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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
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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.
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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
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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 ;
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(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.
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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′).
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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.
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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
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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.
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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].
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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).
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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′#).
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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.
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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.
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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
106
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
109
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
115
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.
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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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