Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps
We consider a complete Riemannian manifold, which consists of a compact interior and one or more 𝜑-cusps: infinitely long ends of a type that includes cylindrical ends and hyperbolic cusps. Here, 𝜑 is a function of the radial coordinate that describes the shape of such an end. Given an action by a c...
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| description | We consider a complete Riemannian manifold, which consists of a compact interior and one or more 𝜑-cusps: infinitely long ends of a type that includes cylindrical ends and hyperbolic cusps. Here, 𝜑 is a function of the radial coordinate that describes the shape of such an end. Given an action by a compact Lie group on such a manifold, we obtain an equivariant index theorem for Dirac operators, under conditions on 𝜑. These conditions hold in the cases of cylindrical ends and hyperbolic cusps. In the case of cylindrical ends, the cusp contribution equals the delocalised 𝜂-invariant, and the index theorem reduces to Donnelly's equivariant index theory on compact manifolds with boundary. In general, we find that the cusp contribution is zero if the spectrum of the relevant Dirac operator on a hypersurface is symmetric around zero.
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| first_indexed | 2026-03-13T10:15:08Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 023, 32 pages
Spectral Asymmetry and Index Theory
on Manifolds with Generalised Hyperbolic Cusps
Peter HOCHS a and Hang WANG b
a) Institute for Mathematics, Astrophysics and Particle Physics, Radboud University,
PO Box 9010, 6500 GL Nijmegen, The Netherlands
E-mail: p.hochs@math.ru.nl
b) School of Mathematical Sciences, East China Normal University,
No. 500, Dong Chuan Road, Shanghai 200241, P.R. China
E-mail: wanghang@math.ecnu.edu.cn
Received June 22, 2022, in final form March 28, 2023; Published online April 20, 2023
https://doi.org/10.3842/SIGMA.2023.023
Abstract. We consider a complete Riemannian manifold, which consists of a compact
interior and one or more φ-cusps: infinitely long ends of a type that includes cylindrical
ends and hyperbolic cusps. Here φ is a function of the radial coordinate that describes the
shape of such an end. Given an action by a compact Lie group on such a manifold, we obtain
an equivariant index theorem for Dirac operators, under conditions on φ. These conditions
hold in the cases of cylindrical ends and hyperbolic cusps. In the case of cylindrical ends,
the cusp contribution equals the delocalised η-invariant, and the index theorem reduces to
Donnelly’s equivariant index theory on compact manifolds with boundary. In general, we
find that the cusp contribution is zero if the spectrum of the relevant Dirac operator on
a hypersurface is symmetric around zero.
Key words: equivariant index; Dirac operator; noncompact manifold; cusp
2020 Mathematics Subject Classification: 58J20; 58D19
1 Introduction
Index theory on noncompact manifolds
Index theory on noncompact, complete manifolds comes up naturally in several different con-
texts. Well-known results include:
(1) the Gromov–Lawson relative index theorem [23] for differences of Dirac operators that are
invertible, and equal, outside compact sets;
(2) the Atiyah–Patodi–Singer (APS) index theorem [6, 21] on compact manifolds with bound-
ary, where an approach is to attach a cylindrical end to the boundary to obtain a complete
manifold without boundary; and
(3) index theorems on noncompact locally symmetric spaces, and manifolds with cusps mod-
elled on such spaces. Some important results include [13, 33, 34, 35, 39].
Other important areas, which are not considered in this paper, are index theory of Callias-type
operators [1, 3, 16, 19, 20, 28], index theory where a group action is used to define an equivariant
index of operators that are not Fredholm in the traditional sense, see, e.g., [5, 17], and index
theory with values in the K-theory of C∗-algebras, see, e.g., [14].
In this paper and in [27], we work towards a common framework for studying the three
types of index problems mentioned. In [27], we considered a complete Riemannian manifold M ,
mailto:p.hochs@math.ru.nl
mailto:wanghang@math.ecnu.edu.cn
https://doi.org/10.3842/SIGMA.2023.023
2 P. Hochs and H. Wang
a Clifford module S → M and a Dirac operator D on Γ∞(S) that is “invertible at infinity” in
the following sense. We assumed that there are a compact subset Z ⊂M and a b > 0 such that
for all s ∈ Γ∞
c (S) supported outside Z,
∥Ds∥L2 ≥ b∥s∥L2 .
Then D is Fredholm as an unbounded operator on L2(S) with a suitable domain [2, 23].
We assumed thatM has a warped product structure outside Z (the φ-cusps as below, without
assumptions on the function φ). Furthermore, we considered an action by a compact group G
on M and S, commuting with D, and a group element g ∈ G. The main result in [27] is an
expression for the value at g of the equivariant index of such an operator, as an Atiyah–Segal–
Singer-type contribution from Z and a contribution from outside Z. This implies an equivariant
version of the second index theorem mentioned at the start, and an equivariant version of the
first for manifolds with the appropriate warped product form at infinity.
In this paper, we give an expression for the contribution from outside Z for manifolds with
specified shapes outside Z, including cylindrical ends and hyperbolic cusps.
φ-cusps
More specifically, let M be a complete Riemannian manifold. Suppose a compact Lie group G
acts isometrically on M , that S → M is a G-equivariant Clifford module, and that D is a G-
equivariant Dirac operator on sections of S. Let a > 0, and let φ ∈ C∞(a,∞). We assume that
there is a compact, G-invariant subset Z ⊂M with smooth boundary N , such that C :=M \Z
is G-equivariantly isometric to the product N × (a,∞), with the Riemannian metric
e2φ
(
BN + dx2
)
,
where BN is a G-invariant Riemannian metric on N , and x is the coordinate in (a,∞). Then we
say that M has φ-cusps. (The results in this paper extend to cases where different functions φ
are used on different connected components of N .)
A natural form of a Dirac operator on C is
e−φc0
(
∂
∂x
)(
∂
∂x
+DN +
dim(M)− 1
2
φ′
)
for a Dirac operator DN on S|N , where c0 is the Clifford action for the product metric BN +dx2
on C. We assume D|C has this form, and, initially, that DN is invertible. Then D2
N ≥ b2 for
some b > 0.
We say that M has weakly admissible φ-cusps if
(1) φ is bounded above, and
(2) there is an α > 0 such that |φ′(x)| ≤ b− α for large enough x,
and strongly admissible φ-cusps if
(1) limx→∞ φ(x) = −∞, and
(2) limx→∞ φ′(x) = 0.
For example, if φ(x) = −µ log(x) for µ ∈ R, then
� M is complete if and only if µ ≤ 1,
� M has weakly admissible φ-cusps if and only if µ ≥ 0, and
� M has strongly admissible φ-cusps if and only if µ > 0.
Furthermore, M has finite volume if and only if µ > 1/ dim(M), but this is not directly relevant
to us here.
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 3
Results
If M has weakly admissible φ-cusps, then our main result, Theorem 2.16 states that D is
Fredholm, and the value of its equivariant index at g ∈ G is
indexG(D)(g) =
∫
Zg
ASg(D)− 1
2
ηφg
(
D+
N
)
. (1.1)
Here
� Zg is the fixed point set of g in Z,
� ASg(D) is the Atiyah–Segal–Singer integrand for D, and
� ηφg
(
D+
N
)
is the φ-cusp contribution associated to D+
N (the restriction of DN to even-graded
sections).
The cusp contribution ηφg
(
D+
N
)
equals
ηφg
(
D+
N
)
= lim
a′↓a
∫ ∞
0
∑
λ∈spec(D+
N )
sgn(λ) tr(g|ker(D+
N−λ))Fφ(a
′, s, |λ|) ds (1.2)
for a function Fφ depending on φ. (Here a is as in the definition of C ∼= N × (a,∞).) This
function is expressed in terms of eigenfunctions of a Sturm–Liouville (or Schrödinger) operator
on the half-line (0,∞), with Dirichlet boundary conditions at 0. See Definition 2.14 for details.
If M has strongly admissible φ-cusps, then this operator has discrete spectrum. If M only
has weakly admissible φ-cusps, then it may have a continuous spectral decomposition, and its
spectral measure also appears in the expression for ηφg
(
D+
N
)
.
There is also a version of (1.1) where the integral over Zg is replaced by an integral over Mg,
if this converges, and the limit in (1.2) is replaced by the limit a′ → ∞.
A possibly interesting feature of the cusp contribution ηφg
(
D+
N
)
is that it equals zero if the
spectrum of D+
N has the equivariant symmetry property that
tr
(
g|ker(D+
N−λ)
)
= tr
(
g|ker(D+
N+λ)
)
(1.3)
for all λ ∈ R. If g = e, then this is exactly symmetry of the spectrum with respect to reflection
in 0, including multiplicities. It is immediate from (1.2) that ηφg
(
D+
N
)
= 0 if the spectrum of D+
N
has this property. As noted in [6], the classical η-invariant also vanishes if D+
N has symmetric
spectrum. So it seems that different ways of measuring spectral asymmetry are relevant to index
theory on manifolds of the type we consider.
After we prove (1.1), we compute the function Fφ in (1.2), and hence the cusp contribu-
tion ηφg
(
D+
N
)
in the case where φ(x) = 0. Then C = N × (a,∞) is a cylindrical end, and
Proposition 4.4 states that η0g
(
D+
N
)
is the equivariant η-invariant [21] of D+
N . This computa-
tion is a spectral version of the geometric computation in [27, Section 5]. Then (1.1) becomes
Donnelly’s equivariant version of the APS index theorem [21].
The eigenfunctions of the Sturm–Liouville operator involved in the expression (1.2) are known
explicitly in several cases besides the cylinder case, such as φ(x) = − log(x)/2 and the hyperbolic
cusp case φ(x) = − log(x). Nevertheless, it seems to be a nontrivial problem to evaluate the cusp
contribution (1.2) explicitly, even when these eigenfunctions are known. Concrete consequences
and special cases of the main Theorem 2.16 are:
(1) the Fredholm property of D,
(2) the fact that, for φ = 0, the cusp contribution η0g
(
D+
N
)
is the delocalised η-invariant of D+
N ,
(3) the fact that, for general φ, the cusp contribution vanishes if the spectrum of D+
N has the
symmetry property (1.3).
Related index theorems were obtained for manifolds with ends of the form N × (a,∞) with
metrics of the form BN,x + dx2, where now BN,x is a Riemannian metric on N depending on
4 P. Hochs and H. Wang
x ∈ (a,∞). In many cases, this family of metrics on N has the form BN,x = ρ2(x)BN , for a fixed
Riemannian metric BN on N and a function ρ on (a,∞). Results in this context include the
ones in [8, 9, 10, 40, 42]. We discuss the relations between these results and (1.1) in Section 2.5.
2 Preliminaries and result
Throughout this paper, M is a p-dimensional Riemannian manifold with p even, and S =
S+ ⊕ S− → M is a Z/2-graded Hermitian vector bundle. We denote the Riemannian density
on M by dm. We also assume that a compact Lie group G acts smoothly and isometrically
on M , that S is a G-equivariant vector bundle and that the action on G preserves the metric
and grading on S. We fix, once and for all, an element g ∈ G.
2.1 φ-cusps
Definition 2.1. The manifold M has (G-invariant) φ-cusps if there are
� a G-invariant compact subset Z ⊂M with smooth boundary N , and
� a number a ≥ 0 and a function φ ∈ C∞(a,∞),
such that
� there is a G-equivariant isometry from C :=M \ Z onto the manifold N × (a,∞) with the
metric
Bφ := e2φ
(
BN + dx2
)
, (2.1)
where BN is the restriction of the Riemannian metric to N and x is the coordinate in
(a,∞), and
� this isometry has a continuous extension to a map C → N × [a,∞), which maps N onto
N × {a}.
The manifold M has strongly admissible φ-cusps or strongly admissible cusps if, in addition,
lim
x→∞
φ(x) = −∞, lim
x→∞
φ′(x) = 0. (2.2)
Remark 2.2. There is no loss of generality in assuming that a = 0 in Definition 2.1. However,
in examples, it may be that φ arises as the restriction to (a,∞) of a function naturally defined
on say (0,∞). Allowing nonzero a then means that we do not need to shift these functions
over a to obtain a function on (0,∞). This does not matter for the results.
Remark 2.3. In Definition 2.1, the hypersurface N may be disconnected. Let N1, . . . , Nk be
its connected components. All results below generalise to the case where the Riemannian metric
on N × (a,∞) is of the form e2φj (BN +dx2) on Nj × (a,∞), for φj ∈ C∞(a,∞) depending on j.
This generalisation is straightforward, and we do not work out details here.
Lemma 2.4. If M has φ-cusps, then it is complete if and only if∫ ∞
a+1
eφ(x) dx = ∞. (2.3)
Proof. For any smooth, increasing γ : [0, 1) → [a + 1,∞) such that γ(0) = a + 1, and any
n ∈ N , consider the curve γn in C given by γn(t) = (n, γ(t)). Then the length of γn is
sgn(γ′)
∫ 1
0
γ′(s)eφ(γ(t)) dt = sgn(γ′)
∫ γ(1)
a+1
eφ(x) dx.
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 5
Here γ(1) := limt→1 γ(t) exists because γ is increasing. The lengths of all such curves with
γ(1) = ∞ are infinite if and only if (2.3) holds. This implies that (2.3) is equivalent to the
condition that any curve in C that goes to infinity has infinite length. Compare also the proof
of Theorem 1 in [36]. ■
From now on, suppose that M is complete and has φ-cusps, and let Z, N , a, φ and BN be
as in Definition 2.1.
It will not directly be important for our results if M has finite or infinite volume. But
because of the relevance of finite-volume manifolds, we note the following fact. Recall that p is
the dimension of M .
Lemma 2.5. The manifold M has finite volume if and only if∫ ∞
a
epφ(x) dx <∞.
Proof. The Riemannian density for Bφ is volBφ = epφ volBN
⊗dx. ■
Lemmas 2.4 and 2.5 are well known and our results do not logically depend on them. We
chose to include their short proofs for the sake of completeness and to illustrate properties of
φ-cusps.
Example 2.6. Suppose that M has φ-cusps, with φ(x) = −µ log(x) for µ ∈ R. Then
� M has strongly admissible cusps if and only if µ > 0,
� M is complete if and only if µ ≤ 1,
� M has finite volume if and only if µ > 1/p.
All three conditions hold in the case µ = 1 of hyperbolic cusps. In the case µ = 0 of a cylindrical
end,M has infinite volume and does not have strongly admissible cusps. But thenM has weakly
admissible cusps as in Definition 2.8 below, which is sufficient for our purposes.
2.2 Dirac operators on φ-cusps
From now on, we suppose that S is a G-equivariant Clifford module, which means that there is
a G-equivariant vector bundle homomorphism c : TM → End(S), with values in the odd-graded
endomorphisms, such that for all v ∈ TM ,
c(v)2 = −∥v∥2 IdS .
Let ∇ be a G-invariant connection on S that preserves the grading. Suppose that for all vector
fields v and w on M ,
[∇v, c(w)] = c
(
∇TM
v w
)
, (2.4)
where ∇TM is the Levi-Civita connection. Consider the Dirac operator
D : Γ∞(S)
∇−→ Γ∞(S ⊗ T ∗M) ∼= Γ∞(S ⊗ TM)
c−→ Γ∞(S). (2.5)
It is odd with respect to the grading on S; we denote its restrictions to even- and odd-graded
sections by D±, respectively.
Suppose that we have a G-equivariant vector bundle isomorphism S|C ∼= S|N × (a,∞) →
N × (a,∞).
6 P. Hochs and H. Wang
We will assume that D has a natural form on M \ Z, (2.7) below. This assumption is
motivated by a special case, Proposition 2.7, which we discuss now.
Let B0 := BN + dx2 be the product metric on N × (a,∞). Then
c0 := e−φc|C : T (N × (a,∞)) → End(S|C)
is a Clifford action for the metric B0. Let {e1, . . . , ep} be a local orthonormal frame for TM
with respect to B0, with ep = ∂
∂x , and {e1, . . . , ep−1} a local orthonormal frame for TN with
respect to BN . (The objects that follow are defined globally by their expressions in terms of
this frame, because they do not depend on the orthonormal frame.)
Because p is even, the operator
γ := (−i)p(p+1)/2c0(e1) · · · c0(ep)
defines a Z/2-grading on S, and c0(v) is odd for this grading for all vector fields v on C. We
suppose that the given grading on S equals γ on S|C . Let ∇0 be a Clifford connection on S|C
with respect to B0 and c0 (i.e., a Hermitian connection satisfying (2.4) for B0 and c0), and
suppose that it preserves the grading γ, and that ∇ ∂
∂x
= ∂
∂x .
Let
DN := −c0(ep)
p−1∑
j=1
c0(ej)∇0
ej .
This is a Dirac operator on S|N with respect to the Clifford multiplication
cN (v) := −c0(ep)c0(v)
for v ∈ TN .
Proposition 2.7. There is a Clifford connection on S|C , with respect to the Clifford action c
and the metric (2.1), such that the resulting Dirac operator is
e−φc0(ep)
(
∂
∂x
+DN +
p− 1
2
φ′
)
. (2.6)
This fact follows from a standard expression for conformal transformations of Dirac operators.
(See, e.g., the proof of Proposition 1.3 in [25] in the Spinc case.) We summarise the arguments
in Appendix A for the sake of completeness.
From now on, we make the two assumptions that
(1) for some Dirac operatorDN on S|N that preserves the grading, and some grading-reversing,
G-equivariant, isometric vector bundle endomorphism σ : S|N → S|N that anti-commutes
with DN ,
D|M\Z = e−φσ
(
∂
∂x
+DN +
p− 1
2
φ′
)
, (2.7)
(2) the Dirac operator DN is invertible.
The first assumption is satisfied for a natural choice of Clifford connection on S, by Proposi-
tion 2.7. We indicate how to remove the second assumption in Section 5.
Because DN is invertible, there is a b > 0 such that D2
N ≥ b2. For the purposes of our index
theorem, the strong admissibility condition in Definition 2.1 may be weakened to the following.
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 7
Definition 2.8. The manifold M has weakly admissible φ-cusps or weakly admissible cusps
(with respect to DN ) if φ is bounded above, and there are a′ ≥ a and α > 0 such that for all
x ∈ (a′,∞),
|φ′(x)| ≤ b− α. (2.8)
Example 2.9. If M has φ-cusps with φ(x) = −µ log(x) as in Example 2.6, then M has weakly
admissible cusps if and only if µ ≥ 0. This includes the case µ = 0 of a cylindrical end,
relevant to the Atiyah–Patodi–Singer index theorem. For cusp metrics of this form, we have the
implications
finite volume ⇒ strongly admissible cusps ⇒ weakly admissible cusps.
For general metrics of the form (2.1), only the second implication always holds.
Example 2.10. If φ is periodic, then φ-cusps are never strongly admissible. But if φ(x) =
φ̃(sin(x)) for some φ̃ ∈ C∞([−1, 1]), then φ-cusps are weakly admissible if |φ̃′| < b.
2.3 Spectral theory for Sturm–Liouville operators
Let q be a real-valued, continuous function on the closed half-line [0,∞). A crucial role will be
played by the spectral theory of Sturm–Liouville operators of the form
∆q := − d2
dy2
+ q,
on [0,∞). We briefly review this theory here, and refer to [29, 41] for details.
For ν ∈ C, let θν ∈ C∞([0,∞)) be the unique solution of
∆qθν = νθν ,
such that
θν(0) = 0, θ′ν(0) = 1. (2.9)
The theory extends to more general boundary conditions, but we will only use the Dirichlet
case. For f ∈ C∞
c (0,∞) and ν ∈ R, define the generalised Fourier transform
Fq(f)(ν) :=
∫ ∞
0
f(y)θν(y) dy.
For a function ρ : R → C, let L2(R,dρ) be the space of square-integrable functions with respect
to the measure dρ, in the sense of Stieltjes integrals. Note that this measure may have singular
points, if ρ is discontinuous.
Theorem 2.11. There exists a unique increasing function ρ : R → R with the following prop-
erties:
(a) The map f 7→ Fq(f) extends to a unitary isomorphism from L2([0,∞)) onto L2(R, dρ).
(b) For all continuous f ∈ L2([0,∞)) such that the integrands on the right-hand side are
well-defined and the integral converges uniformly in y in compact intervals,
f(y) =
∫
R
Fq(f)(ν)θν(y) dρ(ν). (2.10)
8 P. Hochs and H. Wang
See, for example, [29, Theorems 2.1.1 and 2.1.2].
The spectral measure dρ can be computed as follows.
Proposition 2.12. For ν ∈ C, let θ1(−, ν) and θ2(−, ν) be the solutions of ∆qθj = νθj on
[0,∞), such that
θ1(0, ν) = 0, θ′1(0, ν) = 1, θ2(0, ν) = −1, θ′2(0, ν) = 0,
where the prime denotes the derivative with respect to the first variable. There is a function f
on the upper half-plane in C, with negative imaginary part, such that for all ν in the upper
half-plane, θ2(−, ν) + f(ν)θ1(−, ν) ∈ L2([0,∞)). And for all ν ∈ R,
ρ(ν) =
1
π
lim
δ↓0
∫ ν
0
− Im(f(ν ′ + iδ)) dν ′. (2.11)
See [29, Theorem 2.4.1], or [41, Chapter 3], in particular Lemma 3.3 and the theorem on
p. 60. The factor 1
π in (2.11) corresponds to the same factor in the inversion formula in the
theorem on p. 60 of [41], which is not present in (2.10).
2.4 An index theorem for manifolds with admissible φ-cusps
We suppose, as before, that M is complete and has weakly admissible φ-cusps. Let a be as in
Definition 2.1. Consider the function ξ : (a,∞) → (0,∞) defined by
ξ(x) :=
∫ x
a
eφ(x
′) dx′. (2.12)
Then ξ is injective because its derivative is positive, and surjective by Lemma 2.4. For λ ∈ R,
define the functions q±λ ∈ C∞(0,∞) by
q±λ (y) := λ
(
(λ± φ′)e−2φ
)
◦ ξ−1. (2.13)
(We only use q+λ in the current section, but q−λ will also be used in Section 3.7.) In the case
where q = q±λ , we write
∆±
λ := ∆q±λ
. (2.14)
We then write ρλ,± for the function ρ in Theorem 2.11, and θλ,±ν for the function θν in Theo-
rem 2.11.
Example 2.13. Suppose that φ(x) = −µ log(x), for µ ∈ R. If µ = 1, then
q±λ (y) = a2λ2e2y ∓ aλey.
If µ ̸= 1, then
q±λ (y) = λ2
(
(1− µ)y + a1−µ
) 2µ
1−µ ∓ λµ
(
(1− µ)y + a1−µ
) 2µ−1
1−µ .
For a finite-dimensional vector space V with a given representation by G, we write tr(g|V )
for the trace of the action by g on V .
Definition 2.14. The g-delocalised φ-cusp contribution associated toD+
N and a number a′ > a is
ηφg
(
D+
N , a
′) = 2e−pφ(a′)
∫ ∞
0
∑
λ∈spec(D+
N )
sgn(λ) tr
(
g|ker(D+
N−λ)
)
(2.15)
×
∫
R
e−sνθ|λ|,+ν (ξ(a′))
((
θ|λ|,+ν
)′
(ξ(a′)) + |λ|e−φ(a′)θ|λ|,+ν (ξ(a′))
)
dρ|λ|,+(ν) ds,
if the right-hand side converges.
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 9
We will see in Theorem 2.16 that (2.15) indeed converges in the situations we consider.
The notation ηφg and the factor 2 in the definition are motivated by the fact that η0e is the usual
η-invariant of [6], and, more generally, η0g is the delocalised η-invariant [21]. See Proposition 4.4.
Definition 2.15. The spectrum of D+
N is g-symmetric if for all λ ∈ R,
tr
(
g|ker(D+
N−λ)
)
= tr
(
g|ker(D+
N+λ)
)
.
Note that if g = e, then the spectrum of DN is e-symmetric precisely if it is symmetric around
zero, including multiplicities.
For a G-equivariant, odd-graded, self-adjoint Fredholm operator F on L2(S), we denote its
restrictions to even- and odd-graded sections by F+ and F−, respectively. Then
indexG(F ) := [ker(F+)]− [ker(F−)] ∈ R(G)
is the classical equivariant index of F+, in the representation ring R(G) of G. We denote the
value of its character at g by
indexG(F )(g) = tr
(
g|ker(F+)
)
− tr
(
g|ker(F−)
)
.
Let ASg(D) be the Atiyah–Segal–Singer integrand associated to D, see, for example, [15, Theo-
rem 6.16] or [7, Theorem 3.9]. It is a differential form of mixed degree on the fixed-point set Mg
of g. The connected components of Mg may have different dimensions, and the integral of
ASg(D) over Mg is defined as the sum over these connected components of the integral of the
component of the relevant degree.
Theorem 2.16 (index theorem on manifolds with admissible cusps). Suppose thatM has weakly
admissible φ-cusps, that D|C = Dφ as in Proposition 2.7, and that DN is invertible. Then D is
Fredholm, the cusp contribution (2.15) converges for all a′ > a, and
indexG(D)(g) =
∫
Zg∪(Ng×(a,a′])
ASg(D)− 1
2
ηφg
(
D+
N , a
′). (2.16)
Furthermore, lima′↓a η
φ
g
(
D+
N , a
′) converges, and
indexG(D)(g) =
∫
Zg
ASg(D)− 1
2
lim
a′↓a
ηφg
(
D+
N , a
′). (2.17)
If
∫
Mg ASg(D) converges, then lima′→∞ ηφg
(
D+
N , a
′) converges, and
indexG(D)(g) =
∫
Mg
ASg(D)− 1
2
lim
a′→∞
ηφg
(
D+
N , a
′). (2.18)
If the spectrum of D+
N is g-symmetric, then ηφg
(
D+
N , a
′) = 0 for all a′ > a. If M has strongly
admissible φ-cusps, then ∆+
|λ| has discrete spectrum for all λ, so the integral over ν in (2.15)
becomes a sum.
The case (2.18) of Theorem 2.16 applies in some relevant special cases. The following fact
follows from Proposition 4.4. This also follows from the fact that the Â-form on the cylinder
N × (a,∞) is zero.
Lemma 2.17. In the setting of Theorem 2.16, if φ = 0, then (2.18) applies.
The case (2.18) also applies if Mg is compact; this is equivalent to g having no fixed points
on N .
10 P. Hochs and H. Wang
Lemma 2.18. In the setting of Theorem 2.16, if Mg is compact, then (2.18) applies.
IfM has strongly admissible cusps, so that ∆+
|λ| has discrete spectrum, and we assume for sim-
plicity that the eigenspaces are one-dimensional, then unitarity of Fq+|λ|
implies that the mea-
sure with respect to dρ
|λ|,+
ν of every point ν ∈ spec(∆+
|λ|) is 1/∥θ
|λ|,+
ν ∥2L2 . Then (2.15) becomes
ηφg
(
D+
N , a
′) = 2e−pφ(a′)
∫ ∞
0
∑
λ∈spec(D+
N )
sgn(λ) tr
(
g|ker(D+
N−λ)
)
×
∑
ν∈spec(∆+
|λ|)
e−sν
∥θ|λ|,+ν ∥2
L2
θ|λ|,+ν (ξ(a′))
((
θ|λ|,+ν
)′
(ξ(a′)) + |λ|e−φ(a′)θ|λ|,+ν (ξ(a′))
)
ds.
It is generally not possible to normalise the eigenfunctions θ
|λ|,+
ν so that their L2-norms are 1,
because they should satisfy (2.9).
Remark 2.19. The expression (2.15) can be extended to a′ = a, but then it equals zero because
the functions θ
|λ|,+
ν satisfy (2.9). So the limit lima′↓a η
φ
g
(
D+
N , a
′) on the right-hand side of (2.17)
is generally different from the version of (2.15) with a′ = a. See Remark 4.5 for an example,
and also [27, Remarks 2.4 and 5.12].
Remark 2.20. As is the case for the classical η-invariant, the cusp contribution (2.15) measures
(an equivariant version of) spectral asymmetry, in the sense that it is zero if the spectrum of D+
N
is g-symmetric. It is intriguing that even in this more general setting, symmetry or asymmetry of
the spectrum of DN determines if a contribution “from infinity” is required in index theorems on
manifolds of the type we consider. Spectral asymmetry of D+
N can be measured in different ways,
and apparently, the cusp shape function φ determines what measure of spectral asymmetry is
relevant for an index theorem on M .
2.5 Relations to other results
In some cases, the cusp metric (2.1) can be transformed to a metric of the form ρ2BN +du2, for
a function ρ of a radial coordinate u. This helps to clarify the relations between Theorem 2.16
and the results in [8, 9, 10, 40].
Suppose that φ′ has no zeroes. Let ρ be a positive, smooth function defined on (ã, b̃), with
ã := ρ−1
(
eφ(a)
)
and b̃ := limx→∞ ρ−1
(
eφ(x)
)
. Suppose that, on this interval,
ρ′ = φ′ ◦ φ−1 ◦ log(ρ). (2.19)
Then the map (n, u) 7→
(
n, φ−1(log(ρ(u)))
)
is an isometry from N ×
(
ã, b̃
)
, with Riemannian
metric ρ2BN + du2, onto N × (a,∞), with Riemannian metric e2φ
(
BN + dx2
)
.
In the case where φ(x) = −µ log(x), for µ ∈ [0, 1], the function
ρ(u) = (1− µ)µ/(µ−1)uµ/(µ−1)
is a solution of (2.19) if µ ̸= 1. If µ = 1, then a solution is ρ(u) = e−u, a well-known alternative
form of hyperbolic metrics.
In his thesis [40], Stern computed the index of the signature operator on finite-volume mani-
folds with cusps N × (ã,∞) with metric ρ2BN +du2, under certain growth conditions on ρ and
its derivatives. These conditions hold in the case where φ(x) = −µ log(x), for µ ∈ [0, 1].
Baier [8] developed index theory on Spin-manifolds with cusps N × (ã,∞), with metrics of
the form ρ2BN + du2. He considered indices of Spin-Dirac operators, and proved
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 11
(1) vanishing of the index if there is a c > 1 such that u−cρ(u) has a positive lower bound for
large u,
(2) an index formula if there is a c < 1 such that u−cρ(u) has an upper bound for large u,
(3) inequalities satisfied by the index in other cases.
In cylinder case µ = 0 and the hyperbolic case µ = 1, the second of Baier’s results applies.
For µ ∈ (0, 1), neither condition in the first two cases holds, so the inequalities for the index in
the third case apply.
In cases where both Theorem 2.16 and Baier’s index formula apply, Baier’s formula has a sim-
pler contribution from outside Z (the classical η-invariant), but a more complicated contribution
from inside Z, involving the dimension of the kernel of D−|Z with Dirichlet boundary conditions
at N .
Ballmann and Brüning [9] considered two-dimensional manifolds of finite area, with cusps of
a type that includes cusps S1 × (ã,∞) with metrics of the form ρ2BN + du2 mentioned above.
Then they obtained an explicit index formula for Dirac operators.
Ballmann, Brüning and Carron [10] studied manifolds with cusps of a different but related
form. In their setting, the manifold’s ends are diffeomorphic to N × (ã,∞) via the gradient flow
of a function with certain properties. They obtain a general index theorem in this setting, and
more concrete index theorems in the case where the metric on the ends is cuspidal. A metric of
the form ρ2BN + du2, with ρ related to φ via (2.19), is cuspidal if φ(x) = − log(x), but not if
φ(x) = −µ log(x) for µ ∈ [0, 1).
Vaillant [42] obtained an index theorem for manifolds with fibred versions of hyperbolic cusps.
In the case of hyperbolic cusps, he showed that the contribution from infinity equals zero. It is
an interesting question to find conditions on φ that imply vanishing of the cusp contribution in
general.
For some results on the spectrum of Dirac or Laplace operators on manifolds with hyperbolic
cusps, see [11] for Spin-Dirac operators and [31] for the Hodge Laplacian. See [32] for the Hodge
Laplacian on manifolds with a generalisation of hyperbolic cusps.
See [33] for finite-dimensionality of the kernels of Dirac operators on finite-volume hyperbolic
locally symmetric spaces. The Fredholm property in Theorem 2.16 is a stronger version of this,
under the condition that DN is invertible. The latter condition can be removed as in Section 5.
In APS-type index theorems where the metric does not have a product structure near the
boundary, the contribution from the boundary can usually be written as a local contribution
involving a transgression form, and a spectral contribution, the usual η-invariant. See, e.g.,
[22, 24, 38], or [18] for an equivariant version. It is an interesting question if the cusp contribution
in Theorem 2.16, defined in terms of the spectrum of the operator D+
N , can be decomposed in
a similar way, into a local contribution from the manifold N and a possibly simpler spectral
contribution.
3 Proof of Theorem 2.16
We first state an index theorem from [27] for Dirac operators that are invertible outside a compact
set, see Theorem 3.1. Then we deduce Theorem 2.16 from this result.
3.1 Dirac operators that are invertible at infinity
We review the geometric setting and notation needed to formulate Theorem 2.2 in [27], this is
Theorem 3.1 below. The text leading up to Theorem 3.1 is a slight reformulation of corresponding
material from [27]. Compared to Theorem 2.16, some assumptions in the main result from [27],
Theorem 3.1, are weaker, but there are additional assumptions on D, such as invertibility at
12 P. Hochs and H. Wang
infinity. Part of the proof of Theorem 2.16 is to show that these additional assumptions hold in
the setting of manifolds with admissible cusps.
We assume in this subsection and the next that M has φ-cusps, but not that these cusps are
weakly or strongly admissible.
We say that D is invertible at infinity if there are a G-invariant compact subset Z ⊂M and
a constant b > 0 such that for all s ∈ Γ∞
c (S) supported in M \ Z,
∥Ds∥L2 ≥ b∥s∥L2 . (3.1)
We assume in this subsection and the next that D is invertible at infinity, and that the set Z
may be taken as in Definition 2.1. (In [27], we took a = 0, now we allow general a ≥ 0 for
consistency with Theorem 2.16.)
For k = 0, 1, 2, . . . , let W k
D(S) be the completion of Γ∞
c (S) in the inner product
(s1, s2)Wk
D
:=
k∑
j=0
(
Djs1, D
js2
)
L2 .
Because D is invertible at infinity, it is Fredholm as an operator
D : W 1
D(S) → L2(S).
See [2, Theorem 2.1] or [23, Theorem 3.2].
Rather than the more specific form (2.7) of D on C =M \ Z, we assume that
D|C = σ
(
f1
∂
∂x
+ f2DN + f3
)
, (3.2)
where
� σ ∈ End(S|N )G interchanges S+|N and S−|N ,
� f1, f2, f3 ∈ C∞(a, a+ 2),
� DN is a G-equivariant, invertible Dirac operator on S|N that preserves the grading.
Consider the vector bundle
SC := S|C → C.
For k ∈ N at least 1, consider the Sobolev space
W k
D(SC) :=
{
s|C ; s ∈W k
D(S)
}
.
(See [12] for other constructions of such Sobolev spaces on manifolds with boundary.) We denote
the subspaces of even- and odd-graded sections by W k
D
(
S±
C
)
, respectively.
If s ∈ W k+1
D (S), then the restriction of Ds ∈ W k
D(S) to the interior of C is determined by
the restriction of s to the interior of C. Since k ≥ 1, the restriction of Ds to the interior of C
has a unique extension to C. So Ds|C is determined by s|C . In this way, D gives a well-defined,
bounded operator from W k+1
D (SC) to W
k
D(SC), which we denote by DC .
BecauseN is compact, DN has discrete spectrum. LetD±
N be the restriction ofDN to sections
of S±|N . Let L2
(
S+|N
)
>0
be the direct sum of these eigenspaces for positive eigenvalues (recall
that 0 is not an eigenvalue). Consider the orthogonal projection
P+ : L2
(
S+|N
)
→ L2
(
S+|N
)
>0
. (3.3)
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 13
(In [27], more general spectral projections are allowed, but this is the one relevant to the current
setting.) We will also use the projection
P− := σ+P
+σ−1
+ : L2
(
S−|N
)
→ σ+L
2
(
S+|N
)
>0
,
where σ+ := σ|(S+|N ). In general, P− is not necessarily a spectral projection for D−
N , but in
the setting that is relevant to us, we have σD+
N = −D−
Nσ, so P
− is projection onto the negative
eigenspaces of D−
N ; see (3.12). We combine P+ and P− to an orthogonal projection
P := P+ ⊕ P− : L2(S|N ) → L2
(
S+|N
)
>0
⊕ σ+L
2
(
S+|N
)
>0
. (3.4)
We will sometimes omit the superscripts ± from P±.
Consider the spaces
W 1
D
(
S+
C ;P
)
:=
{
s ∈W 1
D
(
S+
C
)
;P+(s|N ) = 0
}
,
W 1
D
(
S−
C ; 1− P
)
:=
{
s ∈W 1
D
(
S−
C
)
; (1− P−)(s|N ) = 0
}
,
W 2
D
(
S+
C ;P
)
:=
{
s ∈W 2
D
(
S+
C
)
;P+(s|N ) = 0, (1− P−)
(
D+
Cs|N
)
= 0
}
,
W 2
D
(
S−
C ; 1− P
)
:=
{
s ∈W 2
D
(
S−
C
)
; (1− P−)(s|N ) = 0, P+
(
D−
Cs|N
)
= 0
}
.
(3.5)
Here we use the fact that there are well-defined, continuous restriction/extension maps
W 1
D(SC) → L2(S|N ).
We assume that
� the operators
D+
C : W 1
D
(
S+
C ;P
)
→ L2
(
S−
C
)
(3.6)
and
D−
C : W 1
D
(
S−
C ; 1− P
)
→ L2
(
S+
C
)
(3.7)
are invertible; and
� the operator D−
CD
+
C on L2
(
S+
C
)
, with domain W 2
D
(
S+
C ;P
)
, and the operator D+
CD
−
C on
L2
(
S−
C
)
, with domain W 2
D
(
S−
C ; 1− P
)
, are self-adjoint.
3.2 An index theorem
Under the assumptions in Section 3.1, we state an index theorem using the following ingredients.
For t > 0, let e
−tD−
CD+
C
P be the heat operator for the operator D−
CD
+
C on L2
(
S+
C
)
, with domain
W 2
D
(
S+
C ;P
)
. By [27, Lemma 4.7], the operator e
−tD−
CD+
C
P D−
C has a smooth kernel λPt . The
contribution from infinity associated to DC and a′ ∈ (a, a+ 2) is
Ag(DC , a
′) := −f1(a′)
∫ ∞
0
∫
N
tr(gλPs (g
−1n, a′;n, a′)) dn ds, (3.8)
defined whenever the integral in (3.8) converges.
The following result is a combination of Theorem 2.2 and Corollary 2.3 in [27].
14 P. Hochs and H. Wang
Theorem 3.1 (index theorem for Dirac operators invertible at infinity). For all a′ > a, the
quantity (3.8) converges, and
indexG(D)(g) =
∫
Zg∪(Ng×(a,a′])
ASg(D) +Ag(DC , a
′). (3.9)
Furthermore, the limit lima′↓aAg(DC , a
′) converges, and
indexG(D)(g) =
∫
Zg
ASg(D) + lim
a′↓a
Ag(DC , a
′).
Theorem 2.16 follows from Theorem 3.1 because of the following three propositions.
Proposition 3.2. In the setting of Theorem 2.16, the conditions of Theorem 3.1 hold.
Proposition 3.3. In the setting of Theorem 2.16, the cusp contribution (2.15) converges for all
a′ > a, and
Ag(DC , a
′) = −1
2
ηφg
(
D+
N , a
′).
If the spectrum of DN is g-symmetric, then the right-hand side is zero.
Proposition 3.4. If M has strongly admissible φ-cusps, then the operators ∆+
|λ| have discrete
spectrum for all λ ∈ spec
(
D+
N
)
.
Proof. It follows from the definition (2.13) of q+λ that for all λ ̸= 0,
lim
y→∞
q+|λ|(y) = ∞
if φ satisfies (2.2). It follows that ∆+
|λ| has discrete spectrum, see [29, Theorem 1.3.1, Lemma 3.1.1
and equation (1.3)]. ■
In the rest of this section, we prove Propositions 3.2 (at the end of Section 3.6) and 3.3 (at
the end of Section 3.7), and thus Theorem 2.16. (The case (2.18) follows immediately from the
case (2.16).)
3.3 Transforming Dirac operators
We return to the setting of Section 2, where D is of the form in Proposition 2.7.
To compute (3.8), we use a Liouville-type transformation to relate DC and D2
C to simpler
operators. At the same time, this allows us to transform the Riemannian density of Bφ to
a product density.
Define Φ ∈ C∞(a,∞) by
Φ(x) := e−
p−1
2
φ(x). (3.10)
(Recall that p = dim(M).) Consider the vector bundle S̃C := S|N × (0,∞) → N × (0,∞). For
a section s of S̃C and n ∈ N and x ∈ (a,∞), define
(Ts)(n, x) := Φ(x)s(n, ξ(x))
with ξ as in (2.12).
Lemma 3.5. The operator T defines a G-equivariant unitary isomorphism
T : L2
(
S̃C , volBN
⊗dy
)
→ L2
(
S|C , volBφ
)
.
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 15
Proof. We have volBφ = volBN
⊗epφdx. And by a substitution y = ξ(x), and the equality
ξ′ = eφ, we have for all f ∈ L2((0,∞),dy),
∥Φ · (f ◦ ξ)∥2L2((a,∞),epφdx) =
∫ ∞
a
|f(ξ(x))|2eφ(x) dx = ∥f∥2L2((0,∞),dy). ■
Consider the function h := e−φ◦ξ−1 ∈ C∞(0,∞), and the Dirac operator
D̃C := σ
(
∂
∂y
+ hDN
)
on Γ∞(S̃C). Here y is the coordinate in (0,∞). Let DC be as in Section 3.1, but viewed as
acting on smooth sections. It is given by (2.6).
Lemma 3.6. The following diagram commutes:
Γ∞(SC)
DC // Γ∞(SC)
Γ∞(S̃C)
T
OO
D̃C // Γ∞(S̃C).
T
OO
Proof. This is a direct computation, based on (2.6) and (2.7). ■
It will be convenient to identify S−|N ∼= S+|N via σ in (2.7). Because signs and gradings
are important in what follows, it is worth being explicit about details here. We write τ :=
σ|S+|N × 1: S̃+
C → S̃−
C , and consider the isomorphism
1⊕ τ : S̃+
C ⊕ S̃+
C
∼=−→ S̃C . (3.11)
The operatorDN preserves the grading on S|N ; letD±
N be its restrictions to even and odd-graded
sections, respectively. Because DN anticommutes with σ, we have
D−
N ◦ τ = −τ ◦D+
N . (3.12)
We will use the operators
D̃±
C := ± ∂
∂y
+ hD+
N : Γ∞(S̃+
C
)
→ Γ∞(S̃+
C
)
.
Lemma 3.7. Under the isomorphism (3.11), the operator D̃C corresponds to the operator(
0 D̃−
C
D̃+
C 0
)
on Γ∞(S̃+
C ⊕ S̃+
C
)
.
Proof. Consider the vector bundle endomorphism
Σ :=
(
0 −1
1 0
)
: S̃+
C ⊕ S̃+
C → S̃+
C ⊕ S̃+
C .
Then the diagram
S̃C
σ // S̃C
S̃+
C ⊕ S̃+
C
Σ //
1⊕τ
OO
S̃+
C ⊕ S̃+
C
1⊕τ
OO
commutes.
16 P. Hochs and H. Wang
Because of (3.12) and the fact that the operator ∂
∂y commutes with τ ,(
∂
∂y
+ hDN
)
◦ (1⊕ τ) =
(
∂
∂y + hD+
N 0
0 ∂
∂y + hD−
N
)(
1 0
0 τ
)
=
(
1 0
0 τ
)( ∂
∂y + hD+
N 0
0 ∂
∂y − hD+
N
)
.
We find that under the isomorphism (3.11), the operator D̃C corresponds to
Σ
(
∂
∂y + hD+
N 0
0 ∂
∂y − hD+
N
)
=
(
0 − ∂
∂y + hD+
N
∂
∂y + hD+
N 0
)
. ■
For later use, we record expressions for the operators D̃∓
CD̃
±
C . Consider the Laplace-type
operators
∆± := − ∂2
∂y2
+ e−2φ◦ξ−1(
D+
N
)2 ± (φ′ ◦ ξ−1
)
e−2φ◦ξ−1
D+
N on Γ∞(S̃C).
Lemma 3.8. We have D̃∓
CD̃
±
C = ∆±.
Proof. By a direct computation,
D̃∓
CD̃
±
C = − ∂2
∂y2
+ e−2φ◦ξ−1(
D+
N
)2 ± (φ ◦ ξ−1
)′
e−φ◦ξ−1
D+
N .
The right-hand side equals ∆±, because(
φ ◦ ξ−1
)′
=
(
φ′ ◦ ξ−1
)
e−φ◦ξ−1
. ■
3.4 APS-boundary conditions
Analogously to the Sobolev spaces defined in Section 3.2, we use the APS-type projection (3.3)
to define the following Hilbert spaces. Here we identify S̃−
C
∼= S̃+
C via the map τ , and use the
unitary isomorphism T from Lemma 3.5.
W k
D̃C
(
S̃±
C
)
:= T−1
(
W k
D(S
±
C )
)
,
W 1
D̃C
(
S̃+
C ;P
)
:=
{
s ∈W 1
D̃C
(
S̃+
C
)
;P+(s|N ) = 0
}
,
W 1
D̃C
(
S̃+
C ; 1− P
)
:=
{
s ∈W 1
D̃C
(
S̃+
C
)
; (1− P+)(s|N ) = 0
}
,
W 2
D̃C
(
S̃+
C ;P
)
:=
{
s ∈W 2
D̃C
(
S̃+
C
)
;P+(s|N ) = 0, (1− P+)
(
D̃+
Cs|N
)
= 0
}
,
W 2
D̃C
(
S̃+
C ; 1− P
)
:=
{
s ∈W 2
D̃C
(
S̃+
C
)
; (1− P+)(s|N ) = 0, P+
(
D̃−
Cs|N
)
= 0
}
.(
Analogously to (3.5), we use continuous restriction maps W 1
D̃C
(
S̃C
)
→ L2
(
S|N
)
.
)
By Lem-
ma 3.6, we have the bounded operators
D̃+
C : W 1
D̃C
(
S̃+
C ;P
)
→ L2
(
S̃+
C
)
, (3.13)
D̃−
C : W 1
D̃C
(
S̃+
C ; 1− P
)
→ L2
(
S̃+
C
)
, (3.14)
D̃−
CD̃
+
C : W 2
D̃C
(
S̃+
C ;P
)
→ L2
(
S̃+
C
)
, (3.15)
D̃+
CD̃
−
C : W 2
D̃C
(
S̃+
C ; 1− P
)
→ L2
(
S̃+
C
)
. (3.16)
The results in the previous subsection lead to the following conclusion, which shows that the
conditions of Theorem 3.1 are equivalent to corresponding properties of the operator D̃C in this
setting.
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 17
Proposition 3.9.
(a) The operator D is invertible at infinity in the sense of (3.1) if and only if there are b0 > 0
and u ≥ a such that for all s ∈ Γ∞
c
(
S̃C
)
supported in (u,∞),∥∥D̃Cs
∥∥
L2(S̃C ,volBN
⊗dy)
≥ b0∥s∥L2(S̃C ,volBN
⊗dy)
.
(b) The operator (3.6) is invertible if and only if the operator (3.13) is.
(c) The operator (3.7) is invertible if and only if the operator (3.14) is.
(d) The operator D−
CD
+
C on L2
(
S+
C
)
, with domain W 2
D
(
S+
C ;P
)
, is self-adjoint if and only if
the operator D̃−
CD̃
+
C on L2
(
S̃+
C
)
, with domain W 2
D̃C
(
S̃+
C ;P
)
is.
(e) The operator the operator D+
CD
−
C on L2
(
S−
C
)
, with domain W 2
D
(
S−
C ; 1− P
)
is self-adjoint
if and only if the operator D̃+
CD̃
−
C on L2
(
S̃+
C
)
, with domain W 2
D̃C
(
S̃+
C ; 1− P
)
is.
Proof. Part (a) follows from Lemmas 3.5 and 3.6.
For any s ∈ Γ∞(S̃C), we have
(Ts)|N = Φ(a)s|N ,
because we view N as embedded into C as N ×{a} and into N × [0,∞) as N ×{0}. Therefore,
for any such section, P (s|N ) = 0 if and only if P (Ts|N ) = 0, and similarly for 1−P . Hence the
operator T , together with (3.11), defines unitary isomorphisms
T : W 1
D̃C
(
S̃+
C ;P
) ∼=−→W 1
D
(
S+
C ;P
)
,
T ◦ τ : W 1
D̃C
(
S̃+
C ; 1− P
) ∼=−→W 1
D
(
S−
C ; 1− P
)
,
T : W 2
D̃C
(
S̃+
C ;P
) ∼=−→W 2
D
(
S+
C ;P
)
,
T ◦ τ : W 2
D̃C
(
S̃+
C ; 1− P
) ∼=−→W 2
D
(
S−
C ; 1− P
)
.
(3.17)
In the second and fourth lines, we use the fact that by definition (3.4) of P , we have
P |L2(S−|N ) = τ+P
+τ−1
+ ,
so that the isomorphisms T ◦ τ preserve the given boundary conditions.
Under the isomorphisms (3.17), the pairs of operators in parts (b)–(e) correspond to each
other. Here we again used Lemmas 3.5 and 3.6, and also Lemma 3.7. ■
3.5 Lower bounds and invertibility
Two kinds of invertibility at infinity ofD are assumed in Theorem 3.1: there is the condition (3.1)
on D, and invertibility of (3.6) and (3.7). To verify these conditions in the context of manifolds
with φ-cusps, we use Lemma 3.10 and Proposition 3.12 below.
The proof of the following lemma is the only place where we use the condition (2.8) in the
definition of weakly admissible cusps.
Lemma 3.10. Suppose that M has weakly admissible cusps, i.e., φ has the properties in Defi-
nition 2.8. Suppose that DN is invertible. Then there are u, b0 > 0 such that for all s ∈ Γ∞
c
(
S̃C)
supported in N × (u,∞),∥∥D̃Cs
∥∥
L2 ≥ b0∥s∥L2 . (3.18)
18 P. Hochs and H. Wang
Proof. By Lemma 3.8,
D̃2
C ≥ e−2φ◦ξ−1(
D2
N −
∣∣φ′ ◦ ξ−1
∣∣|DN |
)
= e−2φ◦ξ−1 |DN |
(
|DN | −
∣∣φ′ ◦ ξ−1
∣∣). (3.19)
Because DN is invertible, there is a b > 0 such that D2
N ≥ b2. Because M has weakly
admissible cusps, there is an upper bound β for φ, and there are a′ > 0 and α > 0 such that for
all x > a′,
b− |φ′(x)| ≥ α.
Now |DN | ≥ b, so on N × (ξ(a′),∞),
|DN | −
∣∣φ′ ◦ ξ−1
∣∣ ≥ α, and e−2φ◦ξ−1 |DN | ≥ e−2βb.
Hence, on N × (ξ(a′),∞), the right-hand side of (3.19) is greater than or equal to
b20 := αbe−2β.
Here we used that the various operators are self-adjoint and commute. ■
Remark 3.11. By a small adaptation of the proof of Lemma 3.10, we can show that if M
has strongly admissible φ-cusps and DN is invertible, then for all b0 > 0, there is a u > 0
such that (3.18) holds for all s ∈ Γ∞
c
(
S̃C
)
supported in N × (u,∞). By [4, Theorem SD] and
Lemmas 3.5 and 3.6, this implies that D has discrete spectrum. This is an analogous result to
Proposition 3.4.
Proposition 3.12. If h has a positive lower bound, then the operators (3.13) and (3.14) are
invertible.
Lemma 3.13. Let λ ∈ R, ζ ∈ C∞
c (0,∞) and h ∈ C∞[0,∞). For u, v ≥ 0, define
Hλ(u, v) := exp
(
λ
∫ v
u
h(s) ds
)
.
Define f ∈ C∞[0,∞) by
f(u) :=
∫ u
0
Hλ(u, v)ζ(v) dv if λ ≥ 0,
−
∫ ∞
u
Hλ(u, v)ζ(v) dv if λ < 0.
(3.20)
Then
(1) f ′ + λhf = ζ,
(2) if ζ = f̃ ′ + λhf̃ for some f ∈ C∞
c (0,∞), then f = f̃ ,
(3) f(0) = 0 if λ ≥ 0,
(4) if h ≥ ε > 0, then f ∈ L2(0,∞), and |λ| ∥f∥L2 ≤ 2
ε∥ζ∥L2.
Proof. The first two points follow from computations, the third point is immediate from the
definition of f .
For the fourth point, note that because h ≥ ε,
Hλ(u, v) ≤ eελ(v−u), (3.21)
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 19
if either
� λ ≥ 0 and v ≤ u, or
� λ < 0 and v ≥ u.
For λ′ ∈ R, define
fλ
′
(u) :=
∫ u
0
eλ
′(v−u)ζ(v) dv if λ′ ≥ 0,
−
∫ ∞
u
eλ
′(v−u)ζ(v) dv if λ′ < 0.
It is shown in the proof of Proposition 2.5 in [6] that fλ
′ ∈ L2(0,∞), and |λ′|∥fλ′∥L2 ≤ 2∥ζ∥L2 .
First suppose that λ ≥ 0 and ζ ≥ 0. Then (3.21) implies that |f | ≤
∣∣f ελ∣∣, so f ∈ L2(0,∞),
and
ελ∥f∥L2 ≤ ελ
∥∥f ελ∥∥
L2 ≤ 2∥ζ∥L2 .
If ζ also takes negative values, we decompose it as a difference of two nonnegative functions,
and reach the same conclusion.
Now suppose that λ < 0 and ζ ≤ 0. Then again, (3.21) implies that |f | ≤
∣∣f ελ∣∣, and
ε|λ| ∥f∥L2 ≤ ε|λ|
∥∥f ελ∥∥
L2 ≤ 2∥ζ∥L2 .
This also extends to ζ with positive values by a decomposition of ζ into nonpositive functions. ■
For all λ ∈ spec
(
D+
N
)
, let
{
φ1
λ, . . . , φ
mλ
λ
}
be an orthonormal basis of ker
(
D+
N − λ
)
. Then{
φj
λ;λ ∈ spec
(
D+
N
)
, j = 1, . . . ,mλ
}
(3.22)
is a Hilbert basis of L2
(
S+|N
)
of eigensections of D+
N .
Proof of Proposition 3.12. We prove the claim for (3.13), the proof for (3.14) is similar.
Let ζ ∈ Γ∞
c
(
S̃+
C
)
. Write
ζ =
∑
λ∈spec(D+
N )
mλ∑
j=1
ζjλ ⊗ φj
λ, (3.23)
where ζjλ ∈ C∞
c (0,∞). For every λ ∈ spec
(
D+
N
)
and j, define the function f jλ on (0,∞) as the
function f in (3.20), with ζ replaced by ζjλ. We claim that the series
∑
λ∈spec(D+
N )
mλ∑
j=1
f jλ ⊗ φj
λ (3.24)
converges to an element f ∈W 1
D̃C
(
S̃+
C ;P
)
. We set Qζ := f for any such ζ. Then the first point
in Lemma 3.13 implies that D̃+
Cf = ζ. The second point in Lemma 3.13 implies that QD̃+
C f̃ = f̃
for any f̃ ∈W 1
D̃C
(
S̃+
C ;P
)
. The third point in Lemma 3.13 implies that P (f |N×{0}) = 0.
To prove convergence of (3.24), we note that by the first point in Lemma 3.13,∥∥f jλ ⊗ φj
λ
∥∥2
W 1
D̃C
=
∥∥f jλ∥∥2L2 +
∥∥ζjλ∥∥2L2 .
20 P. Hochs and H. Wang
Since |λ| is bounded away from zero, the fourth point in Lemma 3.13 implies that∥∥f jλ ⊗ φj
λ
∥∥
W 1
D̃C
≤ B
∥∥ζjλ∥∥L2 (3.25)
for a constant B > 0 independent of λ. So convergence of (3.23) in L2
(
S̃+
C
)
implies convergence
of (3.24) in W 1
D̃C
(
S̃+
C
)
.
We have just seen that ∥Qζ∥W 1
D̃C
≤ B∥ζ∥L2 , thus Q extends continuously to an inverse
of (3.13). ■
3.6 Adjoints
Proposition 3.14. If h has a positive lower bound, then the two operators (3.13) and (3.14)
are each other’s adjoints.
Proof. We claim that for all sP ∈W 1
D̃C
(
S̃+
C ;P
)
and s1−P ∈W 1
D̃C
(
S̃+
C ; 1− P
)
,(
D̃+
CsP , s1−P
)
L2 =
(
sP , D̃
−
Cs1−P
)
L2 . (3.26)
Indeed, suppose that
sP = sNP ⊗ fP , s1−P = sN1−P ⊗ f1−P ,
for sNP , s
N
1−P ∈ L2
(
S+|N
)
and fP , f1−P ∈ L2[0,∞) such that sP ∈ W 1
D̃C
(
S̃+
C ;P
)
and s1−P ∈
W 1
D̃C
(
S̃+
C ; 1 − P
)
. Then by self-adjointness of D+
N and integration by parts, the left-hand side
of (3.26) equals
(
sNP , D
+
Ns
N
1−P
)
L2(S+|N )
((
fP f1−P
)
|∞0 +
∫ ∞
0
fP (x)
(
−f ′1−P (x) + h(x)f1−P (x)
)
dx
)
. (3.27)
If we further decompose the expressions with respect to eigenspaces of D+
N , and use that the
components of fP for positive eigenvalues equal zero at zero, and the components of f1−P for
negative eigenvalues equal zero at zero, then we find that the components for all eigenvalues
of fP f1−P are zero at zero. So the term
(
fP f1−P
)
|∞0 in (3.27) vanishes, and (3.27) equals the
right-hand side of (3.26).
Now if σP := D̃+
CsP and σ1−P := D̃−
Cs1−P , then (3.26) becomes(
σP ,
(
D̃−
C
)−1
σ1−P
)
L2 =
((
D̃+
C
)−1
σP , σ1−P
)
L2 .
The sections σP and σ1−P of this type are dense in L2
(
S̃+
C
)
. And the inverse operators
(
D̃+
C
)−1
and
(
D̃−
C
)−1
are bounded, so we find that((
D̃+
C
)−1)∗
=
(
D̃−
C
)−1
.
This implies that
(
D̃+
C
)∗
= D̃−
C . ■
Proposition 3.15. If h has a positive lower bound, then the operators (3.15) and (3.16) are
self-adjoint.
Proof. We prove the claim for (3.15), the proof for (3.16) is similar.
The operator (3.13) is invertible by Proposition 3.12. It maps the subspace
W 2
D̃C
(
S̃+
C ;P
)
⊂W 1
(
S̃+
C ;P
)
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 21
onto
W 1
(
S̃+
C ; 1− P
)
⊂ L2
(
S̃+
C
)
.
So we obtain an invertible operator
D̃+
C : W 2
D̃C
(
S̃+
C ;P
)
→W 1
(
S̃+
C ; 1− P
)
.
So, again by Proposition 3.12, the composition
W 2
D̃C
(
S̃+
C ;P
) D̃+
C−−→W 1
(
S̃+
C ; 1− P
) D̃−
C−−→ L2
(
S̃+
C
)
(3.28)
is invertible, with bounded inverse. The adjoint of the bounded operator(
D̃+
C
)−1
: W 1
D̃C
(
S̃+
C ; 1− P
)
→W 2
D̃C
(
S̃+
C ;P
)
is the restriction of the adjoint of(
D̃+
C
)−1
: L2
(
S̃+
C
)
→W 1
D̃C
(
S̃+
C ;P
)
to W 2
D̃C
(
S̃+
C ;P
)
. By Proposition 3.14, this is
(
D̃−
C
)−1|
W 2
D̃C
(S̃+
C ;P )
.
Again applying Proposition 3.14, we find that the inverse of (3.28) has adjoint
W 2
D̃C
(
S̃+
C ;P
) (D̃−
C )−1
−−−−−→W 1
D̃C
(
S̃+
C ; 1− P
) (D̃+
C )−1
−−−−−→ L2
(
S̃+
C
)
.
Hence, as maps from W 2
D̃C
(
S̃+
C ;P
)
to L2
(
S̃+
C
)
,
((
D̃−
CD̃
+
C
)−1)∗
=
(
D̃−
CD̃
+
C
)−1
.
This implies that
(
D̃−
CD̃
+
C
)∗
= D̃−
CD̃
+
C . ■
Remark 3.16. Proposition 3.15 can also be deduced from Proposition 3.14 via [37, Theo-
rem X.25].
If M has weakly admissible φ-cusps, then φ has an upper bound, so h = e−φ◦ξ has a pos-
itive lower bound. Therefore, Proposition 3.2 follows from Proposition 3.9, Lemma 3.10 and
Propositions 3.12 and 3.15.
3.7 Cusp contributions
To prove Proposition 3.3, we express the contribution from infinity in Theorem 3.1 in terms of
the operator D̃C . Recall the definition of the function Φ in (3.10).
Lemma 3.17. Suppose that the operator (3.15) is self-adjoint, and let λ̃Ps be the Schwartz kernel
of e
−sD̃−
C D̃+
C
P D̃−
C . Then for all n, n′ ∈ N , x, x′ ∈ (a,∞) and s > 0,
λPs (n, x;n
′, x′) = e(1−p)φ(x′) Φ(x)
Φ(x′)
λ̃Ps (n, ξ(x);n
′, ξ(x′)).
22 P. Hochs and H. Wang
Proof. The kernel λPs is defined with respect to the Riemannian density epφ dn dx on C,
whereas λ̃Ps is defined with respect to the Riemannian density dndy on N×(0,∞). Furthermore,
Lemma 3.6 and the third isomorphism on (3.17) imply that
e
−sD−
CD+
C
P D−
C = T ◦ e−sD̃−
C D̃+
C
P D̃−
C ◦ T−1.
We find that for all n ∈ N and x′ ∈ (a,∞) and s ∈ Γ∞
c (SC),∫
N
∫ ∞
a
λPs (n, x;n
′, x′)s(n′, x′)epφ(x
′) dx′ dn′
=
(
e
−sD−
CD+
C
P D−
Cs
)
(n, x) = T
(
e
−sD̃−
C D̃+
C
P D̃−
CT
−1s
)
(n, x)
= Φ(x)
∫
N
∫ ∞
0
λ̃Ps (n, ξ(x);n
′, y′)
1
Φ(ξ−1(y′))
s(n′, ξ−1(y′)) dn′ dy′.
By a substitution x′ = ξ−1(y′), the latter integral equals
Φ(x)
∫
N
∫ ∞
a
λ̃Ps (n, ξ(x);n
′, ξ(x′))
1
Φ(x′)
s(n′, x′)eφ(x
′) dn′ dx′.
Here we used that ξ′ = eφ. ■
Recall the choice of the Hilbert basis (3.22) of L2
(
S+|N
)
of eigensections of D+
N . Let ρλ,±
and θλ,±ν be as ρ and θν in Theorem 2.11, with q = q±λ as in (2.13).
Lemma 3.18. For all s > 0, the Schwartz kernel λ̃Ps in Lemma 3.17 equals
∑
λ>0
mλ∑
j=1
(∫
R
e−sνθλ,+ν ⊗
(
d
dy
+ λe−φ◦ξ−1
)
θλ,+ν dρλ,+(ν)
)
⊗
(
φj
λ ⊗ φj
λ
)
+
∑
λ<0
mλ∑
j=1
(∫
R
e−sν
(
− d
dy
+ λe−φ◦ξ−1
)
θλ,−ν ⊗ θλ,−ν dρλ,−(ν)
)
⊗
(
φj
λ ⊗ φj
λ
)
. (3.29)
Here we identify S+
N
∼=
(
S+
N
)∗
using the metric, so we view φj
λ⊗φj
λ as a section of S+
N ⊠
(
S+
N
)∗
.
Proof. We extend the projection (3.3) to a projection
P : L2
(
S̃+
C
) ∼= L2
(
S+|N
)
⊗ L2(0,∞)
P⊗1−−−→ L2(S+|N )>0 ⊗ L2(0,∞) ↪→ L2
(
S̃+
C
)
.
Then [27, Proposition 3.5] states that
e
−sD̃−
C D̃+
C
P D̃−
C = e
−sD̃−
C D̃+
C
F D̃−
CP + D̃−
Ce
−sD̃+
CD̃−
C
F (1− P ). (3.30)
For s > 0, let e
−sD̃∓
C D̃±
C
F be the heat operator for the Friedrichs extension of
D̃∓
CD̃
±
C : Γ∞
c
(
S̃+
C
)
→ L2
(
S̃+
C
)
.
Let κF,±s be its Schwartz kernel. By (2.14), (2.13) and Lemma 3.8, the restriction of D̃∓
CD̃
±
C
to ker
(
D+
N − λ
)
⊗ L2(0,∞) equals ∆q±λ
. So by Theorem 2.11, the Schwartz kernel κF,±λ,s of the
restriction of e
−sD̃∓
C D̃±
C
F to ker
(
D+
N − λ
)
⊗ L2(0,∞) is
κF,±λ,s =
mλ∑
j=1
(∫
R
e−sνθλ,±ν ⊗ θλ,±ν dρλ,±(ν)
)
⊗
(
φj
λ ⊗ φj
λ
)
.
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 23(
Note that
∑mλ
j=1 φ
j
λ ⊗ φj
λ is the identity operator on ker
(
D+
N − λ
)
.
)
The Schwartz kernel of
D̃−
Ce
−sD̃+
CD̃−
C
F equals D̃−
C applied to the first entry of κF,−s , so its restriction to ker
(
D+
N − λ
)
⊗
L2([0,∞)) is
mλ∑
j=1
(∫
R
e−sν
(
− d
dy
+ λe−φ◦ξ−1
)
θλ,−ν ⊗ θλ,−ν dρλ,−(ν)
)
⊗
(
φj
λ ⊗ φj
λ
)
. (3.31)
The Schwartz kernel of e
−sD̃−
C D̃+
C
F D̃−
C equals the adjoint D̃+
C of D̃−
C applied to the second entry
of κF,+s , so its restriction to ker
(
D+
N − λ
)
⊗ L2(0,∞) is
mλ∑
j=1
(∫
R
e−sνθλ,+ν ⊗
(
d
dy
+ λe−φ◦ξ−1
)
θλ,+ν dρλ,+(ν)
)
⊗
(
φj
λ ⊗ φj
λ
)
. (3.32)
The claim follows from (3.30) and the expressions (3.31) and (3.32) for the relevant Schwartz
kernels on eigenspaces of D+
N . ■
Lemma 3.19. In the situation of Lemma 3.17, we have for all a′ > a,∫
N
tr
(
gλPs
(
g−1n, a′;n, a′
))
dn = e(1−p)φ(a′)
∑
λ∈spec(D+
N )
sgn(λ) tr(g|ker(D+
N−λ))
×
∫
R
e−sνθ|λ|,+ν (ξ(a′))
((
d
dy
+ |λ|e−φ◦ξ−1
)
θ|λ|,+ν
)
(ξ(a′)) dρ|λ|,+(ν). (3.33)
Proof. It follows directly from (2.13) that for all λ ∈ R,
q+λ = q−−λ.
This implies that, for all λ ∈ R and ν ∈ C, with notation as in (2.14) and below,
∆+
λ = ∆−
−λ, θλ,+ν = θ−λ,−
ν , ρλ,+ = ρ−λ,−.
The last two relations imply in particular that for all λ < 0,
θλ,−ν = θ|λ|,+ν , ρλ,− = ρ|λ|,+, − d
dy
+ λe−φ◦ξ−1
= sgn(λ)
(
d
dy
+ |λ|e−φ◦ξ−1
)
.
These equalities, together with Lemmas 3.17 and 3.18 imply that for all n ∈ N ,
gλPs
(
g−1n, a′;n, a′
)
= e(1−p)φ(a′)
∑
λ>0
mλ∑
j=1
sgn(λ)
(∫
R
e−sνθ|λ|,+ν (ξ(a′))
(
d
dy
+ |λ|e−φ◦ξ−1
)
× θ|λ|,+ν (ξ(a′)) dρλ,+(ν)
)(
gφj
λ(g
−1n)⊗ φj
λ(n)
)
.
This equality and
mλ∑
j=1
∫
N
tr
(
gφj
λ
(
g−1n
)
⊗ φj
λ(n)
)
dn =
mλ∑
j=1
(
g · φj
λ, φ
j
λ
)
L2 = tr
(
g|ker(D+
N−λ)
)
together imply (3.33). ■
24 P. Hochs and H. Wang
Proof of Proposition 3.3. By Proposition 2.7, the function f1 in (3.2) equals e−φ in our
situation. So it follows from Lemma 3.19 that for all a′ > a
Ag(DC , a
′) = −e−pφ(a′)
∫ ∞
0
∑
λ∈spec(D+
N )
sgn(λ) tr
(
g|ker(D+
N−λ)
)
×
∫
R
e−sνθ|λ|,+ν (ξ(a′))
((
d
dy
+ |λ|e−φ◦ξ−1
)
θ|λ|,+ν
)
(ξ(a′)) dρ|λ|,+(ν) ds
= −1
2
ηφg
(
D+
N , a
′). (3.34)
In particular, because Ag(DC , a
′) converges by Theorem 3.1, so does ηφg
(
D+
N , a
′).
Vanishing of ηφ
(
D+
N , a
′) when D+
N has g-symmetric spectrum around zero follows directly
from the definition (2.15): then the term corresponding to λ ∈ spec(DN ) equals minus the term
corresponding to −λ. ■
Proposition 3.4 was proved at the start of this section, and Proposition 3.2 was proved at
the end of Section 3.6. So Propositions 3.2–3.4 are proved, and the proof of Theorem 2.16 is
complete.
4 Cylinders
If φ = 0, then the metric (2.1) is the cylinder metric BN + dx2. We show that the cusp con-
tribution η0g
(
D+
N , a
′) then equals Donnelly’s g-delocalised version of the Atiyah–Patodi–Singer
η-invariant, for all a′ > a. The computation in this subsection is a spectral counterpart of the
geometric computation in [27, Section 4].
We start by recalling [27, Proposition 5.1].
Proposition 4.1. Let (λj)
∞
j=1 and (aj)
∞
j=1 be sequences in R such that |λ1| > 0, and |λj | ≤ |λj+1|
for all j, and such that there are c1, c2, c3, c4 > 0 such that for all j,
|λj | ≥ c1j
c2 , |aj | ≤ c3j
c4 .
Then for all a′ > 0,∫ ∞
0
∞∑
j=1
sgn(λj)aj
e−λ2
jse−a′2/s
√
s
(
a′
s
− |λj |
)
ds = 0.
For a function f ∈ L1(R), we write
f̌(x) :=
∫
R
eixζf(ζ) dζ
for its inverse Fourier transform (up to a possible power of 2π).
Lemma 4.2. Let f ∈ L1(R), and α, β ∈ R. If f is even, then∫ ∞
0
sin(αµ) sin(βµ)f(µ) dµ =
1
4
(
−f̌(α+ β) + f̌(α− β)
)
.
If f is odd, then∫ ∞
0
sin(αµ) cos(βµ)f(µ) dµ =
1
4i
(
f̌(α+ β) + f̌(α− β)
)
.
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 25
Lemma 4.3. If φ = 0, then the spectral measure dρ|λ|,+ in (2.15) equals
dρ|λ|,+(ν) =
1
π
√
ν − λ2 dν if ν ≥ λ2,
0 if ν < λ2.
Proof. The proof is analogous to the computation in [41, Section 4.1] in the Neumann case.
With notation as in Proposition 2.12, we now have
θ1(x, ν) =
1
µ
sin(µx), θ2(x, ν) = − cos(µx),
with µ :=
√
ν − λ2. If ν has positive imaginary part, then we choose the square root with
positive real and imaginary parts. Then f(ν) = −iµ has negative imaginary part, and
θ2(x, ν) + f(ν)θ1(x, ν) = −eiµx
defines a function in L2([0,∞)) if Im(ν) > 0. With our choice of square roots, we have for all
ν ∈ R,
lim
δ↓0
− Im(f(ν + iδ)) =
{
µ if ν ≥ λ2,
0 if ν < λ2.
This implies the claim via Proposition 2.12. ■
The g-delocalised η-invariant of D+
N [21, 26, 30] is
ηg
(
D+
N
)
=
1√
π
∫ ∞
0
Tr
(
gD+
Ne−s(D+
N )2
) 1√
s
ds.
If g = e, this equals the classical η-invariant of D+
N .
Proposition 4.4. If φ = 0, then for all a′ > a
η0g
(
D+
N , a
′) = ηg
(
D+
N
)
.
Proof. We apply the definition (2.15) of cusp contributions with φ = 0. We look for solutions
of −θ′′ + λ2 = νθ satisfying
θ(0, ν) = 0, θ′(0, ν) = 1,
and find the eigenfunctions θ
|λ|,+
ν (y) = 1√
ν−λ2
sin
(√
ν − λ2y
)
. Let a′ > a, and set a′′ :=
a′ − a = ξ(a′). Then by Lemma 4.3, (2.15) becomes
η0g
(
D+
N , a
′) = 2
π
∫ ∞
0
∑
λ∈spec(D+
N )
sgn(λ) tr
(
g|ker(D+
N−λ)
) ∫ ∞
λ2
e−sν sin
(√
ν − λ2a′′
)
√
ν − λ2
×
(
cos
(√
ν − λ2a′′
)
+ |λ|
sin
(√
ν − λ2a′′
)
√
ν − λ2
)√
ν − λ2 dν ds. (4.1)
The change of variables µ =
√
ν − λ2 and dν = 2
√
ν − λ2 dµ reduces it to
η0g
(
D+
N , a
′) = 4
π
∫ ∞
0
∑
λ∈spec(D+
N )
sgn(λ) tr
(
g|ker(D+
N−λ)
)
×
∫ ∞
0
e−s(µ2+λ2) sin(µa′′)
(
µ cos(µa′′) + |λ| sin(µa′′)
)
dµds.
26 P. Hochs and H. Wang
Applying Lemma 4.2 and using f(µ) = e−sµ2
, so f̌(x) =
√
π
s e
−x2
4s and (µ 7→ µf(µ))∨ = 1
i (f̌)
′,
we have∫ ∞
0
sin2(µa′′)e−sµ2
dµ =
√
π
4
√
s
(
1− e−
a′′2
s
)
,∫ ∞
0
sin(µa′′) cos(µa′′)µe−sµ2
dµ =
√
πa′′
4s
3
2
e−
a′′2
s .
Therefore,
η0g
(
D+
N , a
′) = 1√
π
∫ ∞
0
∑
λ∈spec(D+
N )
sgn(λ) tr
(
g|ker(D+
N−λ)
)
e−sλ2
(
a′′
s
3
2
e−
a′′2
s +
|λ|√
s
− |λ|√
s
e−
a′′2
s
)
ds
=
1√
π
∫ ∞
0
∑
λ∈spec(D+
N )
tr
(
g|ker(D+
N−λ)
)e−sλ2
λ√
s
ds
− 1√
π
∫ ∞
0
∑
λ∈spec(D+
N )
sgn(λ) tr
(
g|ker(D+
N−λ)
)e−sλ2
e−
a′′2
s
√
s
(
a′′
s
− |λ|
)
ds. (4.2)
The first term equals ηg
(
D+
N
)
. For the second term, we use Proposition 4.1, and take λj to be
the jth eigenvalue of D+
N (ordered by absolute values), and aj := tr
(
g|ker(D+
N−λj)
)
. Then Weyl’s
law for D+
N shows that λj has the growth behaviour assumed in the proposition. And
|aj | ≤ dim
(
ker
(
D+
N − λj
))
grows at most polynomially by Weyl’s law. Hence Proposition 4.1 applies, and implies that the
second term in (4.2) is zero. ■
Remark 4.5. By Proposition 4.4, the cusp contribution η0g
(
D+
N , a
′) is independent of a′ > a in
this case. Furthermore, we see directly from (4.1) that a version of η0g
(
D+
N , a
′) with a′ replaced
by a equals zero. This illustrates the fact that the limit on the right-hand side of (2.17) does
not equal the expression (2.15) with a′ replaced by a.
As a consequence of Theorem 2.16 and Proposition 4.4, we obtain Donnelly’s equivariant
APS index theorem [21] for the index of D|Z with APS boundary conditions at N :
indexAPS
G (D|Z)(g) =
∫
Zg
ASg(D)− 1
2
ηg
(
D+
N
)
.
Indeed, if D+
N is invertible, then the left-hand side equals indexG(D)(g). In general, one replaces
Theorem 2.16 by Theorem 5.3 below.
5 Non-invertible DN
In this section, we do not assume that DN is invertible. Because N is compact, D+
N has discrete
spectrum. Let ε > 0 be such that
spec
(
D+
N
)
∩ (−2ε, 2ε) ⊂ {0}. (5.1)
Let w ∈ C∞(M) be a function such that for all x ≥ a and all n ∈ N ,
w(n, x) = x. (5.2)
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 27
Consider the operator
Dεw := e−εwDeεw. (5.3)
This operator equals (2.6) on C, with D+
N replaced by the invertible operator D+
N +ε. Therefore,
much of the proof of Theorem 2.16 applies to Dεw, apart from the fact that this operator and
D+
N+ε are not Dirac operators of the form (2.5). This affects the limits as t ↓ 0 of heat operators
associated to these operators.
In the proof of Theorem 3.1, given in [27], heat kernel asymptotics were used that may not
apply to D+
N+ε. We therefore start from a version of this theorem where the hard cutoff between
Zg ∪
(
Ng × (a, a′]
)
and the contribution from infinity Ag(DC , a
′) is replaced by a smooth cutoff
function. Let ψ ∈ C∞(M) be such that
ψ|Z ≡ 1, ψ|N×[a+1,∞) ≡ 0. (5.4)
For t > 0, define
At
g(DC , ψ) :=
∫ ∞
0
∫
N
∫ ∞
a
tr
(
gλPs
(
g−1n, x;n, x
))
ψ′(x)f1(x) dx dn ds,
and
ηφ,tg (D+
N , ψ) =− 2
∫ ∞
t
∫ ∞
a
ψ′(x)e−pφ(x)
∑
λ∈spec(D+
N )
sgn(λ) tr
(
g|ker(D+
N−λ)
)
×
∫
R
e−sνθ|λ|,+ν (ξ(x))
((
θ|λ|,+ν
)′
(ξ(x)) + |λ|e−φ(x)θ|λ|,+ν (ξ(x))
)
dρ|λ|,+(ν) dx ds.
We use the following standard regularisation method.
Definition 5.1. For a function f(t) that has an asymptotic expansion in t as t ↓ 0, the regularised
limit LIMt↓0 f(t) is the coefficient of t0 in such an asymptotic expansion.
The regularised g-delocalised φ-cusp contribution associated to D+
N + ε and ψ is
ηφ,regg
(
D+
N + ε, ψ
)
:= LIMt↓0 η
φ,t
g
(
D+
N + ε, ψ
)
.
The condition (2.8) with b replaced by ε implies that Dεw is Fredholm, via Lemma 3.10.
Theorem 4.14 in [27] then becomes
indexG(D
εw)(g) = LIMt↓0
(
Tr
(
g ◦ e−tD̃−
εwD̃+
εwψ
)
− Tr
(
g ◦ e−tD̃+
εwD̃−
εwψ
)
+At
g
(
Dεw
C , ψ
))
.(5.5)
Here D̃εw is an extension of Dεw to a closed manifold containing M \ (N × (a+ 1,∞)), and we
used the fact that the left-hand side is independent of t. The proof of (5.5) is a direct analogy
of the proof of Theorem 4.14 in [27].
Example 5.2. Suppose that N is the circle, and D+
N = i ddθ . Then D+
N + 1/2 is invertible and
has symmetric spectrum. So
At
e
(
Dεw
C , ψ
)
= ηφ,te
(
D+
N + 1/2, ψ
)
= 0
for all t. Hence (5.5) becomes
index
(
Dw/2
)
= LIMt↓0
(
Tr
(
e
−tD̃−
w/2
D̃+
w/2ψ
)
− Tr
(
e
−tD̃+
w/2
D̃−
w/2ψ
))
.
28 P. Hochs and H. Wang
Theorem 5.3. Suppose that M has weakly admissible φ-cusps, where (2.8) holds on an interval
(a′,∞), with a′ > a, and b replaced by ε. Then Dεw is Fredholm, and its index is independent
of ε and w with the properties mentioned. And for ψ ∈ C∞(M) satisfying (5.4),
indexG
(
Dεw
)
(g) =
∫
Mg
ψ|Mg ASg(D)− 1
2
lim
ε↓0
ηφ,regg
(
D+
N + ε, ψ
)
. (5.6)
Proof. As noted above (5.5), the operator Dεw is Fredholm ifM has weakly admissible φ-cusps
with respect to the spectral gap 2ε of this operator. If ε′ > 0 has the same property (5.1) as ε,
then
Dεw −Dε′w = (ε′ − ε)c(dw).
This is a bounded vector bundle endomorphism, so the linear path between Dεw and Dε′w is
continuous. And all operators on this path are Fredholm, so index(Dεw) = index(Dε′w).
If w′ ∈ C∞(M) has the same property (5.2) as w, then
Dεw −Dεw′
= −εc(d(w − w′)).
Because w − w′ = 0 outside a compact set, Dεw′
is a compact perturbation of Dεw, when
viewed as acting on the relevant Sobolev space. Hence index(Dεw) = index(Dεw′
). We find that
index(Dεw) is independent of ε and w.
By the arguments leading up to (3.34), with integrals over s replaced by integrals from t > 0
to ∞, we have
At
g
(
Dεw
C , ψ
)
= −1
2
ηφ,tg
(
D+
N + ε, ψ
)
.
Hence (5.5) becomes
indexG
(
Dεw
)
(g) = LIMt↓0
(
Tr
(
g ◦ e−tD̃−
εwD̃+
εwψ
)
− Tr
(
g ◦ e−tD̃+
εwD̃−
εwψ
))
− 1
2
ηφ,regg
(
D+
N + ε, ψ
)
. (5.7)
The coefficients of the heat operator e−sD̃2
εw are continuous in ε. And standard heat kernel
asymptotics and localisation apply to e−sD̃2
0 , the analogous operator with ε = 0. These imply
that
lim
ε↓0
LIMt↓0
(
Tr
(
g ◦ e−tD̃−
εwD̃+
εwψ
)
− Tr
(
g ◦ e−tD̃+
εwD̃−
εwψ
))
=
∫
Mg
ψ|Mg ASg(D).
Because the left-hand side of (5.7) is independent of ε, the claim follows. ■
Remark 5.4. The arguments of [27, Section 4.5] showing that ψ may be replaced by a step
function involve an actual limit t ↓ 0, not the regularised limit LIMt↓0. For this reason, it is not
immediately obvious to us if a version of Theorem 5.3 with ψ replaced by a step function is true.
Example 5.5. If φ is the zero function, then the left-hand side of (5.6) is the equivariant index
of the restriction ofD toM\C, with Atiyah–Patodi–Singer boundary conditions at ∂C. Then for
all suitable ψ, a slight modification of the proof of Proposition 4.4 shows that ηφ,regg
(
D+
N + ε, ψ
)
is the regularised g-delocalised η-invariant of D+
N + ε. Hence
lim
ε↓0
ηφ,regg
(
D+
N + ε, ψ
)
= tr
(
g|ker(D+
N )
)
+ ηregg
(
D+
N
)
.
This fact is standard; see, for example, [26, Lemma 6.7].
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 29
A Conformal transformations of Dirac operators
Let M be a manifold of dimension p. Let B0 be a Riemannian metric on M . Let S → M
be a Clifford module for this metric, with Clifford action c0 : TM → End(S). Fix a Clifford
connection ∇0 on S preserving a Hermitian metric on S, and let D0 = c0 ◦∇0 be the associated
Dirac operator. Let φ ∈ C∞(M), and consider the Riemannian metric Bφ := e2φB0. We denote
the gradient operator for B0 by grad.
Proposition A.1. There are a Clifford action cφ by TM on S, with respect to Bφ, and a Clifford
connection ∇φ on S, with respect to cφ and Bφ, such that the associated Dirac operator Dφ =
cφ ◦ ∇φ equals
Dφ = e−φ
(
D0 +
p− 1
2
c0(gradφ)
)
= e−
p+1
2
φD0e
p−1
2
φ. (A.1)
Remark A.2. The operator c0(gradφ) in (A.1) is fibrewise antisymmetric. But the opera-
tor Dφ is symmetric with respect to the L2-inner product defined with the Riemannian density
associated to Bφ. This follows, for example, from Proposition A.1 and the usual argument why
Dirac operators are symmetric.
We write X(M) for the space of smooth vector fields on M . Let ∇TM,0 be the Levi-Civita
connection for B0, and let ∇TM,φ be the Levi-Civita connection for Bφ.
Lemma A.3. For all v, w ∈ X(M),
∇TM,φ
v w = ∇TM,0
v w + v(φ)w + w(φ)v −B0(v, w) gradφ.
Proof. This is a computation based on the Koszul formulas for ∇TM,φ and ∇TM,0. ■
Consider the Clifford action cφ := eφc0 with respect to Bφ. For A ∈ Ω1(M ; End(S)), consider
the connection ∇A := ∇0 +A on S. For v ∈ X(M), let Av ∈ End(S) be the pairing of A and v.
Lemma A.4. The connection ∇A is a Clifford connection for cφ and Bφ if and only if for all
v, w ∈ X(M),
[Av, c0(w)] = w(φ)c0(v)−B0(v, w)c0(gradφ).
Proof. For all v, w ∈ X(M),[
∇A
v , cφ(w)
]
= eφ[∇v, c0(w)] + eφ[Av, c0(w)] + eφv(φ)c0(w). (A.2)
And by Lemma A.3,
cφ
(
∇TM,φ
v w
)
= eφc0
(
∇TM,0
v w
)
+eφv(φ)c0(w) + eφw(φ)c0(v)− eφB0(v, w)c0(gradφ). (A.3)
Because ∇0 is a Clifford connection for c0 and B0, (A.2) and (A.3) are equal if and only if
eφ[Av, c0(w)] + eφv(φ)c0(w) = eφv(φ)c0(w) + eφw(φ)c0(v)− eφB0(v, w)c0(gradφ). ■
Lemma A.5. For all u, v, w ∈ X(M),
[c0(u)c0(v), c0(w)] = −2B0(v, w)c0(u) + 2B0(u,w)c0(v).
Proof. This is a straightforward computation, involving the equality
c0(v1)c0(v2) + c0(v2)c0(v1) = −2B0(v1, v2)
for all v1, v2 ∈ X(M). ■
30 P. Hochs and H. Wang
Let f ∈ C∞(M), and define Aφ,f ∈ Ω1(M ; End(S)) by
Aφ,f
v :=
1
2
c0(gradφ)c0(v) + fB0(gradφ, v).
We write ∇φ,f := ∇Aφ,f
.
Lemma A.6. For all f ∈ C∞(M), the connection ∇φ,f is a Clifford connection for cφ and Bφ.
Proof. Lemma A.5 implies that Aφ,f satisfies the condition in Lemma A.4. ■
Lemma A.7. The connection ∇φ,f preserves the metric on S if and only if f |supp(gradφ) =
1
2 .
Proof. Because ∇0 preserves the metric on S, ∇φ,f preserves the same metric if and only
if Aφ,f
v is anti-Hermitian for any vector field v. And because c0(w) is anti-Hermitian for any
vector field w,(
Aφ,f
v
)∗
= −Aφ,f + (2f − 1)B0(gradφ, v). ■
Proof of Proposition A.1. Let cφ and ∇φ, 1
2 as defined above, where f ≡ 1/2. Then ∇φ, 1
2 is
a Clifford connection and preserves the metric by Lemmas A.6 and A.7.
Let {e1, . . . , ep} be a local orthonormal frame for TM with respect to B0. Then the frame{
e−φe1, . . . , e
−φep
}
is a local orthonormal frame for TM with respect to Bφ. So
Dφ =
p∑
j=1
cφ
(
e−φej
)
∇φ, 1
2
e−φej
= e−φ
p∑
j=1
c0(ej)∇0
ej + e−φ
p∑
j=1
c0(ej)A
φ, 1
2
ej .
The first term on the right-hand side equals e−φD0, and the second term equals
1
2
e−φ
p∑
j=1
c0(ej)c0(gradφ)c0(ej) +
1
2
e−φ
p∑
j=1
c0(ej)B0(gradφ, ej)
=
1
2
e−φ
p∑
j=1
(
c0(gradφ)− 2B0(ej , gradφ)c0(ej)
)
+
1
2
c0(gradφ)
=
p− 1
2
e−φc0(gradφ). ■
Acknowledgements
We thank Mike Chen for a helpful discussion, and Christian Bär for pointing out a useful
reference. We are grateful to the referees for several helpful comments and corrections. In
particular, we thank the referee who pointed out an error in the previous version of [27], on
which the current paper builds, which has since been fixed. PH is partially supported by the
Australian Research Council, through Discovery Project DP200100729. HW is supported by
NSFC-11801178 and Shanghai Rising-Star Program 19QA1403200.
References
[1] Anghel N., Remark on Callias’ index theorem, Rep. Math. Phys. 28 (1989), 1–6.
[2] Anghel N., An abstract index theorem on noncompact Riemannian manifolds, Houston J. Math. 19 (1993),
223–237.
[3] Anghel N., On the index of Callias-type operators, Geom. Funct. Anal. 3 (1993), 431–438.
https://doi.org/10.1016/0034-4877(89)90022-0
https://doi.org/10.1007/BF01896237
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps 31
[4] Anghel N., Fredholmness vs. spectral discreteness for first-order differential operators, Proc. Amer. Math.
Soc. 144 (2016), 693–701.
[5] Atiyah M.F., Elliptic operators and compact groups, Lecture Notes in Math., Vol. 401, Springer, Berlin,
1974.
[6] Atiyah M.F., Patodi V.K., Singer I.M., Spectral asymmetry and Riemannian geometry. I, Math. Proc.
Cambridge Philos. Soc. 77 (1975), 43–69.
[7] Atiyah M.F., Singer I.M., The index of elliptic operators. III, Ann. of Math. 87 (1968), 546–604.
[8] Baier P.D., An index theorem for open manifolds whose ends are warped products, Abh. Math. Sem. Univ.
Hamburg 72 (2002), 269–282.
[9] BallmannW., Brüning J., On the spectral theory of surfaces with cusps, in Geometric Analysis and Nonlinear
Partial Differential Equations, Springer, Berlin, 2003, 13–37.
[10] Ballmann W., Brüning J., Carron G., Index theorems on manifolds with straight ends, Compos. Math. 148
(2012), 1897–1968, arXiv:1011.2290.
[11] Bär C., The Dirac operator on hyperbolic manifolds of finite volume, J. Differential Geom. 54 (2000),
439–488, arXiv:math.DG/0010233.
[12] Bär C., Ballmann W., Boundary value problems for elliptic differential operators of first order, Surv. Differ.
Geom., Vol. 1, Int. Press, Boston, MA, 2012, 1–78, arXiv:1101.1196.
[13] Barbasch D., Moscovici H., L2-index and the Selberg trace formula, J. Funct. Anal. 53 (1983), 151–201.
[14] Baum P., Connes A., Higson N., Classifying space for proper actions and K-theory of group C∗-algebras, in
C∗-Algebras: 1943–1993 (San Antonio, TX, 1993), Contemp. Math., Vol. 167, Amer. Math. Soc., Providence,
RI, 1994, 240–291.
[15] Berline N., Getzler E., Vergne M., Heat kernels and Dirac operators, Grundlehren Text Editions, Springer,
Berlin, 2004.
[16] Bott R., Seeley R., Some remarks on the paper of Callias: “Axial anomalies and index theorems on open
spaces”, Comm. Math. Phys. 62 (1978), 235–245.
[17] Braverman M., Index theorem for equivariant Dirac operators on noncompact manifolds, K-Theory 27
(2002), 61–101.
[18] Braverman M., Maschler G., Equivariant APS index for Dirac operators of non-product type near the
boundary, Indiana Univ. Math. J. 68 (2019), 435–501, arXiv:1702.08105.
[19] Bunke U., A K-theoretic relative index theorem and Callias-type Dirac operators, Math. Ann. 303 (1995),
241–279.
[20] Callias C., Axial anomalies and index theorems on open spaces, Comm. Math. Phys. 62 (1978), 213–234.
[21] Donnelly H., Eta invariants for G-spaces, Indiana Univ. Math. J. 27 (1978), 889–918.
[22] Gilkey P.B., On the index of geometrical operators for Riemannian manifolds with boundary, Adv. Math.
102 (1993), 129–183.
[23] Gromov M., Lawson Jr. H.B., Positive scalar curvature and the Dirac operator on complete Riemannian
manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196.
[24] Grubb G., Heat operator trace expansions and index for general Atiyah–Patodi–Singer boundary problems,
Comm. Partial Differential Equations 17 (1992), 2031–2077.
[25] Hitchin N., Harmonic spinors, Adv. Math. 14 (1974), 1–55.
[26] Hochs P., Wang B.L., Wang H., An equivariant Atiyah–Patodi–Singer index theorem for proper actions I:
The index formula, Int. Math. Res. Not. 2023 (2023), 3138–3193, arXiv:1904.11146.
[27] Hochs P., Wang H., An absolute version of the Gromov–Lawson relative index theorem, arXiv:2110.00376.
[28] Kucerovsky D., A short proof of an index theorem, Proc. Amer. Math. Soc. 129 (2001), 3729–3736.
[29] Levitan B.M., Sargsjan I.S., Sturm–Liouville and Dirac operators, Math. Appl. (Sov. Series), Vol. 59, Kluwer
Academic Publishers Group, Dordrecht, 1991.
[30] Lott J., Delocalized L2-invariants, J. Funct. Anal. 169 (1999), 1–31, arXiv:dg-ga/9612003.
[31] Mazzeo R., Phillips R.S., Hodge theory on hyperbolic manifolds, Duke Math. J. 60 (1990), 509–559.
[32] Moroianu S., Fibered cusp versus d-index theory, Rend. Semin. Mat. Univ. Padova 117 (2007), 193–203,
arXiv:math.DG/0610723.
https://doi.org/10.1090/proc12741
https://doi.org/10.1090/proc12741
https://doi.org/10.1007/BFb0057821
https://doi.org/10.1017/S0305004100049410
https://doi.org/10.1017/S0305004100049410
https://doi.org/10.2307/1970717
https://doi.org/10.1007/BF02941677
https://doi.org/10.1007/BF02941677
https://doi.org/10.1007/978-3-642-55627-2_2
https://doi.org/10.1112/S0010437X12000401
https://arxiv.org/abs/1011.2290
https://doi.org/10.4310/jdg/1214339790
https://arxiv.org/abs/math.DG/0010233
https://doi.org/10.4310/SDG.2012.v17.n1.a1
https://arxiv.org/abs/1101.1196
https://doi.org/10.1016/0022-1236(83)90050-2
https://doi.org/10.1090/conm/167/1292018
https://doi.org/10.1007/BF01202526
https://doi.org/10.1023/A:1020842205711
https://doi.org/10.1512/iumj.2019.68.7621
https://arxiv.org/abs/1702.08105
https://doi.org/10.1007/BF01460989
https://doi.org/10.1007/BF01202525
https://doi.org/10.1512/iumj.1978.27.27060
https://doi.org/10.1006/aima.1993.1063
https://doi.org/10.1007/BF02953774
https://doi.org/10.1080/03605309208820913
https://doi.org/10.1016/0001-8708(74)90021-8
https://doi.org/10.1093/imrn/rnab324
https://arxiv.org/abs/1904.11146
https://arxiv.org/abs/2110.00376
https://doi.org/10.1090/S0002-9939-01-06164-0
https://doi.org/10.1007/978-94-011-3748-5
https://doi.org/10.1007/978-94-011-3748-5
https://doi.org/10.1006/jfan.1999.3451
https://arxiv.org/abs/dg-ga/9612003
https://doi.org/10.1215/S0012-7094-90-06021-1
https://arxiv.org/abs/math.DG/0610723
32 P. Hochs and H. Wang
[33] Moscovici H., L2-index of elliptic operators on locally symmetric spaces of finite volume, in Operator Alge-
bras and K-theory (San Francisco, Calif., 1981), Contemp. Math., Vol. 10, Amer. Math. Soc., Providence,
R.I., 1982, 129–137.
[34] Müller W., Spectral theory for Riemannian manifolds with cusps and a related trace formula, Math. Nachr.
111 (1983), 197–288.
[35] Müller W., Manifolds with cusps of rank one. Spectral theory and L2-index theorem, Lecture Notes in Math.,
Vol. 1244, Springer, Berlin, 1987.
[36] Nomizu K., Ozeki H., The existence of complete Riemannian metrics, Proc. Amer. Math. Soc. 12 (1961),
889–891.
[37] Reed M., Simon B., Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Aca-
demic Press, New York, 1975.
[38] Salomonsen G., Geometric heat kernel coefficients for APS-type boundary conditions, in Journées “Équations
aux Dérivées Partielles” (Saint-Jean-de-Monts, 1998), Nantes Université, Nantes, 1998, Exp. No. XI,
19 pages.
[39] Stern M., L2-index theorems on locally symmetric spaces, Invent. Math. 96 (1989), 231–282.
[40] Stern M.A., L2-index theorems on warped products, Ph.D. Thesis, Princeton University, 1984.
[41] Titchmarsh E.C., Eigenfunction expansions associated with second-order differential equations. Part I, 2nd
ed., Clarendon Press, Oxford, 1962.
[42] Vaillant B., Index- and spectral theory for manifolds with generalized fibred cusps, Bonner Mathematische
Schriften, Vol. 344, Universität Bonn, Bonn, 2001, arXiv:math.DG/0102072.
https://doi.org/10.1002/mana.19831110109
https://doi.org/10.1007/BFb0077660
https://doi.org/10.2307/2034383
https://doi.org/10.1007/BF01393964
https://arxiv.org/abs/math.DG/0102072
1 Introduction
2 Preliminaries and result
2.1 varphi-cusps
2.2 Dirac operators on varphi-cusps
2.3 Spectral theory for Sturm–Liouville operators
2.4 An index theorem for manifolds with admissible varphi-cusps
2.5 Relations to other results
3 Proof of Theorem 2.16
3.1 Dirac operators that are invertible at infinity
3.2 An index theorem
3.3 Transforming Dirac operators
3.4 APS-boundary conditions
3.5 Lower bounds and invertibility
3.6 Adjoints
3.7 Cusp contributions
4 Cylinders
5 Non-invertible D_N
A Conformal transformations of Dirac operators
References
|
| id | nasplib_isofts_kiev_ua-123456789-211920 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T10:15:08Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Hochs, Peter Wang, Hang 2026-01-16T11:19:44Z 2023 Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps. Peter Hochs and Hang Wang. SIGMA 19 (2023), 023, 32 pages 1815-0659 2020 Mathematics Subject Classification: 58J20; 58D19 arXiv:2110.00390 https://nasplib.isofts.kiev.ua/handle/123456789/211920 https://doi.org/10.3842/SIGMA.2023.023 We consider a complete Riemannian manifold, which consists of a compact interior and one or more 𝜑-cusps: infinitely long ends of a type that includes cylindrical ends and hyperbolic cusps. Here, 𝜑 is a function of the radial coordinate that describes the shape of such an end. Given an action by a compact Lie group on such a manifold, we obtain an equivariant index theorem for Dirac operators, under conditions on 𝜑. These conditions hold in the cases of cylindrical ends and hyperbolic cusps. In the case of cylindrical ends, the cusp contribution equals the delocalised 𝜂-invariant, and the index theorem reduces to Donnelly's equivariant index theory on compact manifolds with boundary. In general, we find that the cusp contribution is zero if the spectrum of the relevant Dirac operator on a hypersurface is symmetric around zero. We thank Mike Chen for a helpful discussion, and Christian B¨ar for pointing out a useful reference. We are grateful to the referees for several helpful comments and corrections. In particular, we thank the referee who pointed out an error in the previous version of [27], on which the current paper builds, which has since been fixed. PH is partially supported by the Australian Research Council, through the Discovery Project DP200100729. HW is supported by NSFC-11801178 and Shanghai Rising-Star Program 19QA1403200. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps Article published earlier |
| spellingShingle | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps Hochs, Peter Wang, Hang |
| title | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps |
| title_full | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps |
| title_fullStr | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps |
| title_full_unstemmed | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps |
| title_short | Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps |
| title_sort | spectral asymmetry and index theory on manifolds with generalised hyperbolic cusps |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211920 |
| work_keys_str_mv | AT hochspeter spectralasymmetryandindextheoryonmanifoldswithgeneralisedhyperboliccusps AT wanghang spectralasymmetryandindextheoryonmanifoldswithgeneralisedhyperboliccusps |