∞c (Zr, E) with respect to this norm. We refer

to (3.2.17) as the H1D(Zr)-norm.

81

In general, for any integer k ≥ 1, let HkD(Zr, E) be the completion of C∞c (Zr, E) with

respect to the HkD(Zr)-norm given by

‖u‖2HkD(Zr,E) := ‖u‖2

L2(Zr,E) + ‖(∂t)ku‖2L2(Zr,E) + ‖Aku‖2

L2(Zr,E). (3.2.18)

Note that H1D(Zr, E) ⊂ H1

loc(Zr, E)∩L2(Zr, E). Moreover, we have the following analogue

of the Rellich embedding theorem:

Lemma 3.2.14. The inclusion map H1D(Zr, E) ↪→ L2(Zr, E) is compact.

Proof. Let B be the unit ball about the origin in H1D(Zr, E) and let B denote its closure in

L2(Zr, E). We need to prove that B is compact. By Proposition 3.2.8 it is enough to show

that for every ε > 0 there exists a finite dimensional subspace Yε ∈ L2(Zr, E) such that every

u ∈ B is within distance ε from Yε.

Let λj and uj be as in Subsection 3.2.5. As in the proof of Theorem 3.2.7 we set Λn :=

min{λ2n, λ

2−n}. Choose N > 0 such that

1 + Λn >8

ε2for all n ≥ N. (3.2.19)

Let H1([0, r)) denote the Sobolev space of complex-valued functions on the interval [0, r)

with norm

‖a‖2H1([0,r)) := ‖a‖2

L2([0,r)) + ‖a′‖2L2([0,r)).

Let B′ ⊂ H1([0, r)) denote the unit ball about the origin in H1([0, r)) and let B′ be its

closure in L2([0, r)). By the classical Rellich embedding theorem B′ is compact in L2([0, r)).

Hence, for every ε > 0 there exists a finite set Xε such that every a ∈ B′ is within distance

ε√16N+8

from Xε.

We now define the finite dimensional space

Yε :={ N∑j=−N

aj(t) uj : aj(t) ∈ Xε

}⊂ L2(Zr, E).

82

We claim that every u ∈ B is within distance ε from Yε. Indeed, let u ∈ B. We choose

u =∑∞

j=−∞ bj(t)uj ∈ B such that

‖u− u‖ <ε

2. (3.2.20)

Since {uj} is an orthonormal basis of L2(∂M,E∂M), we conclude from (3.2.17) that

‖u‖2H1D(Zr,E) =

∞∑j=−∞

((1 + λ2

j) ‖bj‖2L2([0,r)) + ‖b′j‖2

L2([0,r))

).

Since ‖u‖2H1D(Zr,E)

≤ 1, for all j ∈ Z

(1 + λ2j) ‖bj‖2

L2([0,r)) + ‖b′j‖2L2([0,r)) ≤ 1.

Hence,

‖bj‖2L2([0,r)) + ‖b′j‖2

L2([0,r)) ≤ 1 =⇒ bj ∈ B′, for all j ∈ Z; (3.2.21)∑|j|>N

‖bj‖2L2([0,r)) <

ε2

8, (3.2.22)

where in the second inequality we use (3.2.19).

From (3.2.21) we conclude that for every j ∈ Z, there exists aj ∈ Xε such that

‖bj − aj‖L2([0,r)) ≤ε√

16N + 8.

Hence,N∑

j=−N

‖bj − aj‖2L2([0,r)) ≤ (2N + 1)

ε2

16N + 8=

ε2

8. (3.2.23)

Set u′ :=∑N

j=−N aj(t)uj ∈ Yε. Then from (3.2.22) and (3.2.23) we obtain

‖u− u′‖2L2(Zr,E) =

∥∥ ∑|j|>N

bjuj +N∑

j=−N

(bj − aj)uj∥∥2

L2(Zr,E)

;≤∑|j|>N

‖bj‖2L2([0,r)) +

N∑j=−N

‖bj − aj‖2L2([0,r)) ≤

ε2

8+

ε2

8=

ε2

4.

Combining this with (3.2.20) we obtain

‖u− u′‖L2(Zr,E) ≤ ‖u− u‖L2(Zr,E) + ‖u− u′‖L2(Zr,E) ≤ε

2+

ε

2= ε,

i.e., u is within distance ε from Yε.

83

Lemma 3.2.15. For all u ∈ C∞c (Zr, E) with PA(0,∞)(u∂M) = 0, we have estimate

1√2‖u‖D ≤ ‖u‖H1

D(Zr,E) ≤ ‖u‖D. (3.2.24)

Proof. Since D = c(τ)(∂t +A) on Zr, we obtain

‖u‖2D ≤ ‖u‖2

L2(Zr,E) + 2(‖∂tu‖2

L2(Zr,E) + ‖Au‖2L2(Zr,E)

)≤ 2 ‖u‖2

H1D(Zr,E),

for all u ∈ C∞c (Zr, E). This proves the first inequality in (3.2.24).

Suppose that u ∈ C∞c (Zr, E) with PA(0,∞)(u∂M) = 0. We want to show the converse

inequality.

We can write u =∑

j aj(t)uj. Then aj(r) = 0 for all j and aj(0) = 0 for all j such that

λj > 0. The latter condition means that∑j

λj |aj(0)|2 ≤ 0.

Then

‖Du‖2L2(Zr,E) =

∑j

∫ r

0

|a′j(t) + aj(t)λj|2dt

=∑j

(∫ r

0

|a′j(t)|2dt+ λ2j

∫ r

0

|aj(t)|2dt+ λj

∫ r

0

(a′j(t)aj(t) + aj(t)a′j(t))dt)

=∑j

(∫ r

0

|a′j(t)|2dt+ λ2j

∫ r

0

|aj(t)|2dt+ λj

∫ r

0

d

dt|aj(t)|2dt

)=∑j

(∫ r

0

|a′j(t)|2dt+ λ2j

∫ r

0

|aj(t)|2dt+ λj(|aj(r)|2 − |aj(0)|2))

≥∑j

(∫ r

0

|a′j(t)|2dt+ λ2j

∫ r

0

|aj(t)|2dt)

= ‖∂tu‖2L2(Zr,E) + ‖Au‖2

L2(Zr,E).

(3.2.25)

Hence

‖u‖2D := ‖u‖2

L2(Zr,E) + ‖Du‖2L2(Zr,E)

;≥ ‖u‖2L2(Zr,E) + ‖∂tu‖2

L2(Zr,E) + ‖Au‖2L2(Zr,E) =: ‖u‖2

H1D(Zr,E).

84

Remark 3.2.16. In particular, the two norms are equivalent on C∞cc (Zr, E).

3.2.10 The trace theorem

The following “trace theorem” establishes the relationship between HkD(Zr, E) and the

Sobolev spaces on the boundary.

Theorem 3.2.17 (The trace theorem). For all k ≥ 1, the restriction map (or trace map)

R : C∞c (Zr, E) → C∞c (∂M,E∂M), R(u) := u∂M

extends to a continuous linear map

R : HkD(Zr, E) → H

k−1/2A (∂M,E∂M).

Proof. Let u(t, x) =∑

j aj(t)uj(x) ∈ C∞c (Zr, E). Then R(u) = u∂M(x) =∑

j aj(0)uj(x),

and we want to show that

‖u∂M‖2

Hk−1/2A (∂M,E∂M )

≤ C(k) ‖u‖2HkD(Zr,E) (3.2.26)

for some constant C(k) > 0.

Applying inverse Fourier transform to aj(t) yields that

aj(t) =

∫Reit·ξ aj(ξ) dξ,

where aj(ξ) is the Fourier transform of aj(t). (Here we use normalized measure to avoid the

coefficient 2π.) So

aj(0) =

∫Raj(ξ) dξ.

By Holder’s inequality,

|aj(0)|2 =(∫

Raj(ξ) dξ

)2

≤(∫

R|aj(ξ)| (1 + λ2

j + ξ2)k/2 (1 + λ2j + ξ2)−k/2 dξ

)2

≤∫R|aj(ξ)|2 (1 + λ2

j + ξ2)k dξ ·∫R(1 + λ2

j + ξ2)−k dξ,

85

where λj is the eigenvalue of A corresponding to index j. We do the substitution ξ =

(1 + λ2j)

1/2τ to get∫R(1 + λ2

j + ξ2)−k dξ = (1 + λ2j)−k+1/2

∫R(1 + τ 2)−k dτ.

It’s easy to see that the integral on the right hand side converges when k ≥ 1 and depends

only on k. Therefore

|aj(0)|2(1 + λ2j)k−1/2 ≤ C1(k)

∫R|aj(ξ)|2 (1 + λ2

j + ξ2)k dξ

≤ C(k)

∫R|aj(ξ)|2 (1 + λ2k

j + ξ2k) dξ

≤ C(k)(∫

R|aj(t)|2 dt+

∫R|aj(ξ)|2ξ2k dξ +

∫R|aj(t)|2λ2k

j dt),

(3.2.27)

where we use Plancherel’s identity from line 2 to line 3. Recall the differentiation property

of Fourier transform (∂t)kaj(t)(ξ) = aj(ξ)ξk. So again by Plancherel’s identity∫

R|aj(ξ)|2 ξ2k dξ =

∫R| (∂t)k aj(t)(ξ)|2 dξ =

∫R|(∂t)k aj(t)|2 dt

Now summing inequality (3.2.27) over j gives (3.2.26) and the theorem is proved.

3.2.11 The space H1D(M,E)

Recall that the cut-off function χ is defined in (3.2.14). By a slight abuse of notation we

also denote by χ the induced function on M . Define

H1D(M,E) := domDmax ∩

{u ∈ L2(M,E) : χu ∈ H1

D(Zr, E)}. (3.2.28)

It is a Hilbert space with the H1D-norm

‖u‖2H1D(M,E) := ‖u‖2

L2(M,E) + ‖Du‖2L2(M,E) + ‖χu‖2

H1D(Zr,E).

As one can see from Remark 3.2.16, a different choice of the cut-off function χ leads to an

equivalent norm. The H1D-norm is stronger than the graph norm of D in the sense that

86

it controls in addition the H1D-regularity near the boundary. We call it H1

D-regularity as

it depends on our concrete choice of the norm (3.2.17), unlike the case in [11], where the

boundary is compact.

Lemma 3.2.15 and Theorem 3.2.17 extend from Zr to M . By the definition of H1D(M,E)

and the fact that D is complete, we have

Lemma 3.2.18. (i) C∞c (M,E) is dense in H1D(M,E);

(ii) C∞cc (M,E) is dense in {u ∈ H1D(M,E) : u∂M = 0}.

The following statement is an immediate consequence of Remark 3.2.16 and Lemma

3.2.18.(ii).

Corollary 3.2.19. domDmin = {u ∈ H1D(M,E) : u∂M = 0}.

3.2.12 Regularity of the maximal domain

We now state the main result of this section which extends Theorem 6.7 of [11] to manifolds

with non-compact boundary.

Theorem 3.2.20. Assume that D is a strongly Callias-type operator. Then

(i) C∞c (M,E) is dense in domDmax with respect to the graph norm of D.

(ii) The trace map R : C∞c (M,E) → C∞c (∂M,E∂M) extends uniquely to a surjective

bounded linear map R : domDmax → H(A).

(iii) H1D(M,E) = {u ∈ domDmax : Ru ∈ H1/2

A (∂M,E∂M)}.

The corresponding statements hold for dom(D∗)max (with A replaced with A]). Furthermore,

for all sections u ∈ domDmax and v ∈ dom(D∗)max, we have

(Dmaxu, v

)L2(M,E)

−(u, (D∗)maxv

)L2(M,E)

= −(c(τ)Ru,Rv

)L2(∂M,E∂M )

. (3.2.29)

87

Remark 3.2.21. In particular, (ii) of Theorem 3.2.20 says that C∞c (∂M,E∂M) is dense in

H(A).

Proof. The proof goes along the same line as the proof of Theorem 6.7 in [11] but some extra

care is needed because of non-compactness of the boundary.

(i) Let M be the double of M formed by gluing two copies of M along their boundaries.

Then M is a complete manifold without boundary. One can extend the Riemannian metric

gM , the Dirac bundle E and the Callias-type operator D on M to a Riemannian metric gM , a

Dirac bundle E and a Callias-type operator D on M . Notice that now dom Dmax = dom Dmin

by [43].

Lemma 3.2.22. If u ∈ dom Dmax, then u := u|M ∈ H1D(M,E).

Proof. Let Z(−r,r) be the double of Zr in M . Clearly, it suffices to consider the case when the

support of u is contained in Z(−r,r). Since dom Dmax = dom Dmin, it suffices to show that if

a sequence un ∈ C∞c (Z(−r,r), E) converges to u in the graph norm of D then un|M converges

in H1D(M,E). This follows from the following estimate

∥∥u|M‖H1D(M,E) ≤ ‖u‖D, u ∈ C∞c (Z(−r,r), E), (3.2.30)

which we prove below.

Since D is a product on Zr, we obtain from (3.2.4) that on Z(−r,r)

D∗D = − ∂2t + A2.

Hence, on compactly supported sections u we have

∥∥Du‖2L2(M,E)

=(D∗Du, u

)L2(M,E)

= ‖∂tu‖2L2(M,E)

+ ‖Au‖2L2(M,E)

.

We conclude that

‖u‖2D := ‖u‖2

L2(M,E)+ ‖Du‖2

L2(M,E)

= ‖u‖2L2(M,E)

+ ‖∂tu‖2L2(M,E)

+ ‖Au‖2L2(M,E)

≥ ‖u|M‖2H1D(M,E).

88

Let Dc denote the operator D with domain C∞c (M,E). Let (Dc)ad denote the adjoint

of Dc in the sense of functional analysis. Note that (Dc)ad ⊂ (D∗)max, where, as usual, we

denote by D∗ the formal adjoint of D.

Fix an arbitrary u ∈ dom(Dc)ad and let u ∈ L2(M, E) and v ∈ L2(M, E) denote the

sections whose restriction to M \M are equal to 0 and whose restriction to M are equal to

u and (Dc)adu respectively.

Let w ∈ C∞c (M, E). The restriction of w = w|M ∈ domDc. Since u|M\M = v|M\M = 0

we obtain

(Dw, u)L2(M,E) = (Dcw, u)L2(M,E) =(w, (Dc)adu

)L2(M,E)

= (w, v)L2(M,E).

Hence, u is a weak solution of the equation D∗u = v ∈ L2(M, E). By elliptic regularity

u ∈ H1loc(M, E). It follows that u|∂M = u|∂M = 0. Also, by Lemma 3.2.22, u ∈ H1

D∗(M,E).

By Corollary 3.2.19, u is in the domain of the minimal extension (D∗)min of (D∗)cc. Since u

is an arbitrary section in dom(Dc)ad, we conclude that (Dc)ad ⊂ (D∗)min. Hence the closure

Dc of Dc satisfies

Dc ⊂ Dmax =((D∗)min

)ad ⊂((Dc)ad

)ad= Dc.

Hence, Dc = Dmax as claimed in part (i) of the theorem.

(ii) By (i) C∞c (M,E) is dense in domDmax. Hence, it follows from Lemma 3.2.12 that

the extension exists and unique. To prove the surjectivity recall that the space Hfin(A),

defined in (3.2.12), is dense in H(A). Fix u ∈ H(A) and let ui → u be a sequence of

sections ui ∈ Hfin(A) which converges to u in H(A). Then, by Lemma 3.2.11, the sequence

E ui ∈ domDmax is a Cauchy sequence and, hence, converges to an element v ∈ domDmax.

Then Rv = u.

(iii) The inclusion

H1D(M,E) ⊂ {u ∈ domDmax : Ru ∈ H1/2

A (∂M,E∂M)}

89

follows directly from (3.2.28) and the Trace Theorem 3.2.17.

To show the opposite inclusion, choose u ∈ domDmax with Ru ∈ H1/2A (∂M,E∂M) and set

v := PA(0,∞)Ru. Then

u = E v + (u− E v).

Using (3.2.16) we readily see that E v ∈ H1D(M,E). Since PA(0,∞)R(u − E v) = 0 it follows

from (i) and Lemma 3.2.15, that u− E v ∈ H1D(M,E). Thus u ∈ H1

D(M,E) as required.

Finally, (3.2.29) holds for u, v ∈ C∞c (M,E) by (3.1.2). Since, by (i), C∞c (M,E) is dense

in both domDmax and dom(D∗)max, the equality for u ∈ domDmax and v ∈ dom(D∗)max

follows now from (i) and (ii) and Lemma 3.2.10.

3.3 Boundary value problems

Moving on from last section, we study boundary value problems of a strongly Callias-type

operator D whose restriction to the boundary is A. We introduce boundary conditions and

elliptic boundary conditions for D as certain closed subspaces of H(A). In particular, we

take a close look at an important elliptic boundary condition – the Atiyah–Patodi–Singer

boundary condition and obtain some results about it.

3.3.1 Boundary conditions

Let D be a strongly Callias-type operator. If ∂M = ∅, then the minimal and maximal

extensions of D coincide, i.e., Dmin = Dmax. But when ∂M 6= ∅ these two extensions are

not equal. Indeed, the restrictions of elements of Dmin to the boundary vanish identically by

Corollary 3.2.19, while the restrictions of elements of Dmax to the boundary form the whole

space H(A), cf. Theorem 3.2.20. The boundary value problems lead to closed extensions

lying between Dmin and Dmax.

90

Definition 3.3.1. A closed subspace B ⊂ H(A) is called a boundary condition for D. We

will use the notations DB,max and DB for the operators with the following domains

dom(DB,max) = {u ∈ domDmax : Ru ∈ B},

domDB = {u ∈ H1D(M,E) : Ru ∈ B}

= {u ∈ domDmax : Ru ∈ B ∩H1/2A (∂M,E∂M)}.

We remark that if B = H(A) then DB,max = Dmax. Also if B = 0 then DB,max = DB =

Dmin.

By Theorem 3.2.20.(ii), dom(DB,max) is a closed subspace of domDmax. Since the trace

map extends to a bounded linear map R : H1D(M,E)→ H

1/2A (∂M,E∂M) andH

1/2A (∂M,E∂M) ↪→

H(A) is a continuous embedding, domDB is also a closed subspace of H1D(M,E). We equip

dom(DB,max) with the graph norm of D and domDB the H1D-norm.

In particular, DB,max is a closed extension of D. Moreover, it follows immediately from

Definition 3.3.1 that B ⊂ H1/2A (∂M,E∂M) if and only if DB = DB,max. Thus in this case

domDB = domDB,max is a complete Banach space with respect to both the H1D-norm and

the graph norm. From [60, p. 71] we now obtain the following analogue of [11, Lemma 7.3]:

Lemma 3.3.2. Let B be a boundary condition. Then B ⊂ H1/2A (∂M,E∂M) if and only if

DB = DB,max, and in this case the H1D-norm and graph norm of D are equivalent on domDB.

3.3.2 Adjoint boundary conditions

For any boundary condition B, we have Dcc ⊂ DB,max. Hence the L2-adjoint operators

satisfy

(DB,max)ad ⊂ (Dcc)ad = (D∗)max.

From (3.2.29), we conclude that

dom(DB,max)ad ={v ∈ dom(D∗)max :

(c(τ)Ru,Rv

)= 0 for all u ∈ domDB,max

}.

91

By Theorem 3.2.20.(ii), for any u ∈ B there exists u ∈ dom(DB,max) with Ru = u.

Therefore

(DB,max)ad = (D∗)Bad,max

with

Bad :={

v ∈ H(A]) :(c(τ)u,v

)= 0 for all u ∈ B

}. (3.3.1)

By Lemma 3.2.10, Bad is a closed subspace of H(A]), thus is a boundary condition for D∗.

Definition 3.3.3. The spaceBad, defined by (3.3.1), is called the adjoint boundary condition

to B.

3.3.3 Elliptic boundary conditions

We adopt the same definition of elliptic boundary conditions as in [11] for the case of non-

compact boundary:

Definition 3.3.4. A boundary condition B is said to be elliptic if B ⊂ H1/2A (∂M,E∂M) and

Bad ⊂ H1/2

A] (∂M,E∂M).

Remark 3.3.5. One can see from Lemma 3.3.2 that when B is an elliptic boundary condition,

DB,max = DB, (D∗)Bad,max = D∗Bad and the two norms are equivalent. Definition 3.3.4 is also

equivalent to saying that domDB ⊂ H1D(M,E) and domD∗

Bad ⊂ H1D∗(M,E).

The following two examples of elliptic boundary condition are the most important to our

study (compare with Examples 7.27, 7.28 of [11]).

Example 3.3.6 (Generalized Atiyah–Patodi–Singer boundary conditions). For any a ∈ R,

let

B = B(a) := H1/2(−∞,a)(A). (3.3.2)

This is a closed subspace of H(A). In order to show that B is an elliptic boundary condition,

we only need to check that Bad ⊂ H1/2

A] (∂M,E∂M). By Lemma 3.2.10, c(τ) maps H1/2(−∞,a)(A)

92

to the subspace H1/2(−a,∞)(A]) of H(A]). Since there is a perfect pairing between H(A]) and

H(A]), we see that

Bad = H1/2(−∞,−a](A

]). (3.3.3)

Therefore B is an elliptic boundary condition. It is called the generalized Atiyah–Patodi–

Singer boundary conditions (or generalized APS boundary conditions for abbreviation). In

particular, B = H1/2(−∞,0)(A) will be called the Atiyah–Patodi–Singer boundary condition and

B = H1/2(−∞,0](A) will be called the dual Atiyah–Patodi–Singer boundary condition.

Remark 3.3.7. One can see from (3.3.3) that the adjoint of the APS boundary condition for

D is the dual APS boundary condition for D∗.

Example 3.3.8 (Transmission conditions). Let M be a complete manifold. For simplicity,

first assume that ∂M = ∅. Let N ⊂ M be a hypersurface such that cutting M along N we

obtain a manifold M ′ (connected or not) with two copies of N as boundary. So we can write

M ′ = (M \N) tN1 tN2.

Let E → M be a Dirac bundle over M and D : C∞(M,E) → C∞(M,E) be a strongly

Callias-type operator. They induce Dirac bundle E ′ →M ′ and strongly Callias-type operator

D′ : C∞(M ′, E ′)→ C∞(M ′, E ′) on M ′. We assume that all structures are product near N1

and N2. Let A be the restriction of D′ to N1. Then −A is the restriction of D′ to N2 and,

thus, the restriction of D′ to ∂M ′ is A′ = A⊕−A.

For u ∈ H1D(M,E) one gets u′ ∈ H1

D′(M′, E ′) (by Lemma 3.2.22) such that u′|N1 = u′|N2 .

We use this as a boundary condition for D′ on M ′ and set

B :={

(u,u) ∈ H1/2A (N1, EN1) ⊕ H

1/2−A(N2, EN2)

}. (3.3.4)

Lemma 3.3.9. The subspace (3.3.4) is an elliptic boundary condition, called the transmission

boundary condition.

Proof. First we show that B is a boundary condition, i.e. is a closed subspace of H(A′).

Clearly B is a closed subspace of H1/2A′ (∂M ′, E∂M ′). Thus it suffices to show that the H

1/2A′ -

93

norm and H(A′)-norm are equivalent on B. Since any two norms are equivalent on the

finite-dimensional eigenspace of A′ associated to eigenvalue 0, we may assume that 0 is not

in the spectrum of A′. Write

u = PA(−∞,0)u + PA(0,∞)u =: u− + u+.

Notice that PA′

I = PAI ⊕ P−AI = PAI ⊕ PA−I for any subset I ⊂ R. We have

PA′

(−∞,0)(u,u) = (u−,u+), PA′

(0,∞)(u,u) = (u+,u−).

Notice also that

‖u+‖H±1/2A (N1)

= ‖u+‖H±1/2−A (N2)

and similar equality holds for u−. It follows that

‖(u,u)‖2

H±1/2

A′ (∂M ′)= 2 ‖u‖2

H±1/2A (N1)

= 2 ‖u‖2

H±1/2−A (N2)

.

Using the above equations we get

‖(u,u)‖2H(A′) = ‖(u−,u+)‖2

H1/2

A′ (∂M ′)+ ‖(u+,u−)‖2

H−1/2

A′ (∂M ′)

= ‖u−‖2

H1/2A (N1)

+ ‖u+‖2

H1/2−A(N2)

+ ‖u+‖2

H−1/2A (N1)

+ ‖u−‖2

H−1/2−A (N2)

= ‖u‖2

H1/2A (N1)

+ ‖u‖2

H−1/2A (N1)

=1

2

(‖(u,u)‖2

H1/2

A′ (∂M ′)+ ‖(u,u)‖2

H−1/2

A′ (∂M ′)

)≥ 1

2‖(u,u)‖2

H1/2

A′ (∂M ′).

The other direction of inequality is trivial. So B is also closed in H(A′) and hence is a

boundary condition.

In order to show that B is an elliptic boundary condition we need to prove that Bad ⊂

H1/2

A′#(∂M ′, E∂M ′). It’s easy to see that

Bad ={

(v,−v) ∈ H−1/2

A] (N1, EN1) ⊕ H−1/2

−A] (N2, EN2)}∩ H(A′#).

94

Again by decomposing v in terms of v− and v+ like above, one can get that v ∈ H1/2

A] (N1, EN1).

Therefore

Bad ={

(v,−v) ∈ H1/2

A] (N1, EN1) ⊕ H1/2

−A](N2, EN2)}⊂ H

1/2

A′#(∂M ′, E∂M ′). (3.3.5)

Therefore B is an elliptic boundary condition.

If M has nonempty boundary and N is disjoint from ∂M , we assume that an elliptic

boundary condition is posed for ∂M . Then one can apply the same arguments as above to

pose the transmission condition for N1 tN2 and keep the original condition for ∂M .

3.4 Index theory

In this section we show that an elliptic boundary value problem for a strongly Callias-

type operator is Fredholm. As two typical examples, the indexes of APS and transmission

boundary value problems are interesting and are used to prove the splitting theorem, which

allows to compute the index by cutting and pasting.

3.4.1 Fredholmness

Let D : C∞(M,E) → C∞(M,E) be a strongly Callias-type operator. The growth assump-

tion of the Callias potential guarantees that D is invertible at infinity.

Lemma 3.4.1. A strongly Callias-type operator D : C∞(M,E) → C∞(M,E) is invertible

at infinity (or coercive at infinity). Namely, there exist a constant C > 0 and a compact set

K ⊂M such that

‖Du‖L2(M,E) ≥ C ‖u‖L2(M,E),

for all u ∈ C∞cc (M,E) with supp(u) ∩K = ∅.

Remark 3.4.2. Note that this property is independent of the boundary condition of D.

95

Proof. By Definition 0.1.2, for a fixed R > 0, one can find an R-essential support KR ⊂ M

for D, so that

‖Du‖2L2(M,E) = (Du,Du)L2(M,E) = (D∗Du, u)L2(M,E)

= (D2u, u)L2(M,E) +((Φ2 + i [D,Φ])u, u

)L2(M,E)

≥ ‖Du‖2L2(M,E) + R ‖u‖2

L2(M,E) ≥ R ‖u‖2L2(M,E)

for all u ∈ C∞cc (M,E) with support outside KR.

Recall that, for ∂M = ∅, a first-order essentially self-adjoint elliptic operator which is

invertible at infinity is Fredholm (cf. [2, Theorem 2.1]). If ∂M 6= ∅ is compact, an analogous

result (with elliptic boundary condition) is proven in [11, Theorem 8.5, Corollary 8.6]. We

now generalize the result of [11] to the case of non-compact boundary

Theorem 3.4.3. Let DB : domDB → L2(M,E) be a strongly Callias-type operator with

elliptic boundary condition. Then DB is a Fredholm operator.

Proof. A bounded linear operator T : X → Y between two Banach spaces has finite-

dimensional kernel and closed image if and only if every bounded sequence {xn} in X such

that {Txn} converges in Y has a convergent subsequence in X, cf. [48, Proposition 19.1.3].

We show below that both, D : domDB → L2(M,E) and (D∗)Bad : dom(D∗)Bad → L2(M,E)

satisfy this property.

We let {un} be a bounded sequence in domDB such that Dun → v ∈ L2(M,E) and want

to show that {un} has a convergent subsequence in domDB.

Recall that we assume that there is a neighborhood Zr = [0, r)×∂M ⊂M of the boundary

such that the restriction of D to Zr is product. For (t, y) ∈ Zr we set χ1(t, y) = χ(t) where χ

is the cut-off function defined in (3.2.14). We set χ1(x) ≡ 0 for x 6∈ Zr. Then χ1 is supported

on Z2r/3 and identically equal to 1 on Zr/3. We also note that dχ1 is uniformly bounded and

supported in Z2r/3.

96

Let the compact set K ⊂M and a constant C > 0 be as in Lemma 3.4.1. We choose two

more smooth cut-off functions χ2, χ3 : M → [0, 1] such that

• K ′ := supp(χ2) is compact and χ1 + χ2 ≡ 1 on K;

• χ1 + χ2 + χ3 ≡ 1 on M .

As a consequence, dχ3 is uniformly bounded and supp(dχ3) ⊂ Z2r/3 ∪K ′. We denote

κ = sup |dχ3|. (3.4.1)

Lemma 3.2.14 and the classical Rellich Embedding Theorem imply that, passing to a

subsequence, we can assume that the restrictions of un to Z2r/3 and to K ′ are L2-convergent.

Then in the inequality

‖un − um‖L2(M,E)

≤ ‖χ1 (un − um)‖L2(M,E) + ‖χ2 (un − um)‖L2(M,E) + ‖χ3 (un − um)‖L2(M,E)

≤ ‖un − um‖L2(Z2r/3,E) + ‖un − um‖L2(K′,E) + ‖χ3 (un − um)‖L2(M,E) (3.4.2)

the first two terms on the right hand side converge to 0 as n,m → ∞. To show that {un}

is a Cauchy sequence it remains to prove that the last summand converges to 0 as well. We

use Lemma 3.4.1 to get

‖χ3 (un − um)‖L2(M,E) ≤ C−1 ‖Dχ3 (un − um)‖L2(M,E)

≤ C−1 ‖c(dχ3)(un − um)‖L2(M,E) + C−1 ‖χ3 (Dun −Dum)‖L2(M,E)

≤ κC−1(‖un − um‖L2(Z2r/3,E) + ‖un − um‖L2(K′,E)

)+ C−1 ‖Dun −Dum‖L2(M,E),

where in the last inequality we used (3.4.1). Since Dun, un|Z2r/3and un|K′ are all convergent,

χ3(un− um)→ 0 in L2(M,E) as m,n→∞. Combining with (3.4.2) we conclude that {un}

is a Cauchy sequence and, hence, converges in L2(M,E).

Now both {un} and {Dun} are convergent in L2(M,E). Hence {un} converges in the

graph norm of D. Since B is an elliptic boundary condition, by Lemma 3.3.2, the H1D-norm

97

and graph norm of D are equivalent on domDB. So we proved that {un} is convergent in

domDB. Therefore DB has finite-dimensional kernel and closed image. Since D∗ is also a

strongly Callias-type operator, exactly the same arguments apply to (D∗)Bad and we get that

DB is Fredholm.

Definition 3.4.4. Let D be a strongly Callias-type operator on a complete Riemannian

manifold M which is product near the boundary. Let B ⊂ H1/2A (∂M,E∂M) be an elliptic

boundary condition for D. The integer

indDB := dim kerDB − dim ker(D∗)Bad ∈ Z (3.4.3)

is called the index of the boundary value problem DB.

It follows directly from (3.4.3) that

ind(D∗)Bad = − indDB. (3.4.4)

3.4.2 Dependence of the index on the boundary conditions

We say that two closed subspaces X1, X2 of a Hilbert space H are finite rank perturbations

of each other if there exists a finite dimensional subspace Y ⊂ H such that X2 ⊂ X1 ⊕ Y

and the quotient space (X1 ⊕ Y )/X2 has finite dimension. We define the relative index of

X1 and X2 by

[X1, X2] := dim (X1 ⊕ Y )/X2 − dimY. (3.4.5)

One easily sees that the relative index is independent of the choice of Y . We also note

that if X1 and X2 are finite rank perturbations of each other, then X1 and the orthogonal

complement X⊥2 of X2 form a Fredholm pair in the sense of [49, §IV.4.1] and the relative

index [X1, X2] is equal to the index of the Fredholm pair (X1, X⊥2 ) as it is defined in [49, §IV

4.1].

The following lemma follows immediately from the definition of the relative index.

98

Lemma 3.4.5. [X2, X1] = [X⊥1 , X⊥2 ] = − [X1, X2].

Proposition 3.4.6. Let D be a strongly Callias-type operator on M and let B1 and B2 be

elliptic boundary conditions for D. If B1, B2 ∈ H1/2A (∂M,E∂M) are finite rank perturbations

of each other, then

indDB1 − indDB2 = [B1, B2]. (3.4.6)

The proof of the proposition is a verbatim repetition of the proof of Theorem 8.14 of [11].

As an immediate consequence of Proposition 3.4.6 we obtain the following

Corollary 3.4.7. Let A be the restriction of D to ∂M and let B0 = H1/2(−∞,0)(A) and B1 =

H1/2(−∞,0](A) be the APS and the dual APS boundary conditions respectively, cf. Example 3.3.6.

Then

indDB1 = indDB0 + dim kerA. (3.4.7)

More generally, let B(a) = H1/2(−∞,a)(A) and B(b) = H

1/2(−∞,b)(A) be two generalized APS

boundary conditions with a < b. Then

indDB(b) = indDB(a) + dimL2[a,b)(A).

3.4.3 The splitting theorem

We use the notation of Example 3.3.8.

Theorem 3.4.8. Suppose M,D,M ′,D′ are as in Example 3.3.8. Let B0 be an elliptic bound-

ary condition on ∂M . Let B1 = H1/2(−∞,0)(A) and B2 = H

1/2[0,∞)(A) = H

1/2(−∞,0](−A) be the APS

and the dual APS boundary conditions along N1 and N2, respectively. Then D′B0⊕B1⊕B2is a

Fredholm operator and

indDB0 = indD′B0⊕B1⊕B2.

Proof. We assume that ∂M = ∅. The proof of the general case is exactly the same, but the

notation is more cumbersome. Since B1 ⊕ B2 is an elliptic boundary condition for D′, the

99

boundary value problem D′B1⊕B2is Fredholm. We need to show the index identity, which

now is

indD = indD′B1⊕B2. (3.4.8)

Let B denote the transmission condition on ∂M ′. Then, using the canonical pull-back of

sections from E to E ′, we have

domD = {u ∈ H1D′(M

′, E ′) : Ru ∈ B} = domD′B

and

indD = indD′B. (3.4.9)

We now proceed as in the proof of Theorem 8.17 of [11] with minor changes. The main

idea is to construct a deformation of the transmission boundary condition B into the APS

boundary condition B1 ⊕B2 and thus to show that indD′B = indD′B1⊕B2.

Recall that in Example 3.3.8, we express any element (u,u) of B as (u− + u+,u+ + u−),

where u− = PA(−∞,0)u and u+ = PA[0,∞)u. Note that u− ∈ B1 and u+ ∈ B2. For 0 ≤ s ≤ 1,

we define a family of boundary conditions

B1,s :={u− + (1− s)u+ : u ∈ H1/2

A (N1, EN1)}

;

B2,s :={u+ + (1− s)u− : u ∈ H1/2

−A(N2, EN2) ' H1/2A (N1, EN1)

},

and a family of isomorphisms

ks : B → B1,s ⊕B2,s, ks(u,u) := (u− + (1− s)u+,u+ + (1− s)u−).

Here k0 = id and k1 is an isomorphism from B to B1⊕B2. One can follow the arguments of

Lemma 3.3.9 to check that for each s ∈ [0, 1],

Bad1,s ⊕Bad

2,s ={

(v− + (1− s)v+,−v+ − (1− s)v−) ∈ H1/2

A] (N1, EN1)⊕H1/2

−A](N2, EN2)},

where v− ∈ H1/2(−∞,0](A]) and v+ ∈ H

1/2(0,∞)(A]). Thus B1,s ⊕ B2,s is an elliptic boundary

condition for all s ∈ [0, 1] and we get a family of Fredholm operators {D′B1,s⊕B2,s}0≤s≤1.

100

By definition,

(ks1 − ks2)(u,u) = (s2 − s1)(u+,u−).

Notice that ‖(u+,u−)‖H

1/2

A′ (∂M ′,E′)≤ ‖(u,u)‖

H1/2

A′ (∂M ′,E′). Hence, for s1, s2 ∈ [0, 1] with

|s1 − s2| < ε, the operator

ks1 − ks2 : B → H1/2A′ (∂M ′, E ′)

has a norm not greater than ε. This implies that {ks} is a continuous family of isomorphisms

from B to H1/2A′ (∂M ′, E ′). The following steps are basically from [11, Lemma 8.11, Theorem

8.12]. Roughly speaking, one can construct a continuous family of isomorphisms

Ks : domD′B → domD′B1,s⊕B2,s.

Then by composing D′B1,s⊕B2,sand Ks, one gets a continuous family of Fredholm operators

on the fixed domain domD′B. The index is constant. Since K1 is an isomorphism, we have

indD′B = indD′B1⊕B2. (3.4.10)

At last, (3.4.8) follows from (3.4.9) and (3.4.10). This completes the proof.

3.4.4 A vanishing theorem

As a first application of the splitting theorem 3.4.8 we prove the following vanishing result.

Corollary 3.4.9. Suppose that there exists R > 0 such that D has an empty R-essential

support. Let B0 = H1/2(−∞,0)(A) be the APS boundary condition, cf. Example 3.3.6. Then

indDB0 = 0. (3.4.11)

Proof. Since all our structures are product near ∂M and the R-essential support of D is

empty, the R-essential support of the restriction A := A− ic(τ)Φ of D to ∂M is also empty.

In particular, the operator A2 is strictly positive. It follows that 0 is not in the spectrum of

A.

101

First consider the case when M = [0,∞)×N is a cylinder and (3.2.4) holds everywhere

on M . In particular, this means that Φ(t, y) = Φ(0, y) for all t ∈ [0,∞) and all y ∈ N = ∂M .

To distinguish this case from the general case, we denote the Callias-type operator on the

cylinder by D′. Any u ∈ dom(D′B0) can be written as

u =∞∑j=1

aj(t) uj,

where uj is a unit eigensection of A with eigenvalue λj < 0. If D′u = 0 then aj(t) = cje−λjt

for all j. It follows that u 6∈ L2(M,E). In other words, there are no L2-sections in the kernel

of D′B0. Similarly, one proves that the kernel of (D′∗)Bad

0is trivial. Thus

indD′B0= 0. (3.4.12)

Let us return to the case of a general manifold M . Let

M :=((−∞, 0]× ∂M

)∪∂M M

be the extension of M by a cylinder. Then M is a complete manifold without boundary.

Since all our structures are product near ∂M they extend naturally to M . Let D be the

induced strongly Callias-type operator on M . It has an empty R-essential support. Hence,

D∗D > 0 and DD∗ > 0. It follows that

ind D = 0. (3.4.13)

Notice that the restriction of D to the cylinder is the operator D′ whose R-essential support

is empty and whose restriction to the boundary is −A. Let

B′0 := H1/2(−∞,0)(−A) = H

1/2(0,∞)(A)

denote the APS boundary condition for D′. Since A is invertible, B′0 coincides with the dual

APS boundary condition for D′. Hence, by the splitting theorem 3.4.8

ind D = indDB0 + indD′B′0 . (3.4.14)

102

The second summand on the right hand side of (3.4.14) vanishes by (3.4.12). The corollary

follows now from (3.4.13).

103

Chapter 4

The Atiyah–Patodi–Singer Index on

Manifolds with Non-Compact

Boundary: Odd-Dimensional Case

In Chapter 3, we established the theory of boundary value problems for strongly Callias-type

operators. In particular, the index of the Atiyah–Patodi–Singer boundary value problem

is well-defined. In this chapter, we study this APS index on a complete odd-dimensional

manifold M with non-compact boundary. Here we do not introduce any extra assumptions

on manifold (in particular, we do not assume that our manifold is of bounded geometry as

considered in [40]). The story of even-dimensional case is developed in next chapter.

4.1 The outline

4.1.1 An almost compact essential support

Let M be a complete Riemannian manifold with non-compact boundary ∂M and let D =

D + iΦ be a (ungraded) strongly Callias-type operator on M . In the theory of Callias-

type operators on a manifold without boundary the crucial notion is that of the essential

support — a compact set K ⊂ M such that the restriction of D∗D to M \ K is strictly

104

positive. For manifolds with boundary we want an analogous subset, but the one which

has the same boundary as M (so that we can keep the boundary conditions). Such a set

is necessarily non-compact. In Section 4.2, we introduce a class of non-compact manifolds,

called essentially cylindrical manifolds, which replaces the class of compact manifolds in

our study. An essentially cylindrical manifold is a manifold which outside of a compact set

looks like a cylinder [0, ε] × N ′, where N ′ is a non-compact manifold. The boundary of

an essentially cylindrical manifold is a disjoint union of two complete manifolds N0 and N1

which are isometric outside of a compact set.

We say that an essentially cylindrical manifold M1, which contains ∂M , is an almost

compact essential support of D if the restriction of D∗D to M \M1 is strictly positive and

the restriction of D to the cylinder [0, ε] × N ′ is a product, cf. Definition 4.2.3. We show

that every strongly Callias-type operator on M which is a product near ∂M has an almost

compact essential support.

The main result of Section 4.2 is that the index of the APS boundary value problem for

a strongly Callias-type operator D on a complete odd-dimensional manifold M is equal to

the index of the APS boundary value problem of the restriction of D to its almost compact

essential support M1, cf. Theorem 4.2.7.

4.1.2 Index on an essentially cylindrical manifold

In the previous section we reduced the study of the index of the APS boundary value problem

on an arbitrary complete odd-dimensional manifold to index on an essentially cylindrical

manifold. A systematic study of the latter is done in Section 4.3.

Let M be an essentially cylindrical manifold and let D be a strongly Callias-type operator

on M , whose restriction to the cylinder [0, ε]×N ′ is a product. Suppose ∂M = N0tN1 and

denote the restrictions of D to N0 and N1 by A0 and −A1 respectively (the sign convention

means that we think of N0 as the “left boundary“ and of N1 as the “right boundary” of M).

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Our main result here is that the index of the APS boundary value problem for D depends

only on the operators A0 and A1 and not on the interior of the manifold M and the restriction

of D to the interior of M , cf. Theorem 4.3.3. The odd-dimensionality of M is essential, since

the proof uses the Callias-type index theorem on complete manifolds without boundary.

4.1.3 The relative η-invariant

Suppose now that A0 and A1 are self-adjoint strongly Callias-type operators on complete

even-dimensional manifolds N0 and N1 respectively. An almost compact cobordism between

A0 and A1 is an essentially cylindrical manifold M with ∂M = N0 t N1 and a strongly

Callias-type operator D on M , whose restriction to the cylindrical part of M is a product

and such that the restrictions of D to N0 and N1 are equal to A0 and −A1 respectively.

We say that A0 and A1 are cobordant if there exists an almost compact cobordism between

them. Note that this means, in particular, that A0 and A1 are equal outside of a compact

set.

Let D be an almost compact cobordism between A0 and A1. Let B0 and B1 be the APS

boundary conditions for D at N0 an N1 respectively. Let indDB0⊕B1 denote the index of the

APS boundary value problem for D. We define the relative η-invariant by the formula

η(A1,A0) = 2 indDB0⊕B1 + dim kerA0 + dim kerA1.

It follows from the result of the previous section, that η(A1,A0) is independent of the choice

of an almost compact cobordism.

Notice the “shift of dimension” of the manifold compared to the theory of η-invariants

on compact manifolds. This is similar to the “shift of dimension” in the Callias-type in-

dex theorem: on compact manifolds the index of elliptic operators is interesting for even-

dimensional manifolds, while for Callias-type operators it is interesting for odd-dimensional

manifolds. Similarly, the theory of η-invariants on compact manifolds is more interesting

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on odd-dimensional manifolds, while our relative η-invariant is defined on even-dimensional

non-compact manifolds.

If M is a compact odd-dimensional manifold, then the Atiayh–Patodi–Singer index theo-

rem [5] implies that η(A1,A0) = η(A1)−η(A0) (recall that since the dimension of M is odd,

the integral term in the index formula vanishes). In general, for non-compact manifolds,

the individual η-invariants η(A1) and η(A0) might not be defined. However, we show that

η(A1,A0) in many respects behaves like it was a difference of two individual η-invariants. In

particular, we show, cf. Propositions 4.4.8-4.4.9, that

η(A1,A0) = − η(A0,A1), η(A2,A0) = η(A2,A1) + η(A1,A0).

In [40] Fox and Haskell studied the index of a boundary value problem on manifolds

of bounded geometry. They showed that under rather strong conditions on both M and

D (satisfied for natural operators on manifolds with conical or cylindrical ends), the heat

kernel e−t(DB)∗DB is of trace class and its trace has an asymptotic expansion similar to the

one on compact manifolds. In this case the η-invariant can be defined by the usual analytic

continuation of the η-function. We prove, cf. Proposition 4.4.6, that under the assumptions

of Fox and Haskell, our relative η-invariant satisfies η(A1,A0) = η(A1)− η(A0).

More generally, it is often the case that the individual η-functions η(s;A1) and η(s;A0) are

not defined, but their difference η(s;A1)− η(s;A0) is defined and regular at 0. Bunke, [33],

studied the case of the undeformed Dirac operator A and gave geometric conditions under

which Tr(A1e−tA2

1 − A0e−tA2

0) has a nice asymptotic expansion. In this case he defined the

relative η-function using the usual formula, and showed that it has a meromorphic extension

to the whole plane, which is regular at 0. He defined the relative η-invariant as the value of

the relative η-function at 0. There are also many examples of strongly Callias-type operators

for which the difference of heat kernels A1e−tA2

1 −A0e−tA2

0 is of trace class and the relative

η-function can be defined by the formula similar to [33]. We conjecture that in this situation

our relative η-invariant η(A1,A0) is equal to the value of the relative η-function at 0.

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4.1.4 The spectral flow

Atiyah, Patodi and Singer, [6], introduced a notion of spectral flow sf(A) of a smooth family

A := {As}0≤s≤1 of self-adjoint differential operators on closed manifolds as the integer that

counts the net number of eigenvalues that change sign when s changes from 0 to 1. They

showed that the spectral flow computes the variation of the η-invariant η(A1)− η(A0).

In Section 4.5 we consider a family of self-adjoint strongly Callias-type operators A =

{As}0≤s≤1 on a complete even-dimensional Riemannian manifold. We assume that there is

a compact set K ⊂ M such that the restriction of As to M \K is independent of s. Then

all As are cobordant in the sense of Section 4.1.3. Since the spectrum of As is discrete for

all s, the spectral flow can be defined in more or less usual way. We show, Theorem 4.5.9,

that

η(A1,A0) = 2 sf(A).

Moreover, if A0 is another self-adjoint strongly Callias-type operator which is cobordant to

A0 (and, hence, to all As), then

η(A1,A0) − η(A0,A0) = 2 sf(A).

4.2 Reduction to an essentially cylindrical manifold

In this section we reduce the computation of the index of an APS boundary value problem

to a computation on a simpler manifold which we call essentially cylindrical.

Definition 4.2.1. An essentially cylindrical manifold M is a complete Riemannian manifold

whose boundary is a disjoint union of two components, ∂M = N0 tN1, such that

(i) there exist a compact set K ⊂ M , an open manifold N , and an isometry M \ K '

[0, ε]×N ;

(ii) under the above isometry N0 \K = {0} ×N and N1 \K = {ε} ×N .

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Remark 4.2.2. Essentially cylindrical manifolds should not be confused with manifolds with

cylindrical ends. In a manifold M with cylindrical ends there is a compact set K such that

M \K = [0,∞)×N is a cylinder with infinite axis [0,∞) and compact base N . As opposed

to it, in an essentially cylindrical manifold, M \K is a cylinder with compact axis [0, ε] and

non-compact base N .

4.2.1 Almost compact essential support

We now return to the setting of Section 3.2. In particular, M is a complete Riemannian

manifold with non-compact boundary ∂M and there is a fixed isometry between a neigh-

borhood of ∂M and the product Zr = [0, r) × ∂M , cf. (3.2.1); D = D + iΦ is a strongly

Callias-type operator (cf. Definition 0.1.2) whose restriction to Zr is a product (3.2.4).

Definition 4.2.3. An almost compact essential support of D is a smooth submanifold M1 ⊂

M with smooth boundary, which contains ∂M and such that

(i) M1 contains an essential support for D, cf. Definition 0.1.2;

(ii) there exist a compact set K ⊂M and ε ∈ (0, r) such that

M1 \K = (∂M \K)× [0, ε] ⊂ Zr. (4.2.1)

Note that any almost compact essential support is an essentially cylindrical manifold, one

component of whose boundary is ∂M and A has an empty essential support on the other

component of the boundary. Also the restriction of D to the subset (4.2.1) is given by (3.2.4).

Lemma 4.2.4. For every strongly Callias-type operator which is product on Zr there exists

an almost compact essential support.

Proof. Fix R > 0 and let KR be a compact essential support for D. The union

M ′ :=([0, r/2]× ∂M

)∪KR

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satisfies all the properties of an almost compact essential support, except that its boundary

is not necessarily smooth. For small enough δ > 0 the δ-neighborhood

Mδ :={x ∈M : dist(x,M ′) ≤ δ

}of M ′ has a smooth boundary and is an almost compact essential support for D.

4.2.2 The index on an almost compact essential support

Suppose M1 ⊂M is an almost compact essential support for D and let N1 ⊂M be such that

∂M1 = ∂MtN1. The restriction of D to a neighborhood of N1 need not be product. Since in

this chapter we only consider boundary value problems for operators which are product near

the boundary, we first deform D to a product form. Note that if K is as in Definition 4.2.3

then D is product in a neighborhood of N1 \K. It follows that we only need to deform D

in a relatively compact neighborhood of N1 ∩K. More precisely let ε be as in (4.2.1). We

choose δ ∈ (0, ε) and a tubular neighborhood U ⊂M of N1 such that

U \K = (ε− δ, ε+ δ)× (N1 \K) ⊂ Zr. (4.2.2)

We now identify U with the product (ε−δ, ε+δ)×N1 in a way compatible with (4.2.2). The

next lemma shows that one can find a strongly Callias-type operator D′ which is a product

near N1 and differs from D only on a compact set.

Definition 4.2.5. Fix a new Riemannian metric on M and a new Hermitian metric on E

which differ from the original metrics only on a compact set K ′ ⊂ M . Let c′ : T ∗M →

End(E) and let ∇E′ be a Clifford multiplication and a Clifford connection compatible with

the new metrics, which also differ from c and ∇E only on K ′. Let D′ be the Dirac operator

defined by c′ and ∇E′ . Finally, let Φ′ ∈ End(E) be a new Callias potential which is equal

to Φ on M \ K ′. In this situation we say that the operator D′ := D′ + iΦ′ is a compact

perturbation of D.

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Clearly, if D′ is a compact perturbation of D which is equal to D near ∂M , then every

elliptic boundary condition B for D is also elliptic for D′. Then the stability of the index

implies that

indDB = indD′B. (4.2.3)

Lemma 4.2.6. In the situation of Subsection 4.2.2 there exists a compact perturbation D′

of D which is product near ∂M1 and such that there is a compact essential support of D′

contained in M1.

Proof. By Proposition 5.4 of [27] there exists a smooth deformation (ct,∇Et ) of the Clifford

multiplication and the Clifford connection such that

(i) for t = 0 it is equal to (c,∇E);

(ii) for t > 0 it is a product near N1;

(iii) for all t its restriction to M \U is independent of t (and, hence, coincides with (c,∇E)).

Moreover, since all our structures are product near N1 \K, the construction of this deforma-

tion in Appendix A of [27] provides a deformation which is independent of t on M \ (U ∩K).

Thus for all t > 0 the Dirac operator Dt defined by (ct,∇Et ) is a compact perturbation of D.

Let Φt(x) be a smooth deformation of Φ(x) which coincides with Φ at t = 0, is independent

of t for x 6∈ U ∩K, and is product near N1 for all t > 0. Then Dt := Dt + iΦt is a compact

perturbation of D for all t ≥ 0.

Fix R > 0 such that there is an R-essential support of D which is contained in M1. Then

there exists a compact set KR ⊂M1 such that outside of KR the estimate (0.1.3) holds. Since

all our deformations are smooth and compactly supported Φ2t (x) − |[Dt,Φt](x)| ≥ R/2 for

all small enough t > 0. The assertion of the lemma holds now with R′ = R/2 and D′ = Dt

with t > 0 small.

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4.2.3 Reduction of the index problem to an almost compact es-

sential support

Let M1 ⊂ M be an almost compact essential support of D. Let D′ be as in the previous

subsection. Let A be the restriction of D to ∂M . It is also the restriction of D′ (since D′ = D

near ∂M). We denote by −A1 the restriction of D′ to N1. Thus near N1 the operator D′ has

the form c(τ)(∂t−A1). The sign convention is related to the fact that it is often convenient

to view N0 = ∂M as the “left” boundary of M1 and N1 as the “right” boundary. Then

one identifies a neighborhood of N1 in M1 with the product (−r, 0] × N1. With respect

to this identification the restriction of D′ to this neighborhood becomes c(dt)(∂t + A1). In

particular, on the cylindrical part M1 \K we have A1 = A.

Theorem 4.2.7. Suppose M1 ⊂ M is an almost compact essential support of D and let

∂M1 = ∂M tN1. Let D′ be a compact perturbation of D which is product near N1 and such

that there is a compact essential support for D′ which is contained in M1. Let B0 be any

elliptic boundary condition for D and let

B1 = H1/2(−∞,0)(−A1) = H

1/2(0,∞)(A1)

be the APS boundary condition for the restriction of D′ to a neighborhood of N1. Then

B0 ⊕B1 is an elliptic boundary condition for the restriction D′′ := D′|M1 of D′ to M1 and

indDB0 = indD′′B0⊕B1. (4.2.4)

Proof. Let D′′′ denote the restriction of D′ to M\M1. This is a strongly Callias-type operator

with an empty essential support. Notice that its restriction to N1 is equal to A1. Thus the

APS boundary condition for D′′′ is B2 = H1/2(−∞,0)(A1). Since A1 is invertible, B2 coincides

with the dual APS boundary condition for D′. Hence, by the Splitting Theorem 3.4.8,

indD′B0= indD′′B0⊕B1

+ indD′′′B2.

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The last summand on the right hand side of this equality vanishes by Corollary 3.4.9. The

theorem follows now from (4.2.3).

4.3 The index of operators on essentially cylindrical

manifolds

In the previous section we reduced the computation of the index of D to a computation of the

index of the restriction of D to its almost compact essential support (which is an essentially

cylindrical manifold). In this section we consider a strongly Callias-type operator D on an

essentially cylindrical manifold M (these data might or might not come as a restriction of

another operator to its almost compact essential support. In particular, we don’t assume that

the restriction of D to N1 is invertible). From this point on we assume that the dimension

of M is odd.

Let A0 and −A1 be the restrictions of D to N0 and N1 respectively. The main result of

this section is that the index of the APS boundary value problem for D depends only on

A0 and A1. Thus it is an invariant of the boundary. In the next section we will discuss the

properties of this invariant.

4.3.1 Compatible essentially cylindrical manifolds

Let M be an essentially cylindrical manifold and let ∂M = N0 t N1. As usual, we identify

a tubular neighborhood of ∂M with the product

Zr :=(N0 × [0, r)

)t(N1 × [0, r)

)⊂ M.

Definition 4.3.1. We say that another essentially cylindrical manifold M ′ is compatible with

M if there is a fixed isometry between Zr and a neighborhood Z ′r ⊂ M ′ of the boundary of

M ′.

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Note that if M and M ′ are compatible then their boundaries are isometric.

4.3.2 Compatible strongly Callias-type operators

Let M and M ′ be compatible essentially cylindrical manifolds and let Zr and Z ′r be as

above. Let E → M be a Dirac bundle over M and let D : C∞(M,E) → C∞(M,E) be

a strongly Callias-type operator whose restriction to Zr is product and such that M is an

almost compact essential support of D. This means that there is a compact set K ⊂ M

such that M \K = [0, ε]×N and the restriction of D to M \K is product (i.e. is given by

(3.2.4)). Let E ′ → M ′ be a Dirac bundle over M ′ and let D′ : C∞(M ′, E ′) → C∞(M ′, E ′)

be a strongly Callias-type operator, whose restriction to Z ′r is product and such that M ′ is

an almost compact essential support of D′.

Definition 4.3.2. In the situation discussed above we say that D and D′ are compatible if

there is an isomorphism E|Zr ' E ′|Z′r which identifies the restriction of D to Zr with the

restriction of D′ to Z ′r.

Let A0 and −A1 be the restrictions of D to N0 and N1 respectively. Let B0 = H1/2(−∞,0)(A0)

and B1 = H1/2(−∞,0)(−A1) = H

1/2(0,∞)(A1) be the APS boundary conditions for D at N0 and

N1 respectively. Since D and D′ are equal near the boundary, B0 and B1 are also elliptic

boundary conditions for D′.

Theorem 4.3.3. Suppose D is a strongly Callias-type operator on an essentially cylindrical

odd-dimensional manifold M such that M is an almost compact essential support of D.

Suppose that the operator D′ is compatible with D. Let ∂M = N0 t N1 and let B0 =

H1/2(−∞,0)(A0) and B1 = H

1/2(−∞,0)(−A1) = H

1/2(0,∞)(A1) be the APS boundary conditions for D

(and, hence, for D′) at N0 and N1 respectively. Then

indDB0⊕B1 = indD′B0⊕B1. (4.3.1)

The proof of this theorem occupies Subsections 4.3.3–4.3.5.

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4.3.3 Gluing together M and M ′

Let −M ′ denote another copy of manifold M ′. Even though we don’t assume that our man-

ifolds are oriented, it is useful to think of −M ′ as manifold M with the opposite orientation.

We identify the boundary of −M ′ with the product

−Z ′r :=(N0 × (−r, 0]

)t(N1 × (−r, 0]

)and consider the union

M := M ∪N0tN1 (−M ′).

Then Z(−r,r) := Zr ∪ (−Z ′r) is a subset of M identified with the product

(N0 × (−r, r)

)t(N1 × (−r, r)

).

We note that M is a complete Riemannian manifold without boundary.

4.3.4 Gluing together D and (D′)∗

Let E∂M denote the restriction of E to ∂M . The product structure on E|Zr gives an iso-

morphism ψ : E|Zr → [0, r) × E∂M . Recall that we identified Zr with Z ′r and fixed an

isomorphism between the restrictions of E to Zr and E ′ to Z ′r. By a slight abuse of notation

we use this isomorphism to view ψ also as an isomorphism E ′|Z′r → [0, r)× E∂M .

Let E → M be the vector bundle over M obtained by gluing E and E ′ using the isomor-

phism c(τ) : E|∂M → E ′|∂M ′ . This means that we fix isomorphisms

φ : E|M → E, φ′ : E|M ′ → E ′, (4.3.2)

so that

ψ ◦ φ ◦ ψ−1 = id : [0, r)× E∂M → [0, r)× E∂M ,

ψ ◦ φ′ ◦ ψ−1 = 1× c(τ) : [0, r)× E∂M → [0, r)× E∂M .

We denote by c′ : T ∗M ′ → End(E ′) the Clifford multiplication on E ′ and set c′′(ξ) :=

−c′(ξ). We think of c′′ as the Clifford multiplication of T ∗(−M ′) on E ′ (since the dimension

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of M ′ is odd, changing the sign of the Clifford multiplication corresponds to changing the

orientation on M ′). Then E is a Dirac bundle over M with the Clifford multiplication

c(ξ) :=

c(ξ), ξ ∈ T ∗M ;

c′′(ξ) = −c′(ξ), ξ ∈ T ∗M ′.

(4.3.3)

One readily checks that (4.3.3) defines a smooth Clifford multiplication on E. Let D :

C∞(M, E)→ C∞(M, E) be the Dirac operator. Then the isomorphism φ of (4.3.2) identifies

the restriction of D with D, the isomorphism φ′ identifies the restriction of D with −D′, and

isomorphism ψ ◦ φ′ ◦ ψ−1 identifies the restriction of D to −Z ′r with

D|Z′r = −c′(τ) ◦D′Z′r ◦ c′(τ)−1.

Let Φ′ denote the Callias potential of D′, so that D′ = D′ + iΦ′. Consider the bundle

map Φ ∈ End(E) whose restriction to M is equal to Φ and whose restriction to M ′ is equal

to Φ′. We summarize the constructions presented in this subsection in the following

Lemma 4.3.4. The operator D := D + iΦ is a strongly Callias-type operator on M , whose

restriction to M is equal to D and whose restriction to M ′ is equal to −D′ + iΦ′ = −(D′)∗.

The operator D is a strongly Callias-type operator on a complete Riemannian manifold

without boundary. Hence, [1], it is Fredholm.

Lemma 4.3.5. ind D = 0.

Proof. Since M is a union of two essentially cylindrical manifolds, there exists a compact

essential support K ⊂ M of D such that M \ K is of the form S1×N . We can choose K to

be large enough so that the restriction of D to S1×N is a product. We can also assume that

K has a smooth boundary Σ = S1 × L. Then the Callias index theorem [3, Theorem 1.5]

states that the index of D is equal to the index of a certain operator ∂ on Σ. Since all our

structures are product on M \ K, the operator ∂ is also a product on Σ = S1 × L. Thus it

116

has a form

∂ = γ(∂t + A

),

where A is an operator on L. The kernel and cokernel of ∂ can be computed by separation

of variables and both are easily seen to be isomorphic to the kernel of A. Thus the kernel

and the cokernel are isomorphic and ind ∂ = 0.

4.3.5 Proof of Theorem 4.3.3

Recall that we denote by B0 and B1 the APS boundary conditions for D = D|M . Let D′′

denote the restriction of D to −M ′ = M \M and let B′0 and B′1 be the dual APS boundary

conditions for D′′ at N0 and N1 respectively. By the Splitting Theorem 3.4.8

ind D = indDB0⊕B1 + indD′′B′0⊕B′1 .

Since, by Lemma 4.3.5, ind D = 0, we obtain

indDB0⊕B1 = − indD′′B′0⊕B′1 . (4.3.4)

By Lemma 4.3.4, D′′ = −(D′)∗. Thus, by Remark 3.3.7, B′0 ⊕ B′1 is equal to the adjoint

boundary conditions for −D′. Hence, by (3.4.4),

indD′′B′0⊕B′1 = ind(−D′)∗B′0⊕B′1 = − indD′B0⊕B1.

Combining this equality with (4.3.4) we obtain (4.3.1). �

4.4 The relative η-invariant

In the previous section we proved that the index of the APS boundary value problem DB0⊕B1

for a strongly Callias-type operator on an odd-dimensional essentially cylindrical manifold

depends only on the restriction of D to the boundary, i.e. on the operators A0 and −A1.

In this section we use this index to define the relative η-invariant η(A1,A0) and show that

117

it has properties similar to the difference of η-invariants η(A1) − η(A0) of operators on

compact manifolds. For special cases, [40], when the index can be computed using heat kernel

asymptotics, we show that η(A1,A0) is indeed equal to the difference of the η-invariants of

A1 and A0. In the next section we discuss the connection between the relative η-invariant

and the spectral flow.

4.4.1 Almost compact cobordisms

Let N0 and N1 be two complete even-dimensional Riemannian manifolds and let A0 and A1

be self-adjoint strongly Callias-type operators on N0 and N1, respectively.

Definition 4.4.1. An almost compact cobordism betweenA0 andA1 is given by an essentially

cylindrical manifold M with ∂M = N0 t N1 and a strongly Callias-type operator D on M

such that

(i) M is an almost compact essential support of D;

(ii) D is product near ∂M ;

(iii) The restriction of D to N0 is equal to A0 and the restriction of D to N1 is equal to

−A1.

If there exists an almost compact cobordism between A0 and A1 we say that operator A0 is

cobordant to operator A1.

Lemma 4.4.2. If A0 is cobordant to A1 then A1 is cobordant to A0.

Proof. Let −M denote the manifold M with the opposite orientation and let M := M ∪∂M

(−M) denote the double of M . Let D be an almost compact cobordism between A0 and

A1. Using the construction of Section 4.3.4 (with D′ = D) we obtain a strongly Callias-type

operator D on M whose restriction to M is isometric to D. Let D′′ denote the restriction

of D to −M = M \M . Then the restriction of D′′ to N1 is equal to A1 and the restriction

118

of D′′ to N0 is equal to −A0. Hence, D′′ is an almost compact cobordism between A1 and

A0.

Lemma 4.4.3. Let A0,A1 and A2 be self-adjoint strongly Callias-type operators on even-

dimensional complete Riemannian manifolds N0, N1 and N2 respectively. Suppose A0 is

cobordant to A1 and A1 is cobordant to A2. Then A0 is cobordant to A2.

Proof. Let M1 and M2 be essentially cylindrical manifolds such that ∂M1 = N0 t N1 and

∂M2 = N1 t N2. Let D1 be an operator on M1 which is an almost compact cobordism

between A0 and A1. Let D2 be an operator on M2 which is an almost compact cobordism

between A1 and A2. Then the operator D3 on M1 ∪N1 M2 whose restriction to Mj (j = 1, 2)

is equal to Dj is an almost compact cobordism between A0 and A2.

If follows from Lemmas 4.4.2 and 4.4.3 that cobordism is an equivalence relation on the

set of self-adjoint strongly Callias-type operators.

Definition 4.4.4. Suppose A0 and A1 are cobordant self-adjoint strongly Callias-type op-

erators and let D be an almost compact cobordism between them. Let B0 = H1/2(−∞,0)(A0)

and B1 = H1/2(−∞,0)(−A1) be the APS boundary conditions for D. The relative η-invariant is

defined as

η(A1,A0) = 2 indDB0⊕B1 + dim kerA0 + dim kerA1. (4.4.1)

Theorem 4.3.3 implies that η(A1,A0) is independent of the choice of the cobordism D.

Remark 4.4.5. Sometimes it is convenient to use the dual APS boundary conditions B0 =

H1/2(−∞,0](A0) and B2 = H

1/2(−∞,0](−A1) instead of B0 and B1. It follows from Corollary 3.4.7

that the relative η-invariant can be written as

η(A1,A0) = 2 indDB0⊕B1− dim kerA0 − dim kerA1. (4.4.2)

119

4.4.2 The case when the heat kernel has an asymptotic expansion

In [40], Fox and Haskell studied the index of a boundary value problem on manifolds of

bounded geometry. They showed that under certain conditions (satisfied for natural opera-

tors on manifolds with conical or cylindrical ends) on M and D, the heat kernel e−t(DB)∗DB

is of trace class and its trace has an asymptotic expansion similar to the one on compact

manifolds. In this case the η-function, defined by a usual formula

η(s;A) :=∑

λ∈spec(A)

sign(λ) |λ|s, Re s� 0,

is an analytic function of s, which has a meromorphic continuation to the whole complex

plane and is regular at 0. So one can define the η-invariant of A by η(A) = η(0;A).

Proposition 4.4.6. Suppose now that D is an operator on an essentially cylindrical manifold

M which satisfies the conditions of [40]. We also assume that D is product near ∂M =

N0 tN1 and that M is an almost compact essential support for D. Let A0 and −A1 be the

restrictions of D to N0 and N1 respectively. Let η(Aj) (j = 0, 1) be the η-invariant of Aj.

Then

η(A1,A0) = η(A1) − η(A0). (4.4.3)

Proof. Theorem 9.6 of [40] establishes an index theorem for the APS boundary value problem

satisfying conditions discussed above. This theorem is completely analogous to the classical

APS index theorem [5]. In [40] only the case of even-dimensional manifolds is discussed.

However, exactly the same (but somewhat simpler) arguments give an index theorem on

odd-dimensional manifolds as well. In the odd-dimensional case the integral term in the

index formula vanishes identically. Thus, applied to our situation, Theorem 9.6 of [40] gives

indDB0⊕B1 = − dim kerA0 + η(A0)

2− dim kerA1 + η(−A1)

2.

Since η(−A1) = −η(A1) equation (4.4.3) follows now from the definition (4.4.1) of the

relative η-invariant.

120

More generally, Bunke, [33], considered the situation when Aje−tA2j (j = 0, 1) are not of

trace class but their difference A1e−tA2

1 − A0e−tA2

0 is of trace class and its trace has a nice

asymptotic expansion. In this situation one can define the relative η-function by the usual

formula

η(s;A1,A0) :=1

Γ((s+ 1)/2

) ∫ ∞0

ts−1

2 Tr(A1e

−tA21 −A0e

−tA20)dt. (4.4.4)

(See [54] for even more general situation when the relative η-function can be defined.)

Bunke only considered the undeformed Dirac operator A and gave a geometric condition

under which Tr(A1e−tA2

1 −A0e−tA2

0) has a nice asymptotic expansion and the above integral

gives a meromorphic function regular at 0. One can also consider the cases when the heat

kernels of the Callias-type operators Aj are such that Tr(A1e−tA2

1 − A0e−tA2

0) has a nice

asymptotic expansion and the relative η-function can be defined using (4.4.4).

Conjecture 4.4.7. If the relative η-function (4.4.4) is defined, analytic and regular at 0,

then

η(A1,A0) = η(0;A1,A0). (4.4.5)

4.4.3 Basic properties of the relative η-invariant

Proposition 4.4.6 shows that under certain conditions the η-invariants of A0 and A1 are

defined and η(A1,A0) is their difference. We now show that in general case, when η(A0)

and η(A1) do not necessarily exist, η(A1,A0) behaves like it was a difference of an invariant

of N1 and an invariant of N0.

Proposition 4.4.8 (Antisymmetry). Suppose A0 and A1 are cobordant self-adjoint strongly

Callias-type operators. Then

η(A0,A1) = − η(A1,A0). (4.4.6)

Proof. Let D be an almost compact cobordism between A0 and A1 and let D and D′′ be

as in the proof of Lemma 4.4.2. Then D is a strongly Callias-type operator on a complete

121

Riemannian manifold M without boundary and D′′ is an almost compact cobordism between

A1 and A0.

Let

B′0 = H1/2[0,∞)(A0) = H

1/2(−∞,0](−A0);

B′1 = H1/2[0,∞)(−A1) = H

1/2(−∞,0](A1).

be the dual APS boundary conditions for D′′. It is shown in Section 4.3.5 that B′0 ⊕ B′1 is

an elliptic boundary condition for D′′ and, by (4.3.4),

indD′′B′0⊕B′1 = − indDB0⊕B1 . (4.4.7)

Since D′′ is an almost compact cobordism between A1 and A0 we conclude from (4.4.2)

that

η(A0,A1) = 2 indD′′B′0⊕B′1 − dim kerA0 − dim kerA1. (4.4.8)

Combining (4.4.8) and (4.4.7) we obtain (4.4.6).

Note that (4.4.6) implies that

η(A,A) = 0

for every self-adjoint strongly Callias-type operator A.

Proposition 4.4.9 (The cocycle condition). Let A0,A1 and A2 be self-adjoint strongly

Callias-type operators which are cobordant to each other. Then

η(A2,A0) = η(A2,A1) + η(A1,A0). (4.4.9)

Proof. The lemma follows from the Splitting Theorem 3.4.8 applied to the operator D3

constructed in the proof of Lemma 4.4.3.

4.5 The spectral flow

Atiyah, Patodi and Singer, [6], introduced a notion of spectral flow sf(A) of a continuous

family A := {As}0≤s≤1 of self-adjoint differential operators on a closed manifold. They

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showed that the spectral flow computes the variation of the η-invariant η(A1) − η(A0). In

this section we consider a family of self-adjoint strongly Callias-type operators A = {As}0≤s≤1

on a complete even-dimensional Riemannian manifold and show that for any operator A0

cobordant to A0 we have η(A1,A0)− η(A0,A0) = 2 sf(A).

4.5.1 A family of boundary operators

Let EN → N be a Dirac bundle over a complete even-dimensional Riemannian manifold N .

Let A = {As}0≤s≤1 be a family of self-adjoint strongly Callias-type operators

As = As + Ψs : C∞(N,EN) → C∞(N,EN).

Definition 4.5.1. The family A = {As}0≤s≤1 is called almost constant if there exists a

compact set K ⊂ N such that the restriction of As to N \K is independent of s.

Since dimN = 2p is even, there is a natural grading operator Γ : EN → EN , with Γ2 = 1,

cf. [13, Lemma 3.17]. If e1, . . . , e2p is an orthonormal basis of TN ' T ∗N , then

Γ := ip c(e1) · · · c(e2p).

Remark 4.5.2. The operators As anticommute with Γ. Condition (i) of Definition 0.1.6

implies that Ψ anticommutes with c(ej) (j = 1, . . . , 2p) and, hence, commutes with Γ. So

the operators As neither commute, nor anticommute with Γ. This explains why, even though

the dimension of N is even, the spectrum of the operators As is not symmetric about the

origin and the spectral flow of the family A is, in general, not trivial.

We set M := [0, 1]×N , E := [0, 1]×EN and denote by t the coordinate along [0, 1]. Then

E →M is naturally a Dirac bundle over M with c(dt) := iΓ.

Definition 4.5.3. The family A = {As}0≤s≤1 is called smooth if

D := c(dt)(∂t +At

): C∞(M,E)→ C∞(M,E)

is a smooth differential operator on M .

123

Fix a smooth non-decreasing function κ : [0, 1] → [0, 1] such that κ(t) = 0 for t ≤ 1/3

and κ(t) = 1 for t ≥ 2/3 and consider the operator

D := c(dt)(∂t +Aκ(t)

): C∞(M,E)→ C∞(M,E). (4.5.1)

Then D is product near ∂M . If A is a smooth almost constant family of self-adjoint strongly

Callias-type operators then (4.5.1) is a strongly Callias-type operator for which M is an

almost compact essential support. Hence it is a non-compact cobordism (cf. Definition 4.4.1)

between A0 and A1.

4.5.2 The spectral section

If A = {As}0≤s≤1 is a smooth almost constant family of self-adjoint strongly Callias-type

operators then it satisfies the conditions of the Kato Selection Theorem [49, Theorems II.5.4

and II.6.8], [55, Theorem 3.2]. Thus there is a family of eigenvalues λj(s) (j ∈ Z) which

depend continuously on s. We order the eigenvalues so that λj(0) ≤ λj+1(0) for all j ∈ Z

and λj(0) ≤ 0 for j ≤ 0 while λj(0) > 0 for j > 0.

Atiyah, Patodi and Singer [6] defined the spectral flow sf(A) for a family of operators

satisfying the conditions of the Kato Selection Theorem as an integer that counts the net

number of eigenvalues that change sign when s changes from 0 to 1. Several other equivalent

definitions of the spectral flow based on different assumptions on the family A exist in the

literature. For our purposes the most convenient is the Dai and Zhang’s definition [38] which

is based on the notion of spectral section introduced by Melrose and Piazza [53].

Definition 4.5.4. A spectral section for A is a continuous family P = {P s}0≤s≤1 of self-

adjoint projections such that there exists a constant R > 0 such that for all 0 ≤ s ≤ 1, if

Asu = λu then

P su =

0, if λ < −R;

u, if λ > R.

124

If A satisfies the conditions of the Kato Selection Theorem, then the arguments of the

proof of [53, Proposition 1] show that A admits a spectral section.

Remark 4.5.5. Booss-Bavnbek, Lesch, and Phillips [16] defined the spectral flow for a family

of unbounded operators in an abstract Hilbert space. Their conditions on the family are

much weaker than those of the Kato Selection Theorem. In particular, they showed that a

family of elliptic differential operators on a closed manifold satisfies their conditions if all the

coefficients of the differential operators depend continuously on s. It would be interesting

to find a good practical condition under which a family of self-adjoint strongly Callias-type

operators satisfies the conditions of [16].

4.5.3 The spectral flow

Let P = {P s} be a spectral section for A. Set Bs := kerP s. Let Bs0 := H

1/2(−∞,0)(As) denote

the APS boundary condition defined by the boundary operator As. Recall that the relative

index of subspaces was defined in Section 3.4.2. Since the spectrum of As is discrete, it

follows immediately from the definition of the spectral section that for every s ∈ [0, 1] the

space Bs is a finite rank perturbation of Bs0, cf. Section 3.4.2. Recall that the relative index

[Bs, Bs0] was defined in (3.4.5). Following Dai and Zhang [38] we give the following definition.

Definition 4.5.6. Let A = {As}0≤s≤1 be a smooth almost constant family of self-adjoint

strongly Callias-type operators which admits a spectral section P = {P s}0≤s≤1. Assume

that the operators A0 and A1 are invertible. Let Bs := kerP s and Bs0 := H

1/2(−∞,0)(As). The

spectral flow sf(A) of the family A is defined by the formula

sf(A) := [B1, B10 ] − [B0, B0

0 ]. (4.5.2)

By Theorem 1.4 of [38] the spectral flow is independent of the choice of the spectral

section P and computes the net number of eigenvalues that change sign when s changes from

0 to 1.

125

Remark 4.5.7. The relative index [Bs, Bs0] can also be computed in terms of the orthogonal

projections P s and P s0 with kernels Bs and Bs

0 respectively. Then P s0 defines a Fredholm

operator P s0 : imP s → imP s

0 . Dai and Zhang denote the index of this operator by [P s0 −P s]

and use it in their formula for spectral flow. One easily checks that [P s0 − P s] = [Bs, Bs

0].

Lemma 4.5.8. Let −A denote the family {−As}0≤s≤1. Then

sf(−A) = − sf(A). (4.5.3)

Proof. The lemma is an immediate consequence of Lemma 3.4.5.

4.5.4 Deformation of the relative η-invariant

Let A = {As}0≤s≤1 be a smooth almost constant family of self-adjoint strongly Callias-type

operators on a complete even-dimensional Riemannian manifold N1. Let A0 be another self-

adjoint strongly Callias-type operator, which is cobordant to A0. In Section 4.5.1 we showed

that A0 is cobordant to As for all s ∈ [0, 1]. Hence, by Lemma 4.4.3, A0 is cobordant to A1.

In this situation we say the A0 is cobordant to the family A. The following theorem is the

main result of this section.

Theorem 4.5.9. Suppose A ={As : C∞(N1, E1)→ C∞(N1, E1)

}0≤s≤1

is a smooth almost

constant family of self-adjoint strongly Callias-type operators on a complete Riemannian

manifold N1 such that A0 and A1 are invertible. Then

η(A1,A0) = 2 sf(A). (4.5.4)

If A0 : C∞(N0, E0) → C∞(N0, E0) is an invertible self-adjoint strongly Callias-type op-

erator on a complete even-dimensional Riemannian manifold N0 which is cobordant to the

family A then

η(A1,A0) − η(A0,A0) = 2 sf(A). (4.5.5)

126

Proof. First, we prove (4.5.5). Let M be an essentially cylindrical manifold whose boundary

is the disjoint union of N0 and N1. Let D : C∞(M,E)→ C∞(M,E) be an almost compact

cobordism between A0 and A0.

Consider the “extension of M by a cylinder”

M ′ := M ∪N1

([0, 1]×N1

).

and let E ′ → M ′ be the bundle over M ′ whose restriction to M is equal to E and whose

restriction to the cylinder [0, 1]×N1 is equal to [0, 1]× E1.

We fix a smooth function ρ : [0, 1]× [0, 1]→ [0, 1] such that for each r ∈ [0, 1]

• the function s 7→ ρ(r, s) is non-decreasing.

• ρ(r, s) = 0 for s ≤ 1/3 and ρ(r, s) = r for s ≥ 2/3.

Consider the family of strongly Callias-type operatorsDr : C∞(M ′, E ′)→ C∞(M ′, E ′) whose

restriction to M is equal to D and whose restriction to [0, 1]×N1 is given by

Dr := c(dt)(∂t +Aρ(r,t)

).

Then Dr is an almost compact cobordism between A0 and Ar. In particular, the restriction

of Dr to N1 is equal to −Ar.

Recall that we denote by −A the family {−As}0≤s≤1. Let P = {P s} be a spectral section

for −A. Then for each r ∈ [0, 1] the space Br := kerP r is an elliptic boundary condition for

Dr at {1}×N1. Let B0 := H1/2(−∞,0)(A0) be the APS boundary condition for Dr at N0. Then

B0 ⊕Br is an elliptic boundary condition for Dr.

Recall that the domain domDrB0⊕Br consists of sections u whose restriction to ∂M ′ =

N0 tN1 lies in B0 ⊕Br.

Lemma 4.5.10. indDrB0⊕Br = indD1B0⊕B1 for all r ∈ [0, 1].

127

Proof. For r0, r ∈ [0, 1], let πr0r : Br0 → Br denote the orthogonal projection. Then for

every r0 ∈ [0, 1] there exists ε > 0 such that if |r − r0| < ε then πr0r is an isomorphism. As

in the proof of Theorem 3.4.8, it induces an isomorphism

Πr0r : domDr0B0⊕Br0 → domDrB0⊕Br .

Hence

ind(DrB0⊕Br ◦ Πr0r

)= indDrB0⊕Br . (4.5.6)

Since for |r − r0| < ε

DrB0⊕Br ◦ Πr0r : domDr0B0⊕Br0 → L2(M ′, E ′)

is a continuous family of bounded operators, indDrB0⊕Br ◦ Πr0r is independent of r. The

lemma follows now from (4.5.6).

The space Br0 := H

1/2(−∞,0)(−Ar) is the APS boundary conditions for Dr at {1} ×N1. By

definition, η(A1,A0) = 2 indD1B0⊕B1

0. To finish the proof of Theorem 4.5.9 we note that by

Proposition 3.4.6

indDrB0⊕Br = indDrB0⊕Br0

+ [Br, Br0].

Hence,

η(A1,A0)− η(A0,A0))

2= indD1

B0⊕B10− indD0

B0⊕B00

=(

indD1B0⊕B1 − [B1, B1

0 ])−(

indD0B0⊕B0 − [B0, B0

0 ])

Lemma 4.5.10========== −[B1, B1

0 ] + [B0, B00 ] = − sf(−A)

Lemma 4.5.8========= sf(A).

This proves (4.5.5). Now, by Propostion 4.4.9,

η(A1,A0) = η(A1,A0) − η(A0,A0) = 2 sf(A).

128

Chapter 5

The Atiyah–Patodi–Singer Index on

Manifolds with Non-Compact

Boundary: Even-Dimensional Case

In this chapter we discuss an even-dimensional analogue of Chapter 4. Many parts of the

chapter are parallel to the discussion in Chapter 4. However, there are two important

differences. First, the Atiyah–Singer integrand was, of course, equal to 0 in Chapter 4,

which simplified many formulas. In particular, the relative η-invariant was an integer. As

opposed to it, in the current chapter the Atiyah–Singer integrand plays an important role

and the relative η-invariant is a real number. More significantly, the proof of the main result

in Chapter 4 was based on the application of the Callias index theorem, [3, 34]. This theorem

is not available in our current setting. Consequently, a completely different proof is proposed

in Section 5.1.

Let D = D + Ψ be a graded strongly Callias-type operator on a manifolds M with non-

compact boundary. We adhere to the setting and notations of Chapter 3. In particular, we

denote A (resp. A]) to be the restriction of D+ (resp. D−) to the boundary ∂M .

Like in Section 4.2, we can reduce the APS index to an essentially cylindrical manifold.

To be precise, let M1 ⊂ M be an almost compact essential support of D+. Let (D′)+ be a

129

compact perturbation of D+ which is product near the boundary (cf. Section 4.2.2). Let A

be the restriction of D+ to ∂M . It is also the restriction of (D′)+. We denote by −A1 the

restriction of (D′)+ to N1. Theorem 4.2.7 claims that:

Theorem 5.0.1. Suppose M1 ⊂ M is an almost compact essential support of D+ and let

∂M1 = ∂M tN1. Let (D′)+ be a compact perturbation of D+ which is product near N1 and

such that there is a compact essential support for (D′)+ which is contained in M1. Let B0 be

a generalized APS boundary condition for D+. View (D′)+ as an operator on M1 and let

B1 = H1/2(−∞,0)(−A1) = H

1/2(0,∞)(A1)

be the APS boundary condition for (D′)+ at N1. Then

indD+B0

= ind(D′)+B0⊕B1

. (5.0.1)

5.1 The index of operators on essentially cylindrical

manifolds

In this section we discuss the index of strongly Callias-type operators on even-dimensional

essentially cylindrical manifolds. It is parallel to Section 4.3, where the odd-dimensional case

was considered.

From now on we assume that M is an oriented even-dimensional essentially cylindrical

manifold whose boundary ∂M = N0 tN1 is a disjoint union of two non-compact manifolds

N0 and N1. Let D+ : C∞(M,E+) → C∞(M,E−) be a strongly Callias-type operator as in

Definition 0.1.6 (these data might or might not come as a restriction of another operator to

its almost compact essential support. In particular, we don’t assume that the restriction of

D+ to N1 is invertible). Let A0 and −A1 be the restrictions of D+ to N0 and N1, respectively.

Let M and M ′ be compatible essentially cylindrical manifolds (cf. Definition 4.3.1) and

let Zr and Z ′r be as in Definition 4.3.1. Let E →M be a Z2-graded Dirac bundle over M and

130

let D+ : C∞(M,E+)→ C∞(M,E−) be a strongly Callias-type operator whose restriction to

Zr is product and such that M is an almost compact essential support of D+. This means

that there is a compact set K ⊂M such that M \K = [0, ε]×N and the restriction of D+ to

M \K is product (i.e. is given by (3.2.4)). Let E ′ → M ′ be a Z2-graded Dirac bundle over

M ′ and let (D′)+ : C∞(M ′, (E ′)+) → C∞(M ′, (E ′)−) be a strongly Callias-type operator,

whose restriction to Z ′r is product and such that M ′ is an almost compact essential support

of (D′)+.

Definition 5.1.1. In the situation discussed above we say that D+ and (D′)+ are compat-

ible if there is an isomorphism E|Zr ' E ′|Z′r of graded Dirac bundles which identifies the

restriction of D+ to Zr with the restriction of (D′)+ to Z ′r.

Let B0 = H1/2(−∞,0)(A0) and B1 = H

1/2(−∞,0)(−A1) = H

1/2(0,∞)(A1) be the APS boundary con-

ditions for D+ at N0 and N1 respectively. Since D+ and (D′)+ are equal near the boundary,

B0 and B1 are also APS boundary conditions for (D′)+.

We denote by αAS(D+) the Atiyah–Singer integrand of D+. It can be written as

αAS(D+) := (2πi)− dimM A(M) ch(E/S)

where A(M) and ch(E/S) are the differential forms representing the A-genus of M and the

relative Chern character of E, cf. [13, §4.1].

Since outside of a compact set K, M and E are product, the interior multiplication by

∂/∂t annihilates αAS. Hence, the top degree component of αAS vanishes on M \ K. We

conclude that the integral∫MαAS(D+) is well-defined and finite. Similarly,

∫M ′αAS((D′)+)

is well-defined.

Theorem 5.1.2. Suppose D+ is a strongly Callias-type operator on an oriented even-dimensional

essentially cylindrical manifold M such that M is an almost compact essential support of

D+. Suppose that the operator (D′)+ is compatible with D+. Let ∂M = N0 t N1 and let

131

B0 = H1/2(−∞,0)(A0) and B1 = H

1/2(−∞,0)(−A1) = H

1/2(0,∞)(A1) be the APS boundary conditions

for D+ (and, hence, for (D′)+) at N0 and N1 respectively. Then

indD+B0⊕B1

−∫M

αAS(D+) = ind(D′)+B0⊕B1

−∫M ′αAS((D′)+). (5.1.1)

In particular, indD+B0⊕B1

−∫MαAS(D+) depends only on the restrictions A0 and A1 of D+

to the boundary.

The rest of the section is devoted to the proof of this theorem.

Remark 5.1.3. In Chapter 4 the odd dimensional version of Theorem 5.1.2 was considered.

Of course, in this case αAS vanishes identically and Theorem 4.3.3 states that the indexes

of compatible operators are equal. The proof there is based on application of a Callias-type

index theorem and can not be adjusted to our current situation. Consequently, a completely

different proof is proposed below.

5.1.1 Gluing together the data

We follow Subsections 4.3.3 and 4.3.4 to glue M with M ′ and D+ with (D′)+.

Let −M ′ denote the manifold M ′ with the opposite orientation. We identify the boundary

of −M ′ with the product

−Z ′r :=(N0 × (−r, 0]

)t(N1 × (−r, 0]

)and consider the union

M := M ∪N0tN1 (−M ′).

Then Z(−r,r) := Zr ∪ (−Z ′r) is a subset of M identified with the product

(N0 × (−r, r)

)t(N1 × (−r, r)

).

We note that M is a complete Riemannian manifold without boundary.

Let E∂M = E+∂M ⊕ E−∂M denote the restriction of E = E+ ⊕ E− to ∂M . The product

structure on E|Zr gives a grading-respecting isomorphism ψ : E|Zr → [0, r) × E∂M . Recall

132

that we identified Zr with Z ′r and fixed an isomorphism between the restrictions of E to Zr

and E ′ to Z ′r. By a slight abuse of notation we use this isomorphism to view ψ also as an

isomorphism E ′|Z′r → [0, r)× E∂M .

Let E → M be the vector bundle over M obtained by gluing E and E ′ using the isomor-

phism c(τ) : E|∂M → E ′|∂M ′ . This means that we fix isomorphisms

φ : E|M → E, φ′ : E|M ′ → E ′, (5.1.2)

so that

ψ ◦ φ ◦ ψ−1 = id : [0, r)× E∂M → [0, r)× E∂M ,

ψ ◦ φ′ ◦ ψ−1 = 1× c(τ) : [0, r)× E∂M → [0, r)× E∂M .

Note that the grading of E is preserved while the grading of E ′ is reversed in this gluing

process. Therefore E = E+ ⊕ E− is a Z2-graded bundle.

We denote by c′ : T ∗M ′ → End(E ′) the Clifford multiplication on E ′ and set c′′(ξ) :=

−c′(ξ). Then E is a Dirac bundle over M with the Clifford multiplication

c(ξ) :=

c(ξ), ξ ∈ T ∗M ;

c′′(ξ) = −c′(ξ), ξ ∈ T ∗M ′.

(5.1.3)

One readily checks that (5.1.3) defines a smooth odd-graded Clifford multiplication on E. Let

D : C∞(M, E)→ C∞(M, E) be the Z2-graded Dirac operator. Then the isomorphism φ of

(5.1.2) identifies the restriction of D± with D±, the isomorphism φ′ identifies the restriction

of D± with −(D′)∓, and isomorphism ψ ◦ φ′ ◦ ψ−1 identifies the restriction of D± to −Z ′r

with

D±|Z′r = −c′(τ) ◦ (D′)±Z′r ◦ c′(τ)−1.

Let (Ψ′)± denote the Callias potentials of (D′)±, so that (D′)± = (D′)±+(Ψ′)±. Consider

the bundle maps Ψ± ∈ Hom(E±, E∓) whose restrictions to M are equal to Ψ± and whose

restrictions to M ′ are equal to −(Ψ′)∓. The two pieces fit well on Z(−r,r) by Remark 0.1.7.

To sum up the constructions presented in this subsection, we have

133

Lemma 5.1.4. The operators D± := D± + Ψ± are strongly Callias-type operators on M ,

formally adjoint to each other, whose restrictions to M are equal to D± and whose restrictions

to M ′ are equal to −(D′)∓ − (Ψ′)∓ = −(D′)∓.

The operator D+ is a strongly Callias-type operator on a complete Riemannian manifold

without boundary. Hence, [1], it is Fredholm. We again denote by αAS(D+) the Atiyah–

Singer integrand of D+. It is explained in the paragraph before Theorem 5.1.2 that the

integral∫MαAS(D+) is well defined.

Lemma 5.1.5. ind D+ =∫MαAS(D+).

Proof. Since M is a union of two essentially cylindrical manifolds, there exists a compact

essential support K ⊂ M of D such that M \ K is of the form S1 × N . We can choose K

to be large enough so that the restriction of D to a neighborhood W of M \ K ' S1 ×N is

a product of an operator on N and an operator on S1. Then the restriction of αAS to this

neighborhood vanishes. We can also assume that K has a smooth boundary Σ = S1 × L.

Let D+ be a compact perturbation of D+ in W which is product both near Σ and on W

and whose essential support is contained in K. Then

ind D+ = ind D+.

We cut M along Σ and apply the Splitting Theorem 3.4.81 to get

ind D+ = ind D+

K+ ind D+

M\K , (5.1.4)

where ind D+

Kstands for the index of the restriction of D+ to K with APS boundary con-

dition, and ind D+

M\K stands for the index of the restriction of D+ to M \ K with the dual

APS boundary condition.

Since D+ has an empty essential support in M \ K, by the vanishing theorem Corollary

3.4.9, the second summand in the right hand side of (5.1.4) vanishes. The first summand in

1Since Σ is compact we can also use the splitting theorem for compact hypersurfaces, [11, Theorem 8.17].

134

the right hand side of (5.1.4) is given by the Atiyah–Patodi–Singer index theorem [5, Theorem

3.10] (Note that Σ is outside of an essential support of D+ and, hence, the restriction of D+

to Σ is invertible. Hence, its kernel is trivial)

ind D+

K=

∫K

αAS(D+) − 1

2η(0),

where η(0) is the η-invariant of the restriction of D+ to Σ.

As αAS(D+) ≡ 0 on W and D+ ≡ D+ elsewhere, we have∫K

αAS(D+) =

∫K

αAS(D+) =

∫M

αAS(D+).

To finish the proof of the lemma it suffices now to show η(0) = 0.

Let ω be the inward (with respect to K) unit normal vector field along Σ. Recall that

Σ = S1 × L. We denote the coordinate along S1 by θ. Suppose that {ω, dθ, e1, · · · , em}

forms a local orthonormal frame of T ∗M on Σ. Then the restriction of D+ = D+ + Ψ+ to Σ

can be written as

A+Σ = −

m∑i=1

c(ω)c(ei)∇Eei− c(ω)c(dθ)∂θ − c(ω)Ψ+

which maps C∞(Σ, E+|Σ) to itself. We define a unitary isomorphism Θ on the space

C∞(Σ, E|Σ) given by

Θu(θ, y) := −c(ω)c(dθ)u(−θ, y).

One can check that Θ anticommutes with A+Σ. As a result, the spectrum of A+

Σ is symmetric

about 0. Therefore η(0) = 0 and lemma is proved.

5.1.2 Proof of Theorem 5.1.2

Recall that we denote by B0 and B1 the APS boundary conditions for D+ = D+|M at N0

and N1 respectively. Let (D′′)+ denote the restriction of D+ to −M ′ = M \M . Let B0

and B1 be the dual APS boundary conditions for (D′′)+ at N0 and N1 respectively. By the

135

Splitting Theorem 3.4.8,

ind D+ = indD+B0⊕B1

+ ind(D′′)+B0⊕B1

.

By Lemma 5.1.5, we obtain

indD+B0⊕B1

+ ind(D′′)+B0⊕B1

=

∫M

αAS(D+) +

∫M ′αAS((D′′)+),

which means

indD+B0⊕B1

−∫M

αAS(D+) = − ind(D′′)+B0⊕B1

+

∫M ′αAS((D′′)+). (5.1.5)

By Lemma 5.1.4, (D′′)+ = −(D′)−. Thus B0 ⊕ B1 is the adjoint of the APS boundary

condition for (−D′)+ (cf. Example 3.3.6). Therefore,

ind(D′′)+B0⊕B1

= ind(−D′)−B0⊕B1

= − ind(−D′)+B0⊕B1

= − ind(D′)+B0⊕B1

,

where we used (3.4.4) in the middle equality. Also by the construction of local index density,

αAS((D′′)+) = αAS((−D′)−) = αAS((D′)−) = −αAS((D′)+).

Combining these equalities with (5.1.5) we obtain (5.1.1). �

5.2 The relative η-invariant

In the previous section we proved that on an essentially cylindrical manifold M the difference

indDB0⊕B1 −∫MαAS(D) depends only on the restriction of D to the boundary, i.e., on the

operatorsA0 and −A1. Like in Section 4.4, we can similarly use this fact to define the relative

η-invariant η(A1,A0) and show that it has properties similar to the difference of η-invariants

η(A1)−η(A0) of operators on compact manifolds. In this case, A0,A1 are operators on odd-

dimensional manifolds, and the definition of the relative η-invariant is slightly more involved

than the definition in Section 4.4, we show that most of the properties of η(A1,A0) remain

the same.

136

5.2.1 Almost compact cobordisms

Let N0 and N1 be two complete odd-dimensional Riemannian manifolds and let A0 and A1

be self-adjoint strongly Callias-type operators on N0 and N1, respectively. We adapt the

definition of almost compact cobordism to this situation.

Definition 5.2.1. An almost compact cobordism between A0 and A1 is a pair (M,D), where

M is an essentially cylindrical manifold with ∂M = N0 tN1 and D is a graded self-adjoint

strongly Callias-type operator D on M such that

(i) M is an almost compact essential support of D;

(ii) D is product near ∂M ;

(iii) The restriction of D+ to N0 is equal to A0 and the restriction of D+ to N1 is equal to

−A1.

If there exists an almost compact cobordism between A0 and A1 we say that operator A0 is

cobordant to operator A1.

Lemmas 4.4.2 and 4.4.3 hold true without any changes here.

Definition 5.2.2. Suppose A0 and A1 are cobordant self-adjoint strongly Callias-type oper-

ators and let (M,D) be an almost compact cobordism between them. Let B0 = H1/2(−∞,0)(A0)

and B1 = H1/2(−∞,0)(−A1) be the APS boundary conditions for D+. The relative η-invariant

is defined as

η(A1,A0) = 2

(indD+

B0⊕B1−∫M

αAS(D+)

)+ dim kerA0 + dim kerA1. (5.2.1)

Theorem 5.1.2 implies that η(A1,A0) is independent of the choice of the cobordism

(M,D).

137

Remark 5.2.3. If using the dual APS boundary conditions B0 = H1/2(−∞,0](A0) and B2 =

H1/2(−∞,0](−A1) instead of B0 and B1, it follows from Corollary 3.4.7 that the relative η-

invariant can be written as

η(A1,A0) = 2

(indD+

B0⊕B1−∫M

αAS(D+)

)− dim kerA0 − dim kerA1. (5.2.2)

In the case when the heat kernel has an asymptotic expansion, we have the following

analogue of Proposition 4.4.6 with the same proof

Proposition 5.2.4. Suppose now that D is an operator on an essentially cylindrical manifold

M which satisfies the conditions of [40]. We also assume that D is product near ∂M =

N0 tN1 and that M is an almost compact essential support for D. Let A0 and −A1 be the

restrictions of D+ to N0 and N1 respectively. Let η(Aj) (j = 0, 1) be the η-invariant of Aj.

Then

η(A1,A0) = η(A1) − η(A0). (5.2.3)

5.2.2 Basic properties of the relative η-invariant

We show the basic properties of relative η-invariant as in Subsection 4.4.3 with modified

proofs.

Proposition 5.2.5 (Antisymmetry). Suppose A0 and A1 are cobordant self-adjoint strongly

Callias-type operators. Then

η(A0,A1) = − η(A1,A0). (5.2.4)

Proof. Let −M denote the manifold M with the opposite orientation and let M := M ∪∂M

(−M) denote the double of M . Let D be an almost compact cobordism between A0 and

A1. Using the construction of Section 5.1.1 (with D′ = D) we obtain a graded self-adjoint

strongly Callias-type operator D on M whose restriction to M is isometric to D. Let D′′

138

denote the restriction of D to −M = M \M . Then the restriction of (D′′)+ to N1 is equal

to A1 and the restriction of (D′′)+ to N0 is equal to −A0.

Let

B0 = H1/2[0,∞)(A0) = H

1/2(−∞,0](−A0),

B1 = H1/2[0,∞)(−A1) = H

1/2(−∞,0](A1)

be the dual APS boundary conditions for (D′′)+. By (5.1.5),

ind(D′′)+B0⊕B1

−∫M ′αAS((D′′)+) = − indD+

B0⊕B1+

∫M

αAS(D+). (5.2.5)

Since D′′ is an almost compact cobordism between A1 and A0 we conclude from (5.2.2)

that

η(A0,A1) = 2

(ind(D′′)+

B0⊕B1−∫M ′αAS((D′′)+)

)− dim kerA0 − dim kerA1. (5.2.6)

Combining (5.2.6) and (5.2.5) we obtain (5.2.4).

Note that (5.2.4) implies that

η(A,A) = 0

for every self-adjoint strongly Callias-type operator A.

Proposition 5.2.6 (The cocycle condition). Let A0,A1 and A2 be self-adjoint strongly

Callias-type operators which are cobordant to each other. Then

η(A2,A0) = η(A2,A1) + η(A1,A0). (5.2.7)

Proof. Let M1 and M2 be essentially cylindrical manifolds such that ∂M1 = N0 t N1 and

∂M2 = N1 t N2. Let D1 be an operator on M1 which is an almost compact cobordism

between A0 and A1. Let D2 be an operator on M2 which is an almost compact cobordism

between A1 and A2. Then the operator D3 on M1 ∪N1 M2 whose restriction to Mj (j = 1, 2)

is equal to Dj is an almost compact cobordism between A0 and A2.

139

Let B0 and B1 be the APS boundary conditions for D+1 at N0 and N1 respectively. Then

B1 = H1/2[0,∞)(A1) is equal to the dual APS boundary condition for D+

2 . Let B2 be the APS

boundary condition for D+2 at N2. From Corollary 3.4.7 we obtain

η(A2,A1) = 2

(ind(D+

2 )B1⊕B2−∫M

αAS(D+2 )

)− dim kerA1 + dim kerA2. (5.2.8)

By the Splitting Theorem 3.4.8

ind(D+3 )B0⊕B2 = ind(D+

1 )B0⊕B1 + ind(D+2 )B1⊕B2

. (5.2.9)

Clearly, ∫M1∪M2

αAS(D+3 ) =

∫M1

αAS(D+1 ) +

∫M2

αAS(D+2 ). (5.2.10)

Combining (5.2.8), (5.2.9), and (5.2.10) we obtain (5.2.7).

5.3 The spectral flow

Suppose A := {As}0≤s≤1 is a smooth family of self-adjoint elliptic operators on a closed

manifold N . Let η(As) ∈ R/Z denote the mod Z reduction of the η-invariant η(As). Atiyah,

Patodi, and Singer, [6], showed that s 7→ η(As) is a smooth function whose derivative ddsη(As)

is given by an explicit local formula. Further, Atiyah, Patodi and Singer, [6], introduced a

notion of spectral flow sf(A) and showed that it computes the net number of integer jumps

of η(As), i.e.,

2 sf(A) = η(A1) − η(A0) −∫ 1

0

( ddsη(As)

)ds.

In this section we consider a family of self-adjoint strongly Callias-type operators A =

{As}0≤s≤1 on a complete odd-dimensional Riemannian manifold. Assuming that As is fixed

in s outside of a compact subset of N , we show that for any operator A0 cobordant to A0

the mod Z reduction η(As,A0) of the relative η-invariant depends smoothly on s and

2 sf(A) = η(A1,A0) − η(A0,A0) −∫ 1

0

( ddsη(As,A0)

)ds.

140

Note the presence of the last term in the formula compared to the result in Section 4.5, as

the relative η-invariant here is no longer integer-valued.

5.3.1 A family of boundary operators

Let EN → N be a Dirac bundle over a complete odd-dimensional Riemannian manifold

N . We denote the Clifford multiplication of T ∗N on EN by cN : T ∗N → End(EN). Let

A = {As}0≤s≤1 be a family of self-adjoint strongly Callias-type operators As : C∞(N,EN)→

C∞(N,EN).

Definition 5.3.1. The family A = {As}0≤s≤1 is called almost constant if there exists a

compact set K ⊂ N such that the restriction of As to N \K is independent of s.

Consider the cylinder M := [0, 1]×N and denote by t the coordinate along [0, 1]. Set

E+ = E− := [0, 1]× EN .

Then E = E+ ⊕ E− →M is naturally a Z2-graded Dirac bundle over M with

c(dt) :=

0 − idEN

idEN0

and

c(ξ) :=

0 cN(ξ)

cN(ξ) 0

, for ξ ∈ T ∗N.

Definition 5.3.2. The family A = {As}0≤s≤1 is called smooth if

D := c(dt)

∂t +

At 0

0 −At

: C∞(M,E)→ C∞(M,E) (5.3.1)

is a smooth differential operator on M .

141

Fix a smooth non-decreasing function κ : [0, 1] → [0, 1] such that κ(t) = 0 for t ≤ 1/3

and κ(t) = 1 for t ≥ 2/3 and consider the operator

D := c(dt)

∂t +

Aκ(t) 0

0 −Aκ(t)

: C∞(M,E)→ C∞(M,E). (5.3.2)

Then D is product near ∂M . If A is a smooth almost constant family of self-adjoint strongly

Callias-type operators then (5.3.2) is a strongly Callias-type operator for which M is an

almost compact essential support. Hence it is a non-compact cobordism (cf. Definition 5.2.1)

between A0 and A1.

5.3.2 The spectral flow

Let P = {P s} be a spectral section for A (cf. Definition 4.5.4). Set Bs := kerP s. Let

Bs0 := H

1/2(−∞,0)(As) denote the APS boundary condition defined by the boundary operator

As. We recall the definition and a basic property of spectral flow given in Subsection 4.5.3.

Definition 5.3.3. Let A = {As}0≤s≤1 be a smooth almost constant family of self-adjoint

strongly Callias-type operators which admits a spectral section P = {P s}0≤s≤1. Assume

that the operators A0 and A1 are invertible. Let Bs := kerP s and Bs0 := H

1/2(−∞,0)(As). The

spectral flow sf(A) of the family A is defined by the formula

sf(A) := [B1, B10 ] − [B0, B0

0 ]. (5.3.3)

Lemma 5.3.4. Let −A denote the family {−As}0≤s≤1. Then

sf(−A) = − sf(A). (5.3.4)

5.3.3 Deformation of the relative η-invariant

Let A = {As}0≤s≤1 be a smooth almost constant family of self-adjoint strongly Callias-type

operators on a complete odd-dimensional Riemannian manifold N1. Let A0 be another self-

142

adjoint strongly Callias-type operator, which is cobordant to A0. As in Subsection 4.5.4, we

know that A0 is cobordant to the family A.

The main result of this section is the following theorem.

Theorem 5.3.5. Suppose A ={As : C∞(N1, E1)→ C∞(N1, E1)

}0≤s≤1

is a smooth almost

constant family of self-adjoint strongly Callias-type operators on a complete odd-dimensional

Riemannian manifold N1. Assume that A0 and A1 are invertible. Let A0 : C∞(N0, E0) →

C∞(N0, E0) be an invertible self-adjoint strongly Callias-type operator on a complete Rie-

mannian manifold N0 which is cobordant to the family A. Then the mod Z reduction

η(As,A0) ∈ R/Z of the relative η-invariant depends smoothly on s ∈ [0, 1] and

η(A1,A0) − η(A0,A0) −∫ 1

0

( ddsη(As,A0)

)ds = 2 sf(A). (5.3.5)

The proof of this theorem occupies Subsections 5.3.4–5.3.6.

5.3.4 A family of almost compact cobordisms

Let M be an essentially cylindrical manifold whose boundary is the disjoint union of N0

and N1. First, we construct a smooth family Dr (0 ≤ r ≤ 1) of graded self-adjoint strongly

Callias-type operators on the manifold

M ′ := M ∪N1

([0, 1]×N1

), (5.3.6)

such that for each r ∈ [0, 1] the pair (M ′,Dr) is an almost compact cobordism between A0

and Ar.

Let D : C∞(M,E) → C∞(M,E) be an almost compact cobordism between A0 and A0.

Let E0 and E1 denote the restrictions of E to N0 and N1 respectively.

Let M ′ be given by (5.3.6) and let E ′ → M ′ be the bundle over M ′ whose restriction to

M is equal to E and whose restriction to the cylinder [0, 1]×N1 is equal to [0, 1]× E1.

We fix a smooth function ρ : [0, 1]× [0, 1]→ [0, 1] such that for each r ∈ [0, 1]

143

• the function s 7→ ρ(r, s) is non-decreasing.

• ρ(r, s) = 0 for s ≤ 1/3 and ρ(r, s) = r for s ≥ 2/3.

Consider the family of strongly Callias-type operatorsDr : C∞(M ′, E ′)→ C∞(M ′, E ′) whose

restriction to M is equal to D and whose restriction to [0, 1]×N1 is given by

Dr := c(dt)

∂t +

Aρ(r,t) 0

0 −Aρ(r,t)

.

Then Dr is an almost compact cobordism between A0 and Ar. In particular, the restriction

of Dr to N1 is equal to −Ar.

Recall that we denote by −A the family {−As}0≤s≤1. Let P = {P s} be a spectral section

for −A. Then for each r ∈ [0, 1] the space Br := kerP r is a finite rank perturbation of the

APS boundary condition for Dr at {1} ×N1. Let B0 := H1/2(−∞,0)(A0) be the APS boundary

condition for Dr at N0. Then, by Proposition 3.4.6, the operator DrB0⊕Br is Fredholm. Recall

that the domain domDrB0⊕Br consists of sections u whose restriction to ∂M ′ = N0 tN1 lies

in B0 ⊕Br.

Lemma 5.3.6. indDrB0⊕Br = indD1B0⊕B1 for all r ∈ [0, 1].

The proof is the same as that of Lemma 4.5.10.

5.3.5 Variation of the reduced relative η-invariant

By Definition 5.2.2, the mod Z reduction of the relative η-invariant is given by

η(Ar,A0) := −2

∫M ′

αAS(Dr). (5.3.7)

It follows that η(Ar,A0) depends smoothly on r and

d

drη(Ar,A0) = −2

∫M ′

d

drαAS(Dr). (5.3.8)

144

A more explicit local expression for the right hand side of this equation is given in Sec-

tion 5.3.7. For the moment we just note that (5.3.8) implies that∫ 1

0

( ddsη(As,A0)

)ds = −2

∫M ′

(αAS(D1)− αAS(D0)

). (5.3.9)

5.3.6 Proof of Theorem 5.3.5

Since the operators A0,A0, and A1 are invertible, we have

η(Aj,A0) = 2

(indDj

B0⊕Bj0

−∫M ′

αAS(Dj)), j = 0, 1.

Thus, using (5.3.9), we obtain

η(A1,A0) − η(A0,A0) −∫ 1

0

( ddsη(As,A0)

)ds = 2

(indD1

B0⊕B10− indD0

B0⊕B00

).

(5.3.10)

Recall that, by Proposition 3.4.6,

indDrB0⊕Br = indDrB0⊕Br0

+ [Br, Br0].

Hence, from (5.3.10) we obtain

1

2

(η(A1,A0)− η(A0,A0)−

∫ 1

0

( ddsη(As,A0)

)ds

)=(

indD1B0⊕B1 − [B1, B1

0 ])−(

indD0B0⊕B0 − [B0, B0

0 ])

Lemma 5.3.6= −[B1, B1

0 ] + [B0, B00 ] = − sf(−A)

Lemma 5.3.4= sf(A).

�

5.3.7 A local formula for variation of the reduced relative η-invariant

It is well known that there exists a family of differential forms βr (0 ≤ r ≤ 1), called the

transgression form such that

dβr =d

drαAS(Dr). (5.3.11)

145

The transgression form depends on the symbol of Dr and its derivatives with respect to r.

For geometric Dirac operators one can write very explicit formulas for βr. For example, if

Dr is the signature operator (so that Ar is the odd signature operator) corresponding to a

family ∇r of flat connections on E, then βr = L(M) ∧ ddr∇r, where L(M) is the L-genus of

M , cf. for example, [26, Theorem 2.3]. For general Dirac-type operators, a formula for βr is

more complicated, cf. [27, §6].

We note that since the family Ar is constant outside of the compact set K, the form βr

vanishes outside of K. Hence,∫∂M ′

βr is well defined and finite. Thus we obtain from (5.3.8)

that

d

drη(Ar,A0) = −2

∫M ′dβr = −2

∫∂M ′

βr = 2(∫{1}×N1

βr −∫N0

βr

). (5.3.12)

Hence, (5.3.5) expresses η(A1,A0) − η(A0,A0) as a sum of 2 sf(A) and a local differential

geometric expression 2∫∂M ′

(∫ 1

0βr) dr.

146

Chapter 6

Cauchy Data Spaces and

Atiyah–Patodi–Singer Index on

Non-Compact Manifolds

In the last three chapters, we studied the boundary value problems of strongly Callias-type

operators on manifolds with non-compact boundary. In particular, for the Atiyah–Patodi–

Singer (or APS) boundary value problem, we found a formula to compute the APS index.

An interesting term in the formula is the relative eta-invariant. One question that remains

to be answered is a spectral interpretation of this invariant.

Another notion involved in the study of boundary value problems is the space of Cauchy

data. In particular, the APS index (on manifold with compact boundary) can be com-

puted in terms of the projections onto Cauchy data spaces, which provides another way

of understanding the eta invariant. In this chapter, we address the APS index for strongly

Callias-type operators from this perspective. Traditionally, Cauchy data spaces of Dirac-type

operators can be built through the L2-closure of boundary restrictions of smooth solutions

on partitioned (compact) manifolds. This approach involves pseudo-differential calculus, i.e.,

a Cauchy data space is the range of the L2-extension of Caldron projector. (cf. [20, 61].)

A different but more general approach is established on the maximal domain of an operator

on a manifold with boundary by Booss-Bavnbek and Furutani [18]. When the operator is

147

symmetric, there is a symplectic structure on the space of boundary values of sections in

maximal domain. The (maximal) Cauchy data space is a subspace of this boundary value

space. And under natural assumptions, such a Cauchy data space gives rise to Fredholm-

Lagrangian property. A good feature of this treatment is that it gets rid of pseudo-differential

calculus. We refer the reader to [15] for a nice exposition on these two approaches.

We shall adopt the maximal domain approach to study the Cauchy data spaces of strongly

Callias-type operators on manifolds with non-compact boundary. Since we mainly consider

the graded operator, we will care more about the Fredholmness than the Lagrangian. We

give formulas of the APS index through the APS projection and projections onto Cauchy

data spaces (Theorems 6.1.10 and 6.1.11). We also prove the twisted orthogonality of Cauchy

data spaces (Theorem 6.2.4). These results can be compared with the results in [20, 67]. At

last, we interpret certain Cauchy data spaces as elliptic boundary conditions in the sense

of [31] (Theorem 6.2.9). In [10], Ballmann, Bruning and Carron discussed the Cauchy data

spaces on a semi-infinite cylinder model. Since the growth of the potential in our operator

controls the behavior at infinity, we do not need to consider extended solutions. (Compare

Theorem 6.2.9 with [10, Theorem C].)

6.1 Maximal Cauchy data spaces and index formulas

Let D = D + Ψ be a graded strongly Callias-type operators defined in Definition 0.1.6 on

a complete manifolds M with non-compact boundary. The basic setting and notations are

the same as in Chapter 3. In particular, A and A] are the restrictions of D+ and D− to the

boundary, respectively. In this chapter, we denote the APS and dual APS indexes by

indD+APS := dim kerD+

B − dim kerD−Bad ∈ Z,

indD+dAPS := dim kerD+

B− dim kerD−

Bad ∈ Z,(6.1.1)

where B = H1/2(−∞,0)(A) and B = H

1/2(−∞,0](A) are the APS and dual APS boundary condi-

tions; Bad = H1/2(−∞,0](A]) and Bad = H

1/2(−∞,0)(A]) are the corresponding adjoint boundary

148

conditions, cf. Example 3.3.6.

6.1.1 Unique continuation property

We state a well-known property of Dirac-type operators, called the (weak) unique continu-

ation property, as follows

Theorem 6.1.1. Let P be a Dirac-type operator over a (connected) smooth manifold M .

Then any smooth solution s of Ps = 0 which vanishes on an open subset of M also vanishes

on the whole manifold M .

Essentially, this property only depends on the symmetry of the principal symbol of Dirac-

type operators and a nice proof is given in [20, §8], [17]. In particular, the strongly Callias-

type operators introduced earlier satisfy this property.

Corollary 6.1.2. Let D+ be a strongly Callias-type operator. Then the space of interior

solutions

ker0D+max := {u ∈ domD+

max : D+maxu = 0 and R(u) = 0}

contains only 0-sections. The same conclusion is true for D−.

Proof. Proceeding as in [20, §9], one can construct an invertible double D+ of D+ on M ,

the double of M , such that D+|M = D+. Let u be an element of ker0D+max. We extend it

by zero to get a section u on M . For any compactly supported smooth section v on M , by

Green’s formula,(u; (D+)∗v

)L2(M)

=

∫M

⟨u; (D+)∗v|M

⟩=

∫M

⟨D+u; v|M

⟩+

∫∂M

〈c(τ)u|∂M ; v|∂M〉 = 0.

Thus u is a weak solution of D+s = 0. By elliptic regularity, u is smooth. Since u vanishes on

M \M , applying Theorem 6.1.1 to D+ yields that u ≡ 0 on M . Therefore u is a 0-section.

It follows from the corollary that

149

Corollary 6.1.3. The maps R|kerD±max: kerD±max → H(A) (or H(A])) are injective.

Lemma 6.1.4. rangeD+max = L2(M,E−).

Proof. Since rangeD+max ⊃ rangeD+

APS and the latter admits a closed finite-dimensional com-

plementary subspace in L2(M,E−) (by the Fredholmness of D+APS), one gets that rangeD+

max

is closed in L2(M,E−). Therefore

rangeD+max = (kerD−min)⊥ = {0}⊥ = L2(M,E−).

6.1.2 Maximal Cauchy data spaces

Definition 6.1.5. Let D+ be a strongly Callias-type operator on M . We call

C+max := R(kerD+

max) ⊂ H(A)

the Cauchy data space of the maximal extension D+max, where R is the map of restriction to

the boundary (cf. Chapter 3). Similarly,

C−max := R(kerD−max) ⊂ H(A])

is called the Cauchy data space of the maximal extension D−max.

Note that C+max (resp. C−max) is a closed subspace of H(A) (resp. H(A])).

6.1.3 Fredholm pair

We recall the concept of Fredholm pair (cf. [49, §IV.4.1]).

Definition 6.1.6. Let Z be a Hilbert space. A pair (X, Y ) of closed subspaces of Z is called

a Fredholm pair if

(i) dim(X ∩ Y ) <∞;

150

(ii) X + Y is a closed subspace of Z;

(iii) codim(X + Y ) := dimZ/(X + Y ) <∞.

The index of a Fredholm pair (X, Y ) is defined to be

ind(X, Y ) := dim(X ∩ Y )− codim(X + Y ).

Proposition 6.1.7. (H1/2(−∞,0)(A), C+

max) and (H1/2(−∞,0](A]), C−max) are Fredholm pairs in H(A)

and H(A]), respectively. Moreover,

ind(H1/2(−∞,0)(A), C+

max) = indD+APS = − ind(H

1/2(−∞,0](A

]), C−max). (6.1.2)

The idea of the proof is from [18, Proposition 3.5].

Proof. Since indD+APS = − indD−dAPS by (3.4.4) and (6.1.1), we may only prove the conclusion

for the first pair.

Recall that by Example 3.3.6, H1/2(−∞,0)(A) = R(domD+

APS) and by Definition 6.1.5, C+max =

R(kerD+max). We first show that

R(domD+APS ∩ kerD+

max) = R(domD+APS) ∩R(kerD+

max). (6.1.3)

It is clear that the right hand side includes the left hand side. To show the other direction,

let u ∈ R(domD+APS) ∩R(kerD+

max). Then u = R(u1) = R(u2) for some u1 ∈ domD+APS,

u2 ∈ kerD+max. So u1 − u2 ∈ domD+

max and R(u1 − u2) = 0, which implies that u1 − u2 ∈

domD+APS. Hence u2 ∈ domD+

APS and it follows that u2 ∈ domD+APS ∩ kerD+

max. Therefore

u ∈ R(domD+APS ∩ kerD+

max). (6.1.3) is verified.

Since D+APS is a Fredholm operator, it follows from Corollary 6.1.3 that

∞ > dim kerD+APS = dim(domD+

APS ∩ kerD+max)

= dim R(domD+APS ∩ kerD+

max) = dim(H1/2(−∞,0)(A) ∩ C+

max).

(i) of Definition 6.1.6 is proved.

151

Note that the preimage of rangeD+APS under D+ is domD+

APS + kerD+max. Since D+ :

domD+max → L2(M,E−) is continuous,

D+APS Fredholm ⇒ rangeD+

APS is closed in L2(M,E−)

⇒ domD+APS + kerD+

max is closed in domD+max.

Recall that in Chapter 3, we defined a continuous extending map E : H(A) → domD+max

satisfying R◦E = id. If {uj} is a sequence in R(domD+APS+kerD+

max) = H1/2(−∞,0)(A)+C+

max ⊂

H(A) that is convergent to some u ∈ H(A), then {E uj} converges to E u in domD+max. Like

what we argued in proving (6.1.3), using the fact that domD+APS + kerD+

max is a subspace of

domD+max, one can show that E uj ∈ domD+

APS +kerD+max. By the above closedness, E u also

lies in domD+APS + kerD+

max. Therefore u = R(E u) ∈ H1/2(−∞,0)(A) + C+

max. (ii) of Definition

6.1.6 is proved.

To prove Definition 6.1.6.(iii) and equation (6.1.2), note that R induces a bijection be-

tween domD+max/(domD+

APS+kerD+max) and H(A)/(H

1/2(−∞,0)(A)+C+

max). Let π : L2(M,E−)�

(rangeD+APS)⊥ be the orthogonal projection. By Lemma 6.1.4, D+

max : domD+max → L2(M,E−)

is surjective, so

ker(π ◦ D+max) = domD+

APS + kerD+max.

Then

domD+max/(domD+

APS + kerD+max) ∼= (rangeD+

APS)⊥

= L2(M,E−)/ rangeD+APS.

Hence

codim(H1/2(−∞,0)(A) + C+

max) = dim H(A)/(H1/2(−∞,0)(A) + C+

max)

= dim domD+max/(domD+

APS + kerD+max)

= dimL2(M,E−)/ rangeD+APS

= dim cokerD+APS < ∞.

(6.1.4)

Therefore

ind(H1/2(−∞,0)(A), C+

max) = indD+APS.

152

6.1.4 Fredholm pair of projections

A notion that is closely related to Fredholm pair is the Fredholm pair of projections considered

in [9].

Definition 6.1.8. Let Z be a Hilbert space and (X, Y ) be a pair of closed subspaces of Z.

Denote the orthogonal projections from Z onto X, Y by PX , PY , respectively. (PX , PY ) is

called a Fredholm pair of projections if PXPY : rangePY → rangePX is a Fredholm operator.

Its index is defined as ind(PX , PY ) := indPXPY .

We formulate the following standard result about equivalent definitions of Fredholm pairs

and Fredholm pair of projections (cf. [49, §IV.4.2], [20, §24]).

Proposition 6.1.9. Let Z be a Hilbert space and X, Y, PX , PY be as above. Then the fol-

lowing are equivalent:

(i) (X, Y ) is a Fredholm pair;

(ii) (X0, Y 0) is a Fredholm pair, where X0, Y 0 ⊂ Z∗ are the annihilators of X, Y , respec-

tively;

(iii) (X⊥, Y ⊥) is a Fredholm pair, where X⊥, Y ⊥ ⊂ Z are the orthogonal complements of

X, Y , respectively;

(iv) (PX⊥ , PY ) is a Fredholm pair of projections.

In this case, one has

dim(X ∩ Y ) = codim(X0 + Y 0) = codim(X⊥ + Y ⊥) = dim kerPX⊥PY ;

codim(X + Y ) = dim(X0 ∩ Y 0) = dim(X⊥ ∩ Y ⊥) = codim rangePX⊥PY .

In particular,

ind(X, Y ) = − ind(X0, Y 0) = − ind(X⊥, Y ⊥) = ind(PX⊥ , PY ).

153

We return to the Cauchy data spaces. Let Π+(A) be the orthogonal projection H(A)�

H−1/2[0,∞)(A) and P (D+) be the orthogonal projection H(A)� C+

max. Let T := Π+(A)P (D+) :

C+max → H

−1/2[0,∞)(A). The following is a quick consequence of Propositions 6.1.7 and 6.1.9.

Theorem 6.1.10. T is a Fredholm operator and ind T = indD+APS.

6.1.5 L2-situation

We define the L2-Cauchy data space C+ := C+max ∩ L2(∂M,E+|∂M). One can apply the idea

of “criss-cross reduction” in [19] to show that C+ is a closed subspace of L2(∂M,E+|∂M).

We briefly present this argument. First, there exists a closed subspace V ⊂ H(A), such

that C+max can be written as a direct sum of transversal (not necessarily orthogonal) pair of

subspaces

C+max = (H

1/2(−∞,0)(A) ∩ C+

max) + V.

Let π+ (resp. π−) be the projection of V onto H−1/2[0,∞)(A) (resp. H

1/2(−∞,0)(A)) along H

1/2(−∞,0)(A)

(resp. H−1/2[0,∞)(A)). Then π+ is injective and range π+ = range T is closed. By closed graph

theorem, π+ has a bounded inverse ι+ : range π+ → V . We then have a bounded operator

φ := π− ◦ ι+ : rangeπ+ → rangeπ−. This gives another expression of C+max:

C+max = (H

1/2(−∞,0)(A) ∩ C+

max) + graph(φ). (6.1.5)

Let φ be the restriction of φ to L2(∂M,E+|∂M). Then domφ is closed in L2(∂M,E+|∂M).

Viewed as an operator domφ→ L2(∂M,E+|∂M), φ is still bounded. Note that now C+ can

be written as

C+ = (H1/2(−∞,0)(A) ∩ C+

max) + graph(φ).

Since the first summand is finite-dimensional, C+ is closed in L2(∂M,E+|∂M).

Like in Subsection 6.1.4, we define the orthogonal projections

Π+(A) : L2(∂M,E+|∂M)� L2[0,∞)(A) and P (D+) : L2(∂M,E+|∂M)� C+.

154

And let

T := Π+(A)P (D+) : C+ → L2[0,∞)(A).

It is clear that kerT = ker T , and

rangeT = (L2(−∞,0)(A) + C+) ∩ L2

[0,∞)(A)

= (L2(−∞,0)(A) + C+

max) ∩ L2[0,∞)(A)

⊃ (H1/2(−∞,0)(A) + C+

max) ∩ L2[0,∞)(A).

On the other hand, since the L2-norm is stronger than the H-norm on L2[0,∞)(A),

rangeT = (clL2(H1/2(−∞,0)(A)) + C+

max) ∩ L2[0,∞)(A)

⊂ (clH(H1/2(−∞,0)(A)) + C+

max) ∩ L2[0,∞)(A)

⊂ clH(H1/2(−∞,0)(A) + C+

max) ∩ L2[0,∞)(A)

= (H1/2(−∞,0)(A) + C+

max) ∩ L2[0,∞)(A),

where we used Proposition 6.1.7 in the last line. Therefore

rangeT = (H1/2(−∞,0)(A) + C+

max) ∩ L2[0,∞)(A)

= range T ∩ L2[0,∞)(A).

and is a closed subspace of L2[0,∞)(A). Let W be the finite-dimensional orthogonal comple-

ment of range T in H−1/2[0,∞)(A) and let W := W |L2

[0,∞)(A). Then

L2[0,∞)(A) = rangeT + W. (6.1.6)

Taking closure with respect to the H-norm for both sides implies that H−1/2[0,∞)(A) = range T +

W . Hence W = W . It follows that (6.1.6) is a direct sum decomposition. Therefore

codim rangeT = dimW = dim W = codim range T .

To sum up, we obtain an L2-version of Theorem 6.1.10:

Theorem 6.1.11. T is a Fredholm operator and indT = indD+APS.

Corollary 6.1.12. (L2(−∞,0)(A), C+) is a Fredholm pair in L2(∂M,E+|∂M) and

ind(L2(−∞,0)(A), C+) = indD+

APS.

155

6.2 Cauchy data spaces and boundary value problems

6.2.1 Twisted orthogonality of Cauchy data spaces

By Proposition 6.1.9 and Corollary 6.1.12, (L2[0,∞)(A), (C+)0) and (L2

(0,∞)(A]), (C−)0) are

Fredholm pairs in L2(∂M,E+|∂M) and L2(∂M,E−|∂M), respectively. And they satisfy

ind(L2[0,∞)(A), (C+)0) = − ind(L2

(−∞,0)(A), C+),

= ind(L2(−∞,0](A]), C−) = − ind(L2

(0,∞)(A]), (C−)0). (6.2.1)

The following property of Fredholm pairs can be verified easily.

Lemma 6.2.1. Let (X, Y1), (X, Y2) be two Fredholm pairs in a Hilbert space Z. If Y1 ⊂ Y2

and ind(X, Y1) = ind(X, Y2), then Y1 = Y2.

Proposition 6.2.2. Recall that c(τ) induces an isomorphism between L2(∂M,E−|∂M) and

L2(∂M,E+|∂M) (cf. Subsection 3.2.7). Then c(τ)(C−) = (C+)0, c(τ)(C+) = (C−)0.

Proof. We only need to show the first equality. Let v ∈ C−. Then there exists a v ∈ kerD+max

such that R(v) = v. For any u ∈ C+, there again exists a u ∈ kerD+max such that R(u) = u.

By (3.2.29),

0 = (D+maxu; v)L2(M) − (u;D−maxv)L2(M) = (u, c(τ)v)L2(∂M) ⇒ c(τ)v ∈ (C+)0.

Hence c(τ)(C−) ⊂ (C+)0.

Notice that the isomorphism c(τ) maps the Fredholm pair (L2(−∞,0](A]), C−) to the pair

(L1/2[0,∞)(A), c(τ)(C−)). Thus the latter is a Fredholm pair in L2(∂M,E+|∂M) and

ind(L2[0,∞)(A), c(τ)(C−)) = ind(L2

(−∞,0](A]), C−)(6.2.1)

===== ind(L1/2[0,∞)(A), (C+)0).

Using the fact that c(τ)(C−) ⊂ (C+)0 and Lemma 6.2.1, one has c(τ)(C−) = (C+)0.

Remark 6.2.3. In the same way, one can prove that c(τ)(C−max) = (C+max)0, c(τ)(C+

max) =

(C−max)0.

156

Since the pairing between elements of (L2(∂M,E+|∂M))∗ ∼= L2(∂M,E+|∂M) and elements

of L2(∂M,E+|∂M) coincides with the inner product on L2(∂M,E+|∂M), we have (C+)⊥ =

(C+)0. Therefore, we obtian the following L2-decomposition theorem.

Theorem 6.2.4. C+ and c(τ)(C−) are orthogonal complementary subspaces of L2(∂M,E+|∂M).

Similar statement is true for C− and c(τ)(C+).

Consider a bilinear form on L2(∂M,E|∂M) defined by

ω(u,v) := (c(τ)u; v)L2(∂M).

One can check that this is a symplectic form. Then Theorem 6.2.4 indicates the following.

Corollary 6.2.5. The total L2-Cauchy data space C+⊕C− of the total strongly Callias-type

operator D is a Lagrangian subspace of L2(∂M,E|∂M).

Remark 6.2.6. From Remark 6.2.3, one can also show that the total maximal Cauchy data

spaces C+max ⊕ C−max is a Lagrangian subspace of H(A)⊕ H(A]).

6.2.2 Cauchy data spaces as elliptic boundary conditions

In this subsection, we discuss an elliptic boundary condition induced by Cauchy data spaces.

Let

C+1/2 := C+

max ∩H1/2A (∂M,E+|∂M), C−1/2 := C−max ∩H

1/2

A] (∂M,E−|∂M).

Using again the expression (6.1.5) of C+max, like in Subsection 6.1.5, we have

C+1/2 = (H

1/2(−∞,0)(A) ∩ C+

max) + graph(φ1/2), (6.2.2)

where φ1/2 : domφ1/2 → H−1/2(−∞,0)(A) is the restriction of φ to H

1/2[0,∞)(A), and it is still a

bounded operator. So C+1/2 is a closed subspace of H(A), and c(τ)(C+

1/2) is a closed subspace

of H(A]). Similarly, c(τ)(C−1/2) is a closed subspace of H(A).

157

Lemma 6.2.7. (H−1/2(−∞,0)(A), C+

1/2) is a Fredholm pair in H(A) and

ind(H−1/2(−∞,0)(A), C+

1/2) = ind(H1/2(−∞,0)(A), C+

max).

Proof. First,

H−1/2(−∞,0)(A) ∩ C+

1/2 = H−1/2(−∞,0)(A) ∩ C+

max ∩H1/2A (∂M,E+|∂M)

= H1/2(−∞,0)(A) ∩ C+

max.

By (6.2.2),

H−1/2(−∞,0)(A) + C+

1/2 = H−1/2(−∞,0)(A) + graph(φ1/2) = H

−1/2(−∞,0)(A)⊕ domφ1/2,

which is closed in H(A). Then

dim H(A)/(H−1/2(−∞,0)(A) + C+

1/2) = dimH1/2[0,∞)(A)/ domφ1/2

= dimH−1/2[0,∞)(A)/ domφ = dim H(A)/(H

1/2(−∞,0)(A) + C+

max).

The lemma is proved.

Remark 6.2.8. One also has that (H−1/2(−∞,0](A]), C

−1/2) is a Fredholm pair in H(A]) and

ind(H−1/2(−∞,0](A

]), C−1/2) = ind(H1/2(−∞,0](A

]), C−max).

Theorem 6.2.9. c(τ)(C−1/2) is an elliptic boundary condition for D+, whose adjoint boundary

condition is c(τ)(C+1/2) and indD+

c(τ)(C−1/2

)= 0.

Proof. From the discussion above, c(τ)(C−1/2) ⊂ H1/2A (∂M,E+|∂M) and is a boundary condi-

tion. By (3.3.1), to prove the adjoint property, it suffices to show that c(τ)(C+1/2) = (C−1/2)0.

Note that c(τ) maps the Fredholm pair (H−1/2(−∞,0)(A), C+

1/2) of H(A) to a Fredholm pair

(H−1/2(0,∞)(A]), c(τ)(C+

1/2)) of H(A]) and

ind(H−1/2(0,∞)(A

]), c(τ)(C+1/2)) = ind(H

1/2(−∞,0)(A), C+

max)

c(τ)==== ind(H

1/2(0,∞)(A

]), c(τ)(C+max))

Remark 6.2.3========= ind(H

1/2(0,∞)(A

]), (C−max)0)

= − ind(H1/2(−∞,0](A

]), C−max)Remark 6.2.8

========= − ind(H−1/2(−∞,0](A

]), C−1/2)

= ind(H−1/2(0,∞)(A

]), (C−1/2)0).

158

One then uses the argument as in the proof of Proposition 6.2.2 to show that c(τ)(C+1/2) ⊂

(C−1/2)0. Therefore c(τ)(C+1/2) = (C−1/2)0 by Lemma 6.2.1.

By Theorem 6.2.4, one gets

c(τ)(C−1/2) ⊂ c(τ)(C−) = (C+)⊥ =⇒ c(τ)(C−1/2) ∩ C+

= c(τ)(C−1/2) ∩ C+max ∩ L2(∂M,E+|∂M)

= c(τ)(C−1/2) ∩ C+max = {0}.

So kerD+

c(τ)(C−1/2

)= {0}. Also kerD−

c(τ)(C+1/2

)= {0}. Hence

indD+

c(τ)(C−1/2

)= dim kerD+

c(τ)(C−1/2

)− dim kerD−

c(τ)(C+1/2

)= 0.

159

160

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