On the Spectrum of Certain Hadamard Manifolds
We demonstrate the absolute continuity of the spectrum and determine the spectrum as a set for two classes of Hadamard manifolds, as well as for specific domains and quotients of one of these classes.
Gespeichert in:
| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Datum: | 2023 |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2023
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211981 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On the Spectrum of Certain Hadamard Manifolds. Werner Ballmann, Mayukh Mukherjee and Panagiotis Polymerakis. SIGMA 19 (2023), 050, 19 pages |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860261862197690368 |
|---|---|
| author | Ballmann, Werner Mukherjee, Mayukh Polymerakis, Panagiotis |
| author_facet | Ballmann, Werner Mukherjee, Mayukh Polymerakis, Panagiotis |
| citation_txt | On the Spectrum of Certain Hadamard Manifolds. Werner Ballmann, Mayukh Mukherjee and Panagiotis Polymerakis. SIGMA 19 (2023), 050, 19 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We demonstrate the absolute continuity of the spectrum and determine the spectrum as a set for two classes of Hadamard manifolds, as well as for specific domains and quotients of one of these classes.
|
| first_indexed | 2026-03-21T09:05:59Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 050, 19 pages
On the Spectrum of Certain Hadamard Manifolds
Werner BALLMANN a, Mayukh MUKHERJEE b and Panagiotis POLYMERAKIS c
a) Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
E-mail: hwbllmnn@mpim-bonn.mpg.de
b) Indian Institute of Technology Bombay, Powai, 400076 Maharashtra, India
E-mail: mathmukherjee@gmail.com
c) Department of Mathematics, University of Thessaly, 3rd km Old National Road Lamia-Athens,
35100 Lamia, Greece
E-mail: ppolymerakis@uth.gr
Received January 27, 2023, in final form July 14, 2023; Published online July 23, 2023
https://doi.org/10.3842/SIGMA.2023.050
Abstract. We show the absolute continuity of the spectrum and determine the spectrum
as a set for two classes of Hadamard manifolds and for specific domains and quotients of
one of the classes.
Key words: Laplace operator; absolutely continuous spectrum; point spectrum; Hadamard
manifold; asymptotically harmonic manifold
2020 Mathematics Subject Classification: 58J50; 53C20
Dedicated to Jean-Pierre Bourguignon
on the occasion of his 75th birthday
1 Introduction
The spectrum of the Laplacian is a classical invariant in Riemannian geometry. If the underlying
manifold is closed, then the spectrum of its Laplacian consists of eigenvalues of finite multiplicity,
and many studies are concerned with estimates of the eigenvalues and their multiplicities. In this
paper, we investigate the spectrum of (the Laplacian of) non-compact Riemannian manifolds.
Then the structure of the spectrum is quite different in general. For example, the spectrum of
Euclidean spaces does not have eigenvalues, but is, what is called absolutely continuous.
For a complete and connected Riemannian manifold M , we view its Laplacian ∆ as an
unbounded and symmetric operator with domain C∞
c (M) ⊆ L2(M). The closure of ∆, which
we also denote by ∆, is a self-adjoint operator in L2(M). Its spectrum as a subset of R will be
denoted by σ(M).
Functional analysis of self-adjoint operators yields the orthogonal decomposition
L2(M) = Hpp(M)⊕Hac(M)⊕Hsc(M)
of L2(M) into ∆-invariant subspaces such that
(1) Hpp(M) is spanned by eigenfunctions;
(2) Hac(M) consists of functions with absolutely continuous spectral measure;
(3) Hsc(M) consists of functions with singularly continuous spectral measure.
This paper is a contribution to the Special Issue on Differential Geometry Inspired by Mathemati-
cal Physics in honor of Jean-Pierre Bourguignon for his 75th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Bourguignon.html
mailto:hwbllmnn@mpim-bonn.mpg.de
mailto:mathmukherjee@gmail.com
mailto:ppolymerakis@uth.gr
https://doi.org/10.3842/SIGMA.2023.050
https://www.emis.de/journals/SIGMA/Bourguignon.html
2 W. Ballmann, M. Mukherjee and P. Polymerakis
We present some more details about these notions and decompositions in Appendix A, since
they are needed in our discussion of spectra of Riemannian products.
There is the well known decomposition σ(M) = σd(M)∪σess(M) of the spectrum as a disjoint
union of discrete and essential spectrum, where λ ∈ R belongs to σess(M) if ∆ − λ is not
a Fredholm operator. Then λ ∈ σd(M) if and only if λ is an eigenvalue of ∆ of finite multiplicity
and is an isolated point of σ(M). Therefore, the preimage in L2(M) of σd(M) under the spectral
projection associated to ∆ is contained in Hpp(M). In particular, if the essential spectrum of ∆
is empty, then Hac(M) = Hsc(M) = {0}. The essential spectrum of ∆ is determined by the
geometry of M at infinity; see, for example, [4, Proposition 3.6] for a general formulation of
this fact. In particular, the essential spectrum vanishes if M is compact. But there are also
a number of results about the vanishing of the essential spectrum for non-compact manifolds;
see, for example, [6, Theorems 1.1 and 1.2] and [4, Examples 3.7].
Now the vanishing of the essential spectrum implies the vanishing of the absolutely continuous
spectrum, which is not what we are aiming for. In fact, we will obtain conditions which imply the
absolute continuity of the spectrum, Hac(M) = L2(M), or, in other words, the vanishing of the
point and the singular continuous spectrum. As a byproduct, we will also discuss the vanishing of
the point spectrum, Hpp(M) = {0}. The underlying manifolds will be Hadamard manifolds, that
is, complete and simply connected Riemannian manifolds of non-positive sectional curvature.
Recall that a homogeneous Hadamard manifold can be thought of as a simply connected
solvable Lie group S with a left-invariant Riemannian metric and given as a semi-direct product
S = A⋉N , where A is Abelian and N is nilpotent. Our main result is the following
Theorem 1.1. If M is a homogeneous Hadamard manifold, then the spectrum of M is absolutely
continuous with σ(M) =
[
h2/4,∞
)
, where h is the mean curvature of N in S ∼= M .
An explicit formula for h is given in Theorem 5.2, where we also show that the equality
σ(M) =
[
h2/4,∞
)
holds in greater generality, namely for left-invariant Riemannian metrics on
Lie groups G, which are solvable or compact extensions of solvable Lie groups, such that their
derived subgroups are not cocompact in G.
Concerning the identification of homogeneous Hadamard manifolds with certain simply con-
nected solvable Lie groups with left-invariant Riemannian metrics, one may ask, whether repre-
sentation theory could be a tool to study the spectrum of their Laplacians. However, in general,
in our situation the underlying Lie groups are not exponential solvable, see [1, p. 26 and p. 37]
and [2, Remark (ii) on p. 16], so that at least the extension of Kirillov theory by Auslander and
Kostant does not seem to apply.
For Euclidean and symmetric spaces of non-compact type, there is much more precise in-
formation on their spectrum. For example, the Fourier transform yields a unitary equivalence
between the Laplacian in L2(Rm) and multiplication by |x|2 in L2(Rm), which implies that the
spectrum of the Laplacian is absolutely continuous with spectrum [0,∞). Similar, but more
involved, characterizations of the Laplacian are known in the case of symmetric spaces of non-
compact type.
We say that a Hadamard manifold M is asymptotically harmonic about a point ξ ∈ M∞
if there is a number h ≥ 0 such that each horosphere in M with center ξ has constant mean
curvature h. Hyperbolic spaces Hk
F , endowed with Fubini–Study metrics, are asymptotically
harmonic about any point in M∞, where λ0 = h2/4 > 0. Asymptotically harmonic manifolds in
the usual sense are asymptotically harmonic about any point in M∞. Such manifolds arise, for
example, in investigations on the rigidity of geodesic flows. Furthermore, since horospheres are
limits of spheres, harmonic Hadamard manifolds are asymptotically harmonic about any point
in M∞. However, homogeneous Hadamard manifolds without Euclidean factor, which are not
hyperbolic spaces, are asymptotically harmonic about some, but not any point M∞. A second
main result of this article is
On the Spectrum of Certain Hadamard Manifolds 3
Theorem 1.2. If KM < 0 and M is asymptotically harmonic about a point ξ ∈ M∞, then
the spectrum of ∆ on M is absolutely continuous with σ(M) ⊆
[
h2/4,∞
)
, where h > 0 is the
mean curvature of the horospheres with center ξ. Moreover, if the sectional curvature of M
is negatively pinched and the covariant derivative of the curvature tensor of M is uniformly
bounded, then σ(M) =
[
h2/4,∞
)
.
Say that a group of isometries of a Hadamard manifold M is elementary if it is discrete and
without torsion, fixes a point ξ ∈ M∞, and such that it is of one of the following two types:
(a) Γ leaves a geodesic γ in M emanating from ξ invariant.
(b) Γ leaves Busemann functions associated to ξ invariant.
In the first case, C = Γ\γ is a circle and the projection M → γ along horospheres with center ξ
descends to a Riemannian submersion π : N → C, where N = Γ\M . In the second case, any
Busemann function b associated to ξ is invariant under Γ and hence pushes down to N and
defines a Riemannian submersion π = b : N → R.
Theorem 1.3. For M as in Theorem 1.2, if Γ is an elementary group of isometries of M fixing ξ,
then the spectrum of the quotient N = Γ\M is absolutely continuous with σ(N) ⊆
[
h2/4,∞
)
,
where h > 0 denotes the mean curvature of the fibers of π. Moreover,
(1) if N is of type (b) and the fibers of π are of finite volume or
(2) if N is of type (b), the fibers of π are of infinite volume, the sectional curvature of M is
negatively pinched, and the covariant derivative of the curvature tensor of M is uniformly
bounded,
then σ(N) =
[
h2/4,∞
)
.
Since Euclidean space Rm has vanishing curvature and a basis of parallel differential forms, its
Hodge–Laplacian (d+d∗)2 decomposes into parts which are unitarily equivalent to its Laplacian
on functions. In particular, its differential form spectrum is absolutely continuous. One might
ask whether the differential form spectrum of Hadamard manifolds as considered above is also
absolutely continuous. Except for symmetric spaces, we do not know of any such general result.
But there are results about the essential spectrum of the Hodge–Laplacian, given sufficiently
strong pinching of sectional curvature (depending on dimension and degree), see for exam-
ple [9, 12], and, for asymptotically hyperbolic manifolds, on the non-existence of eigenvalues
above the essential spectrum, see [14, Theorem 16].
Regarding the absolute continuity of the spectrum, we rely on the same integral formulae
as the ones used by Xavier [19] and Donnelly–Garofalo [7, 8], using, in particular, arguments
from [19]. One important difference to the latter is that we also use functions which are convex,
but not strictly convex. The determination of the spectra as sets is by different arguments,
where we rely on [15] in the case of homogeneous Hadamard manifolds.
2 Preliminaries
For a Hadamard manifoldM and points ξ ∈ M∞ and x ∈ M , the Busemann function b associated
to ξ with b(x) = 0 is given by
b(y) = lim
n→∞
(d(xn, y)− d(xn, x)),
where (xn) is any sequence in M converging to ξ. Recall that the limit exists and that b is a C2
distance function on M with horoballs and horospheres centered at ξ as sublevel and level sets;
cf. [3] for this and other facts about Hadamard manifolds.
4 W. Ballmann, M. Mukherjee and P. Polymerakis
Observation 2.1. If a Hadamard manifold M is asymptotically harmonic about a point ξ ∈ M∞
or if N = Γ\M is of type (b), then the Cheeger constant of M respectively N is at least h,
where h is the mean curvature of the horospheres with center ξ. In particular, the spectrum
of M respectively N is contained in
[
h2/4,∞
)
.
Throughout the article, we use the absolute value sign to denote the appropriate volumes of
sets under discussion.
Proof of Observation 2.1. Let D be a compact domain in M respectively N with smooth
boundary. Let b be a Busemann function associated to ξ and X = ∇b. Then |X| = 1 and
divX = h and hence
h|D| =
∫
D
divX =
∫
∂D
⟨X, ν⟩ ≤ |∂D|,
where ν is the outer normal field of D along ∂D. The last assertion is just the Cheeger in-
equality. ■
Observation 2.2. If a Hadamard manifold M is asymptotically harmonic about a point ξ ∈ M∞
and b is a Busemann function of M centered at ξ, then
0 ≤ ∇2b(v, v) ≤ h|v|2.
Proof. The claim follows immediately from h = tr∇2b and ∇2b ≥ 0. ■
Suppose that a Lie group G acts properly, freely, and isometrically on a complete Rieman-
nian manifold M . Then the quotient Q = G\M is a smooth manifold. Moreover, Q inherits
a Riemannian metric such that the projection π : M → Q is a Riemannian submersion. The
mean curvature field H of the fibers is π-related to a vector field on Q, which is then the
push-forward π∗H of H. Following [15], we define a Schrödinger operator on Q,
S = ∆+
1
4
|π∗H|2 − 1
2
div π∗H. (2.1)
For the determination of the spectrum σ(M) of a homogeneous Hadamard manifold M , we will
then invoke [15, Theorem 1.3] in the following form.
Theorem 2.3. Suppose that G is amenable and that its component of the identity is unimodular.
Then
λ0(M) = λ0(S) =: λ0 and σ(S) ⊆ σ(M),
where λ0(M) and λ0(S) denote the bottom of the spectrum σ(M) of M and σ(S) of S, respec-
tively. In particular, σ(S) = [λ0,∞) implies that σ(M) = [λ0,∞).
For a boundary condition B of a domain D in a Riemannian manifold M , denote by C∞
c,B(D̄)
the space of smooth functions on D̄ with compact support which satisfy B along the bound-
ary ∂D of D.
Definition 2.4. We say that a boundary condition B for a smooth domain D is non-positive
if u ∈ C∞
c,B(D̄) implies 2u∇u = ∇νu
2 ≤ 0 along ∂D.
Examples 2.5. Dirichlet and Neumann boundary conditions are non-positive. Robin boundary
conditions αu+ β∇νu = 0 are non-positive if αβ > 0. These kinds of boundary conditions are
also self-adjoint and elliptic in the usual sense.
On the Spectrum of Certain Hadamard Manifolds 5
3 Vanishing of point spectrum
For a C1 vector field X and a C2 function u on a Riemannian manifold M , we have the following
identity
2⟨∇∇uX,∇u⟩ = 2⟨X,∇u⟩∆u+ |∇u|2 divX + div
{
2X(u)∇u− |∇u|2X
}
. (3.1)
In the context of spectral theory, this identity was used by Donnelly–Garofalo, see [7, Lemma 2.1],
[8, Proposition 3.1], and Xavier, see [19, identity (1)], but it also occurs in the earlier work of
Kazdan–Warner [13, identity (8.1)]. Donnelly and Garofalo attribute (3.1) to Rellich [17] in the
case of Euclidean spaces; cf. (3) respectively II on page 62 of [17]. For the convenience of the
reader, we give a short proof of the identity.
Proof of (3.1). Since X(u) = ⟨∇u,X⟩ and |∇u|2 = ⟨∇u,∇u⟩ = ∇u(u), we have
2 div(X(u)∇u) = 2∇u(X(u)) + 2X(u) div∇u
= 2X(∇u(u)) + 2[∇u,X](u)− 2X(u)∆u
= 2X
(
|∇u|2
)
+ 2(∇∇uX)(u)− 2(∇X∇u)(u)− 2X(u)∆u
= 2X
(
|∇u|2
)
+ 2⟨∇∇uX,∇u⟩ − 2⟨∇X∇u,∇u⟩ − 2X(u)∆u
= X
(
|∇u|2
)
+ 2⟨∇∇uX,∇u⟩ − 2X(u)∆u
= div
(
|∇u|2X
)
− |∇u|2 divX + 2⟨∇∇uX,∇u⟩ − 2X(u)∆u,
which is the assertion. ■
Corollary 3.1. If ∆u = φu for a C1 function φ on M , then
2⟨∇∇uX,∇u⟩ = φX(u2) + |∇u|2 divX + div
{
2X(u)∇u− |∇u|2X
}
(3.2)
=
(
|∇u|2 − φu2
)
divX −X(φ)u2
+ div
{
2X(u)∇u+
(
φu2 − |∇u|2
)
X
}
. (3.3)
Proof. If ∆u = φu, then
2X(u)∆u = 2φX(u)u = φX
(
u2
)
= X
(
φu2
)
−X(φ)u2
= div
(
φu2X
)
− φu2 divX −X(φ)u2.
Together with (3.1), this gives (3.2) and (3.3). ■
In [8], Donnelly and Garofalo consider eigenfunctions Lu = λu of Schrödinger operators
L = ∆+ V . This corresponds to φ = λ− V in (3.2) and (3.3).
To exhibit the strength of (3.1), we present immediate applications of the above identities.
With somewhat more elaborate techniques, we will obtain stronger results later. Our results are
reminiscent of [17, Satz 2].
Theorem 3.2. Suppose that KM < 0 and that M is asymptotically harmonic about a point
ξ ∈ M∞. Let D be the complement of a horoball in M with center ξ. Then the point spectrum
HB,pp = HB,pp(∆, D̄) vanishes for any self-adjoint elliptic boundary condition B for D, which
is non-positive.
Proof. Let b be the Busemann function associated to ξ such that D = b−1((0,∞)). By as-
sumption, there is a number h, because of negative curvature strictly positive, such that the
6 W. Ballmann, M. Mukherjee and P. Polymerakis
mean curvature of the horospheres b−1(r) is constant, equal to h. In other words, div∇b = h.
Let
X =
(
1− e−hb
)
∇b. (3.4)
Then
divX = h and ∇X = he−hb∇b⊗∇b+
(
1− e−hb
)
∇2b.
Now ∇2b vanishes in the direction of ∇b and is positive definite perpendicular to it. Since
1− e−hb > 0 on D, we conclude that ∇X is positive definite on D. By Observation 2.2, ∇2b is
bounded on M , hence ∇X on D.
Let u be a square-integrable smooth function on D with ∆u = λu, satisfying the boundary
condition B. Since X vanishes on ∂D = b−1(0) and ∇X is bounded on D, integration of (3.3),
with φ = λ, is justified and implies that
2⟨∇∇uX,∇u⟩2 = h
∫
D
(
|∇u|2 − λu2
)
= h
∫
D
(
u∆u− λu2
)
+ h
∫
∂D
u∇νu
= h
∫
∂D
u∇νu ≤ 0, (3.5)
where the inequality is implied by the boundary condition B. Now ∇X is positive definite on D,
therefore ∇u vanishes, and hence u is constant. On the other hand, the volume of D is infinite
and u is square-integrable, thus u vanishes. ■
Restricting to Dirichlet or Neumann boundary conditions, we can allow for more general
kinds of domains. Let M be a Hadamard manifold. Say that a subset D ⊆ M is a shadow if
there is a horosphere H ⊆ M and a subset C ⊆ H such that D is the set of all points x ∈ M
such that the ray from x to the center ξ of H passes through C. Then we also say that D is
the shadow of C, thinking of ξ as a source of light. The exterior of a horoball considered in
Theorem 3.2 above corresponds to C = H.
Theorem 3.3. Suppose that KM < 0 and that M is asymptotically harmonic about a point
ξ ∈ M∞. Let D ⊆ M be the shadow of a smooth domain C in a horosphere H in M with
center ξ. Then the Dirichlet and Neumann point spectra of D vanish.
Proof. Let b be the Busemann function on M associated to ξ which vanishes on H = b−1(0)
and X be the vector field as in (3.4). As in the previous proof, we get that ∇X is positive
definite on b−1((0,∞)), and hence, in particular, on D \ C.
Let u be a square-integrable smooth function on D with ∆u = λu, satisfying the Dirichlet
or the Neumann boundary condition. Now the boundary of D consists of the smooth domain
C ⊆ H and the shadow of ∂C, where X is tangential to it. Since C is smooth, ∂C is the singular
part of ∂D. Now the codimension of ∂C in M is two, and hence we can apply the divergence
formula when integrating (3.2). Since X vanishes on H, the contribution of C vanishes in the
integrated version of (3.3). As for the shadow part, the contribution of the first boundary term
2X(u)∇νu vanishes since then X(u) = 0 in the case of the Dirichlet boundary condition and
∇νu = 0 in the case of the Neumann boundary condition. The contribution of the second term,(
λu2 − |∇u|2
)
X, always vanishes since ⟨X, ν⟩ = 0 in the shadow part. Thus repeating the
computation in (3.5), we get
⟨∇∇uX,∇u⟩2 = h
∫
∂D
u∇νu = 0.
Since ∇X is positive definite on D \ C, we conclude, as above, that u = 0. ■
On the Spectrum of Certain Hadamard Manifolds 7
4 Absolutely continuous spectrum
Let S be a self-adjoint operator in a separable Hilbert space H, and denote by R(z) = (S−z)−1
the resolvent of S. Say that a closed operator T in H is S-smooth if, for each x ∈ H and ε ̸= 0,
R(λ+ iε)x ∈ DomT for almost all λ ∈ R and
sup
|x|=1, ε ̸=0
∫ ∞
−∞
|TR(λ+ iε)x|2 < ∞,
see [16, p. 142]. In our context, the point is that then the image
RanT ∗ ⊆ Hac(S), (4.1)
see [16, Theorem XIII.23]. For a Borel subset B ⊆ R, say that a closed operator T in H is
S-smooth on B if TEB is H-smooth, where EB denotes the spectral projection of S associated
to B. Then
RanEBT
∗ ⊆ Hac(S),
by (4.1). By [16, Theorem XIII.30], T is S-smooth on the closure B̄ of B ⊆ R if
DomT ⊇ DomS and sup
|x|=1, λ∈B,
0<|ε|<1
|ε||TR(λ+ iε)x| < ∞. (4.2)
Xavier [19, p. 581f.] shows that the latter criterion is satisfied if B is bounded and if there is
a constant C such that
|Tx|2 ≤ C|(S − λ)x|(|Sx|+ |x|) (4.3)
for all x in a core of S in H and all λ ∈ B. In fact, he shows that the term in (4.2) is then
bounded by C(1 + |λ| + |ε|), which explains why B is assumed to be bounded; see [19, top
of p. 582].
Guided by [19, Section 3], we return to the geometric situation in Section 3 and let X be a C1
vector field and u be a C2 function on a Riemannian manifold M such that suppX ∩ suppu is
compact. Then (3.1) gives, for any smooth domain D in M with outer normal ν along ∂D,∫
D
⟨X,∇u⟩∆u+
1
2
∫
D
|∇u|2 divX
=
∫
D
⟨∇∇uX,∇u⟩ −
∫
∂D
(
⟨X,∇u⟩⟨∇u, ν⟩ − 1
2
|∇u|2⟨X, ν⟩
)
.
As in [19, p. 583], we use
|∇u|2 = u∆u− 1
2
∆u2
and obtain∫
D
∆u
(
⟨X,∇u⟩+ u
2
divX
)
=
∫
D
(
⟨∇∇uX,∇u⟩+ 1
4
divX∆u2
)
−
∫
∂D
(
⟨X,∇u⟩⟨∇u, ν⟩ − 1
2
|∇u|2⟨X, ν⟩
)
.
Since
PXu = ⟨X,∇u⟩+ u
2
divX
8 W. Ballmann, M. Mukherjee and P. Polymerakis
satisfies
uPXu = div
(
1
2
u2X
)
,
we arrive at∫
D
(∆u− λu)PXu =
∫
D
(
⟨∇∇uX,∇u⟩+ 1
4
divX∆u2
)
−
∫
∂D
(
⟨X,∇u⟩⟨∇u, ν⟩ − 1
2
(
|∇u|2 − λu2
)
⟨X, ν⟩
)
, (4.4)
which corresponds to [19, identity (2)], but with boundary integral included. It will also be
useful to have the latter formula with one of the substitutions∫
D
divX∆u2 =
∫
D
〈
∇ divX,∇u2
〉
−
∫
∂D
divX∇νu
2 (4.5)
=
∫
D
(∆divX)u2 +
∫
∂D
(
(∇ν divX)u2 − divX∇νu
2
)
,
which is Green’s formula applied to the functions divX and u2. Clearly,∫
D
(∆u− λu)PXu ≤ ∥∆u− λu∥2∥PXu∥2 ≤ C∥∆u− λu∥2(∥∆u∥2 + ∥u∥2) (4.6)
if X and divX are bounded on D, and u satisfies a non-positive boundary condition. This will
be important in view of (4.3).
Theorem 4.1. Suppose that KM < 0 and that M is asymptotically harmonic about a point
ξ ∈ M∞. Let D be the complement of a horoball in M with center ξ and B be a self-adjoint
elliptic boundary condition B for D which is non-positive. Then the spectrum of ∆ on D with
respect to B is absolutely continuous.
Proof. We return to the setup in the proof of Theorem 3.2. Let b be the Busemann function
associated to ξ such that D = b−1((0,∞)) and
X =
(
1− e−hb
)
∇b.
Since |∇b| = 1, |X| < 1 on D. Recall that divX = h > 0 and that ∇X is positive definite on D.
As in the proof of [19, Theorem 2], let Y be a smooth vector field on D with compact support
in D and T = TY be differentiation of functions in the direction of Y . Then there are constants
C1, C2 > 0 such that ∇X ≥ C1 on the support of Y and such that |Y |2 ≤ C2.
Now C∞
c,B(D̄), the space of smooth functions on D̄ with compact support in D̄ satisfying
one of the above boundary condition B, is a core for ∆ with boundary condition B. Since
divX = h > 0 is constant and X vanishes along ∂D, (4.4) and (4.5) yield, for u ∈ C∞
c,B(D̄),∫
D
(∆u− λu)PXu =
∫
D
(
⟨∇∇uX,∇u⟩+ h
4
∆u2
)
=
∫
D
⟨∇∇uX,∇u⟩ − h
2
∫
∂D
u∇νu
≥
∫
D
⟨∇∇uX,∇u⟩ ≥ C1
∫
suppY
|∇u|2
≥ C1
C2
∫
D
|Y u|2 = C1
C2
∥TY u∥22.
On the Spectrum of Certain Hadamard Manifolds 9
Now (4.6) applies and shows that TY is S-smooth on any given bounded Borel subset A ⊆ R.
Hence we get from (4.1) that RanEAT
∗
Y ⊆ Hac(∆, B), for any bounded Borel subset A ⊆ R.
Therefore, RanT ∗
Y ⊆ Hac(∆, B).
Now suppose that u ∈ L2(D) is perpendicular to Hac(∆, B). Then u is perpendicular to
RanT ∗
Y , for all vector fields Y as above. But then TY u = 0 in the sense of distributions, for
all such vector fields Y . This implies that u is constant and, therefore, that u vanishes, by
square-integrability. ■
Theorem 4.2. Suppose that KM < 0 and that M is asymptotically harmonic about a point
ξ ∈ M∞. Let D ⊆ M be the shadow of a smooth domain C in a horosphere H in M with
center ξ. Then the Dirichlet and Neumann spectra of ∆ on D are absolutely continuous.
Proof. The proof is similar to the above one, changing the proof of Theorem 3.3, instead of
Theorem 3.2, analogously. ■
Proof of first part of Theorem 1.2. Let b be a Busemann function associated to ξ and
X = ∇b. Then |X| = 1 and divX = h > 0. Furthermore, ∇X is positive definite on the
orthogonal complement of X, that is, in the direction to the horospheres in M with center ξ.
Let Y be a smooth vector field on M with compact support such that Y ⊥ X. Then there
are constants C1, C2 > 0 such that ∇X ≥ C1 on the support of Y and perpendicularly to X
and such that |Y |2 ≤ C2.
Now C∞
c (M) is a core for ∆. Since divX = h > 0 is constant, (4.4) and (4.5) yield, for
u ∈ C∞
c (M),∫
M
(∆u− λu)PXu =
∫
M
⟨∇∇uX,∇u⟩ ≥ C1
∫
M
|∇⊥u|2
≥ C1
C2
∫
M
|Y u|2 = C1
C2
∥TY u∥22,
where ∇⊥u denotes the component of ∇u perpendicular to X. Now (4.6) applies and shows
that TY is S-smooth on any given bounded Borel subset A ⊆ R. Hence (4.1) implies that
RanEAT
∗
Y ⊆ Hac(∆), for any bounded Borel subset A ⊆ R. Therefore, RanT ∗
Y ⊆ Hac(∆).
Now suppose that u ∈ L2(D) is perpendicular to Hac(∆). Then u is perpendicular to RanT ∗
Y ,
for all vector fields Y as above. But then TY u = 0 in the sense of distributions, for all such
vector fields Y . This implies that u is constant along horospheres, that is, levels of u are unions
of horospheres. Now the preimage b−1(B) of any Borel subset B ⊆ R of positive measure has
infinite measure. Therefore, u vanishes, by square-integrability. ■
Proof of part of Theorem 1.3. By the definition of elementary groups of isometries of M ,
the gradient field X of Busemann functions, that is, the field of velocity vectors of unit speed
geodesics emanating from ξ, is invariant under Γ. Notice that X spans the normal space to the
fibers of the Riemannian submersion π. Now the arguments of the previous proof apply and
show that u ∈ L2(N) is perpendicular to Hac(N) if and only if u is constant on the fibers of π.
There are now two cases: The fibers of π have infinite volume. This is always the case when Γ
is of type (a). Then, as in the previous proof, the square-integrability of u implies that u = 0.
In the second case, Γ is of type (b). Then the flow (Ft)t∈R of X induces diffeomorphisms
between the fibers of π = b, and the volume element of horospheres is multiplied by exp(ht)
under Ft. Hence, since the fibers of b have finite volume, their volumes satisfy
exp(ht)
∣∣b−1(s)
∣∣ = exp(hs)
∣∣b−1(t)
∣∣.
Furthermore, the space L2
b(N) of functions onN , which are constant fiberwise, and its orthogonal
complement L2
0(N) are invariant under ∆. That is, the spectrum of ∆ is the combination of the
10 W. Ballmann, M. Mukherjee and P. Polymerakis
spectra of ∆ on L2
b(N) and of ∆ on L2
0(N). By what we said above, if u ∈ L2
0(N) is perpendicular
to Hac(N), then u = 0. Thus L2
0(N) ⊆ Hac(N), and we are left with discussing ∆ on L2
b(N).
Note that u ∈ L2(N) is in L2
b(N) if and only if u is a pull-back b∗v for some function v on R.
We endow R with the measure µ = h0 exp(ht)dt, where h0 =
∣∣b−1(0)
∣∣ and get that
b∗ : L2(R, µ) → L2
b(N)
is a unitary transformation. With respect to b∗, the Laplacian on N corresponds to a diffusion
operator on R,
∆b∗v = b∗
(
−v′′ − hv′
)
.
Let φ be the square root of h0 exp(ht). Then
φ′ =
h
2
φ and φ′′ =
h2
4
φ.
Therefore, under the unitary transformation
L2(R, µ) → L2(R), v 7→ φv,
we obtain
(φv)′′ = φ′′v + 2φ′v′ + φv′′ = φ
(
h2
4
v + hv′ + v′′
)
.
We conclude that the Schrödinger operator Sw = −w′′ + h2w/4 in L2(R) corresponds to the
above diffusion operator on L2(R, µ). Since h2/4 is a constant, the spectrum of the latter is
absolutely continuous and equal to
[
h2/4,∞
)
. By unitary equivalence, the same holds for ∆
on L2
b(N).
By Observation 2.1, σ(N) ⊆
[
h2/4,∞
)
, hence the above shows equality, σ(N) =
[
h2/4,∞
)
,
in the case where the fibers of π are of finite volume. The case where the fibers are of infinite
volume will be discussed in Section 6. ■
5 Homogeneous Hadamard manifolds
The aim of this section is the proof of Theorem 1.1. In contrast to Theorem 1.2, we do not
assume that the sectional curvature is negative. For that reason, we need to investigate the
underlying geometry of the manifolds more carefully and start with the necessary background
material. Our main source is [18], and we refer the reader to it for more details and references.
Another good reference is [10].
Let M be a homogeneous Hadamard manifold. Then there is a solvable Lie group S of
isometries of M which acts simply transitively on M . By choosing an origin x0 of M , the orbit
map S → M , g 7→ gx0, is a diffeomorphism, and the pull-back of the Riemannian metric of M
to S is a left-invariant metric on S. In this way, we identify M with the simply connected
solvable Lie group S, endowed with the above left-invariant Riemannian metric.
Let s be the Lie algebra of S, Then n = [s, s] is nilpotent, the orthogonal complement a = n⊥
of n in s is Abelian, and S = A ⋉ N , where A and N denote the connected Lie subgroups
of S tangent to a and n, respectively. Note that A and N are simply connected and, therefore,
non-compact. By Corollary A.12, we can assume that M respectively S has no Euclidean factor.
This is convenient since it keeps the structure of S more accessible; cf. [18, Section 1].
For v ∈ a, let Dv and Sv be the symmetric and skew-symmetric parts of adv |n.
On the Spectrum of Certain Hadamard Manifolds 11
Proposition 5.1. There are an ad a-invariant orthogonal decomposition
n = n1 ⊕ · · · ⊕ nk
of n into subalgebras nj, roots µj ∈ a∗, and positive definite symmetric operators Dj on nj, for
1 ≤ j ≤ k, such that
(1) for all v ∈ a, the restriction of the symmetric part Dv of adv to nj is given by µj(v)Dj;
(2) the µj are pairwise not collinear and span a∗;
(3) W = {v ∈ a | µj(v) > 0 for all 1 ≤ j ≤ k} ≠ ∅.
Let v ∈ W and γ be the geodesic through the neutral element of S with initial velocity v.
By [18, Proposition 3.3], S fixes ξ = γ(∞). Hence, since all left-translations are isometries of S,
we conclude that S is asymptotically harmonic about ξ.
View elements of s now as left-invariant vector fields on S, where we use V to denote the
one corresponding to v. Since S fixes ξ, the gradient of Busemann functions on S centered at ξ
equals −V . Hence the second derivative of Busemann functions on S centered at ξ equals −∇V .
By (1) and the choice of v, there exists a constant c > 0 such that
⟨[v, z], z⟩ = ⟨adv z, z⟩ = ⟨Dvz, z⟩ ≥ c|z|2
for all z ∈ n. Hence we obtain
−⟨∇ZV,Z⟩ = ⟨[V,Z], Z⟩ ≥ c|Z|2
for any left-invariant vector field Z coming from n. Since a is Abelian,
−⟨∇Y V, Y ⟩ = ⟨[V, Y ], Y ⟩ = 0
for any left-invariant vector field Y coming from a. Finally, for Z coming from n and Y from a,
we compute
⟨∇ZV, Y ⟩+ ⟨∇Y V,Z⟩ = ⟨∇V Z, Y ⟩+ ⟨∇Y V,Z⟩ = ⟨Z, [Y, V ]⟩ = 0,
where we use that n is an ideal and that a is Abelian.
Proof of first part of Theorem 1.1. Now we adapt the arguments from the proof of Theo-
rem 1.2 to the present situation and set X = −V . Recall that V is a gradient field so that ∇X is
symmetric. On the left-invariant distribution Sa coming from a, we have ∇X = 0. With C1 = c,
where c is as above, we have ∇X ≥ C1 on the left-invariant distribution Sn coming from n. Fur-
thermore, ⟨∇SaX,Sn⟩ = 0.
Let Y be a smooth vector field on S with compact support which is perpendicular to Sa.
Choose a constant C2 > 0 such that |Y |2 ≤ C2.
Now C∞
c (M) is a core for ∆. Since divX = h > 0 is constant, (4.4) and (4.5) yields, for
u ∈ C∞
c (M),∫
M
(∆u− λu)PXu =
∫
M
⟨∇∇uX,∇u⟩ ≥ C1
∫
M
|∇⊥u|2
≥ C1
C2
∫
M
|Y u|2 = C1
C2
∥TY u∥22,
where ∇⊥u denotes the component of ∇u perpendicular to Sa. Now (4.6) applies and shows
that TY is S-smooth on any given bounded Borel subset B ⊆ R. Hence (4.1) implies that
RanEBT
∗
Y ⊆ Hac(M), for any bounded Borel subset B ⊆ R. Therefore, RanT ∗
Y ⊆ Hac(M).
12 W. Ballmann, M. Mukherjee and P. Polymerakis
Now suppose that u ∈ L2(M) is perpendicular to Hac(M). Then u is perpendicular to
RanT ∗
Y , for all vector fields Y as above. But then TY u = 0 in the sense of distributions, for
all such vector fields Y . This implies that u is constant along the orbits of N , since the orbits
of N meet the distribution Sa orthogonally. Since N is non-compact, the N -orbits have infinite
(dimN -dimensional) volume. Hence the union of orbits meeting a Borel subset of an A-orbit
of positive (dimA-dimensional) volume has infinite volume in S. Since u is square-integrable,
we conclude that u is identically zero. This concludes the proof that the spectrum of M is
absolutely continuous. ■
The second part of Theorem 1.1 holds under much more general assumptions. Therefore, we
state it as an extra result here.
Theorem 5.2. Let G be a connected Lie group, endowed with a left-invariant Riemannian
metric. Assume that G is solvable or is a compact extension of a solvable Lie group, Let N be
the derived subgroup of G, and assume that Q = N\G is non-compact. Then σ(G) =
[
h2/4,∞
)
,
where h is the mean curvature of N as a submanifold of G, given by
h2 =
∑
j
(tr adXj |n)2 =
∑
j
(tr adXj )
2.
Here (Xj) is an orthonormal basis of the orthogonal complement of n in s.
Proof. We invoke Theorem 2.3. The Riemannian manifold M there corresponds to the Lie
group G here, the Lie group G acting on M there corresponds to N here, acting freely on G
by left-translations. Since N is a closed subgroup of G and the Riemannian metric on G is
left-invariant, the action is proper and isometric.
Since N is the derived subgroup of G, it is a closed and normal subgroup of G such that the
quotient Q = N\G is an Abelian Lie group. As the derived subgroup of a connected Lie group,
N is connected.
Since N is normal in G, the fibration π : G → Q is G-invariant. Therefore, the left-invariant
Riemannian metric on G induces a left-invariant Riemannian metric on Q, such that π is a Rie-
mannian submersion. Since Q is connected and Abelian, the left-invariant metric on M is flat
and Q is a Euclidean space E times a torus T . Since Q is non-compact, dimE > 0. From
Corollary A.10 and Example A.11, we conclude that the Laplace spectrum σ(Q) = [0,∞).
Since π is invariant under left-translation by G, the field H of mean curvature of the fibers is
left-invariant and is π-related to a left-invariant vector field on Q, the push-forward π∗H of H.
Since G is amenable, being a compact extension of a solvable Lie group, we derive that so is
the closed subgroup N . Furthermore, as the derived subgroup of a Lie group, N is unimodular.
Thus we may apply Theorem 2.3 to obtain that σ(M) = [λ0,∞), where λ0 is equal to the bottom
of the spectrum of the Schrödinger operator S on Q defined in (2.1).
Since Q is Abelian and π∗H is left-invariant, π∗H is parallel, hence of constant norm with
vanishing divergence. Therefore, the potential of the Schrödinger operator S in (2.1) is constant
and equal to h2/4 = |π∗H|2/4 = |H|2/4. Since the bottom of the spectrum of the Laplacian
on Q is equal to 0, we get λ0 = h2/4 as claimed.
Let (Yi) be an orthonormal basis of n. Then the mean curvature field H of the fibers of π
is the left-invariant vector field which, at the neutral element, is equal to the component of∑
i∇YiYi perpendicular to n. If (Xj) is an orthonormal basis of the orthogonal complement of n
in s, then we get that
H =
∑
i,j
⟨∇YiYi, Xj⟩Xj =
∑
i,j
⟨[Xj , Yi], Yi⟩Xj =
∑
j
(tr adXj |n)Xj ,
which implies the first asserted equality. The second follows since the orthogonal complement
of n in s does not contribute to tr adXj . ■
On the Spectrum of Certain Hadamard Manifolds 13
6 Determination of the spectrum
In this section we aim at the determination of the spectrum σ(M). We start with the case
where M is a Hadamard manifold with sectional curvature KM < 0 which is asymptotically
harmonic about a point ξ ∈ M∞ with mean curvature of horospheres equal to h > 0. Recall
from the first part of the proof of Theorem 1.3 that the case left is N = Γ\M , where the fibers
of the associated Riemannian submersion π = b : N → R are of infinite volume, where b is the
push-down to N of some Busemann function associated to ξ. We subsume the case N = M here
by including Γ = {id}. From Observation 2.1, we know that σ(N) ⊆
[
h2/4,∞
)
.
For any smooth function v on R, we have
∆b∗v = b∗
(
−v′′ − hv′
)
.
Now φ = φ(t) = exp(ht/2) satisfies φ′ = hφ/2 and φ′′ = h2φ/4. With µ = φ2dt, the unitary
transformation
Φ: L2(R, µ) → L2(R), Φ(v) = φv,
yields
(Φ(v))′′ = (φv)′′ = φ
(
h2
4
v + hv′ + v′′
)
,
and hence Φ intertwines the diffusion operator Lv = −(v′′+hv′) in L2(R, µ) with the Schrödinger
operator Sw = −w′′ + h2w/4 in L2(R). Therefore, the spectra of L and S coincide, where we
recall that Hac(S) = L2(R) with σ(S) =
[
h2/4,∞
)
.
We want to show that σ(S) ⊆ σ(N). In contrast to the case where the volume of the fibers of b
are of finite volume, we cannot simply use pull backs b∗v since we would lose square-integrability
that way. What we will do is to carefully cut off pull-backs.
Lemma 6.1. Suppose that −1 ≤ KM ≤ −a2 < 0 and that ∥∇RM∥∞ < ∞. Let χ0 : b
−1(0) → R
be a non-zero C2 function with compact support. Extend χ0 to N by letting χ = χt be equal
to F ∗
−tχ0 on b−1(t), where (Ft)t∈R denotes the flow of X = ∇b. Then χ is C2 and ∇χ and ∆χ
tend to zero uniformly as t → ∞.
Proof. Since χ is constant along the flow lines of X, the gradient of χ is tangential to the
fibers of b−1(t) of b. By the upper curvature bound −a2, unstable Jacobi field J grow at least
exponentially,
|J(t)| ≥ eat|J(0)| for all t > 0.
Therefore,
|∇χt| ≤ e−at|∇χ0| for all t > 0,
which implies the first assertion.
For the estimate of the Laplacian, we refer to [5, Section 6]. The situation there is that
of a Hadamard manifold X with pinched negative sectional curvature and uniformly bounded
covariant derivative of its curvature tensor, a convex domain C in X, and a function obtained by
extending a given function on ∂C to X \C constantly along minimizing geodesics to C. In our
situation, X corresponds to M , C to the horoball b−1((−∞, 0]), and the given function to χ0.
A short look at the arguments in [5, Section 6] shows that they also apply in the corresponding
situation in N . The estimate obtained in [5] is that
|∆χ| ≤ c0e
−at over b−1(t) for all t > 0,
where c0 is a constant, which depends on bounds for KM , ∇RM , ∇χ0, and ∇2χ0; compare
with [5, formula (6.4) and end of Section 6]. ■
14 W. Ballmann, M. Mukherjee and P. Polymerakis
End of proof of Theorem 1.3. Let λ ∈ σ(S) and ε > 0. Choose a non-vanishing w ∈ C∞
c (R)
such that ∥(S − λ)w∥2 ≤ ε∥w∥2. By shifting w along R if necessary, we can assume that
suppw ⊆ [t,∞), where t > 0 is such that |∆χ| < ε on b−1([t,+∞)). Note that shifting w is
legitimate since Sw is only shifted accordingly. Then v = w/φ also has support in [t,∞) and
satisfies ∥(L− λ)v∥2 ≤ ε∥v∥2, where now the L2-norm is taken with respect to µ = φ2dt. Then
∆(χb∗v) = (∆χ)b∗v − 2⟨∇χ,∇b∗v⟩+ χ∆b∗v = (∆χ)b∗v + χb∗Lv
since ∇χ is perpendicular to X and ∇b∗v is collinear with X and since b∗ intertwines ∆ with L.
Since b is a Riemannian submersion and the flow maps Fs of X induces a diffeomorphism
of b−1(0) with b−1(s), for all s ∈ R and with respective Jacobian φ2(s) = ehs. Thus we iden-
tify x ∈ N with (y, s) ∈ b−1(0)× R, where b(x) = s and Fs(y) = x, and get
∥χb∗v∥22 =
∫
χ2(x)(b∗v(x))2dx
=
∫∫
b−1(0)×R
χ2
0(y)v
2(s)φ2(s)dyds
= ∥χ0∥22
∫
R
v2(s)φ2(s)ds = ∥χ0∥22∥v∥22,
where ∥χ0∥2 denotes the L2-norm of χ0 in L2
(
b−1(0)
)
. Likewise, by the choice of t,
∥(∆χ)b∗v∥22 ≤ ε2| suppχ0|∥v∥22.
and hence
∥(∆χ)b∗v∥22 ≤ ε2
| suppχ0|
∥χ0∥22
∥χb∗v∥22 = ε2c21∥χb∗v∥22,
where c1 depends on the choice of χ0. Hence we obtain
∥(∆− λ)(χb∗v)∥22 ≤ 2∥χb∗(L− λ)v∥22 + 2ε2c21∥χb∗v∥22
= 2∥χ0∥22∥(L− λ)v∥22 + 2ε2c21∥χb∗v∥22
= ε2c22∥χb∗v∥22,
where c2 depends on the choice of χ0. This implies that λ ∈ σ(N). ■
A Some spectral theory
We include a short discussion of spectra of self-adjoint operators and of Riemannian manifolds.
The main aim is the description of the spectra of Riemannian products. The latter is well-known
and clear in the compact case, but we could not single out a reference for the general case as
presented in Appendix A.3. The product case is important in our discussion of homogeneous
Hadamard manifolds, where it is used for the reduction to the case of vanishing Euclidean factor,
which keeps the technicalities of the discussion at a moderate level; cf. Section 5.
A.1 Convolution of measures
Recall that a finite Borel measure ν on R is uniquely a sum of two finite Borel measures νac
and νs on R, where νac is absolutely continuous and νs is singular with respect to Lebesgue
measure λ. Then R is the disjoint, but not unique, union of a Lebesgue measurable set Yac and
a Lebesgue nullset Ys such that νs is concentrated on Ys, that is, νs(R \ Ys) = 0. The countable
On the Spectrum of Certain Hadamard Manifolds 15
set Ypp = {y ∈ R | ν(y) > 0} is a subset of Ys and νs = νpp + νsc, where νpp is concentrated
on Ypp and νsc on Ysc = Ys \ Ypp.
The convolution ν1 ∗ ν2 of finite Borel measures ν1 and ν2 on R is defined to be the push-
forward of the finite Borel measure ν1 ⊗ ν2 on R2 under the map
R2 → R, (x1, x2) 7→ x1 + x2.
In other words, for any Borel set B ⊆ R,
(ν1 ∗ ν2)(B) = (ν1 ⊗ ν2)({(x1, x2) | x1 + x2 ∈ B})
=
∫∫
χB(x1 + x2) dν1(x1)dν2(x2).
Clearly ν1 ∗ ν2 = ν2 ∗ ν1. Furthermore, if ν1 = g1λ with a λ-integrable g1 on R, then
ν1 ∗ ν2 = gλ with λ-integrable function g(y) =
∫
g1(y − z) dν2(z). (A.1)
In particular, if in addition ν2 = g2λ with a λ-integrable g2 on R, then
ν1 ∗ ν2 = gλ with λ-integrable function g(y) =
∫
g1(y − z)g2(z) dλ(z). (A.2)
Notice that the latter g = g1 ∗ g2, the usual convolution of g1 and g2.
The support of a measure ν on R is the closed set of points y ∈ R such that ν(U) > 0 for any
neighborhood U of y.
Proposition A.1. The convolution ν = ν1 ∗ ν2 has support
supp ν = supp ν1 + supp ν2.
If supp ν1 and supp ν2 are bounded from below, then supp ν = supp ν1 + supp ν2.
For Borel measures µ, ν on R, we write µ ≺ ν to indicate that µ is absolutely continuous with
respect to ν.
Proposition A.2. The components of the convolution ν = ν1 ∗ ν2 satisfy
νpp = ν1pp ∗ ν2pp with Ypp = Y1pp + Y2pp;
νac ≻ ν1ac ∗ ν2ac + ν1ac ∗ ν2s + ν1s ∗ ν2ac.
In particular, if ν1 or ν2 is absolutely continuous with respect to λ, then also ν.
In general, we cannot exclude that the absolutely continuous component of the convolution
of λ-singular measures vanishes. Thus ν1s ∗ ν2s may contribute to νac.
Proof of Proposition A.2. The assertion about νpp is clear from
(ν1 ∗ ν2)(y) =
∑
y1+y2=y
ν1(y1)ν2(y2).
The assertion about νac follows immediately from (A.1) and (A.2). ■
16 W. Ballmann, M. Mukherjee and P. Polymerakis
A.2 Decomposition of spectra
Let A be a self-adjoint operator on a (real or complex, separable) Hilbert space H. A second
such operator A′ on a Hilbert space H ′ is viewed as equivalent to A if there is an orthogonal
respectively unitary transformation T : H → H ′ such that T DomA = DomA′ and TA = A′T
on DomA. We say that such an operator T intertwines A and A′. The spectral theorem in
its multiplicative version says that there is a measure space X with a finite measure µ and
a measurable real function f on X such that A is equivalent to the multiplication operator Af
on L2(X,µ), Afφ = fφ, with domain DomAf = {φ ∈ L2(X) | fφ ∈ L2(X)}. Note that
neither X nor µ, given X, are unique.
Suppose that T intertwines A with such a multiplication operator Af . Consider the push-
forward ν = f∗µ, a Borel measure on R. By the definition of f∗µ,
supp ν = essran f,
the essential range of f . Whereas ν is not unique, not even the total mass of ν, the spec-
trum σ(A), the spectral projections πB, and the spectral measures νu, u ∈ H, associated to A
are invariant respectively equivariant under T . For that reason, it will be instructive to identify
them in the case of Af .
Recall that the resolvent set ρ(A) of A consists of all y ∈ R or y ∈ C, respectively, such that
A− y : DomA → H
is bijective. By definition, the spectrum σ(A) is the complement of ρ(A) in R or C, respec-
tively, where we actually have σ(A) ⊆ R, by the self-adjointness of A. The point spectrum
σpp(A) ⊆ σ(A) is the set of eigenvalues of A.
Proposition A.3. In the notation further up,
σ(A) = σ(Af ) = essran f and σpp(A) = σpp(Af ) = {y ∈ R | ν(y) > 0}.
Proposition A.4. The spectral projection of Af associated to a Borel set B ⊆ R is given by
multiplication in L2(X) with f∗χB = χB ◦f , where χB denotes the characteristic function of B.
For any u ∈ H, the spectral measure νu on R associated to u (and A) is given by
νu(B) = ⟨u, πBu⟩2
for any Borel set B ⊆ R, where πB : H → H denotes the spectral projection associated to A.
Since T intertwines the spectral measures associated to A and Af ,
νu(B) = νφ(B) = ⟨φ, (f∗χB)φ⟩2 =
∫
X
(f∗χB)φ̄φdµ
for φ = Tu ∈ L2(X) and any Borel set B ⊆ R.
Let now again ν = f∗µ, decompose ν = νac+νpp+νsc, and write R = Yac∪Ys and Ys = Ypp∪Ysc
as disjoint unions as further up. Then we have, respectively set,
σ(A) = σ(Af ) = essran f,
σac(A) = σac(Af ) = Yac ∩ essran f,
σs(A) = σs(Af ) = Ys ∩ essran f,
σpp(A) = σpp(Af ) = Ypp ∩ essran f = Ypp,
σsc(A) = σsc(Af ) = Ysc ∩ essran f.
We call the latter four the absolutely continuous, singular, point and singular continuous parts
of the spectrum of A or Af .
On the Spectrum of Certain Hadamard Manifolds 17
Proposition A.5. Let L2(X)ac, L2(X)s, L2(X)pp, L2(X)sc be the subspaces of φ ∈ L2(X)
vanishing outside f−1(Yac), f
−1(Ys), f
−1(Ypp), f
−1(Ysc), respectively. Then we have orthogonal
decompositions
L2(X) = L2(X)ac ⊕ L2(X)s and L2(X)s = L2(X)pp ⊕ L2(X)sc.
Moreover, besides zero, these subspaces consist precisely of those φ ∈ L2(X) such that the asso-
ciated spectral measures νφ are absolutely continuous and singular with respect to the Lebesgue
measure respectively are concentrated on σpp(Af ) and σsc(Af ).
Proof. The only non-trivial part is the absolute continuity of νφ with respect to the Lebesgue
measure λ for φ ∈ L2(X) vanishing outside f−1(Yac). To that end, let B ⊂ R be a Lebesgue
nullset. Since φ vanishes outside f−1(Yac), we readily see that
νφ(B) =
∫
f−1(B)
φ̄φdµ =
∫
f−1(B∩Yac)
φ̄φdµ.
From the fact that µ
(
f−1(B∩Yac)
)
= ν(B∩Yac) = νac(B∩Yac) = 0 and the square-integrability
of φ, we conclude that νφ(B) = 0, which shows that νφ is absolutely continuous with respect
to λ. ■
The characterization of the various subspaces in terms of the corresponding spectral measures
leads under intertwining to a corresponding decomposition of H.
Proposition A.6. With respect to A, we have orthogonal decompositions
H = Hac ⊕Hs and Hs = Hpp ⊕Hsc,
where the associated spectral measures νφ are absolutely continuous and singular with respect to
the Lebesgue measure respectively are concentrated on σpp(A) and σsc(A).
For k = 1, 2, letXk be measure spaces with finite measures µk, Let fk be measurable functions
on Xk and Ak be the self-adjoint operator in L2(Xk) given by multiplication with fk. Denote
by νk = fk∗µk the associated Borel measures on R. Consider the product X = X1 × X2 with
the product measure µ = µ1 ⊗ µ2 and the measurable function f on X defined by
f(x1, x2) = f1(x1) + f2(x2).
Let A be the self-adjoint operator on L2(X), given by multiplication with f , and denote by
ν = f∗µ the associated Borel measure on R. By the definition of f and of the convolution of
measures,
ν = ν1 ∗ ν2 (A.3)
since ν(B) = µ({(x1, x2) | f1(x1) + f2(x2) ∈ B}), for any Borel set B ⊆ R.
Recall that L2(X) is equal to the Hilbert tensor product L2(X1)⊗̂L2(X2). Accordingly, for
k = 1, 2, let Ak be a self-adjoint operator on a Hilbert space Hk and define a self-adjoint operator
A on the Hilbert tensor product H = H1⊗̂H2 via
A(u1 ⊗ u2) = (A1u1)⊗ u2 + u1 ⊗ (A2u2).
Then Propositions A.1–A.3 and equation (A.3) yield the following results.
Proposition A.7. In H = H1⊗̂H2, we have
σ(A) = σ(A1) + σ(A2).
If σ(A1) and σ(A2) are bounded from below, then σ(A) = σ(A1) + σ(A2).
18 W. Ballmann, M. Mukherjee and P. Polymerakis
Proposition A.8. In H = H1⊗̂H2, we have
Hpp(A) = Hpp(A1)⊗̂Hpp(A2) with σpp(A) = σpp(A1) + σpp(A2);
Hac(A) ⊇ Hac(A1)⊗̂Hac(A2)⊕Hac(A1)⊗̂Hs(A2)⊕Hs(A1)⊗̂Hac(A2).
Corollary A.9.
(1) If Hpp of A1 or A2 vanishes, then Hpp(A) = {0}.
(2) If Hs of A1 or A2 vanishes, then Hs(A) = {0}.
(3) If Hsc of A1 and A2 vanishes, then Hsc(A) = {0}.
Proposition A.7 is the version of the Ichinose lemma for self-adjoint operators, not necessarily
semi-bounded, on Hilbert spaces; compare with [11, Theorem 4.1] and [16, Section XIII.9] and
references therein.
A.3 Spectra of Riemannian manifolds
In the present paper, we study Laplacians ∆ of complete and connected Riemannian manifoldsM
or domains D as self-adjoint operators in their respective spaces of square-integrable functions.
In the case of Riemannian products, M = M1 ×M2,
∆(u1 ⊗ u2) = (∆u1)⊗ u2 + u1 ⊗ (∆u2).
Since the Laplacian is bounded from below, Proposition A.7 yields
Corollary A.10. We have σ(M) = σ(M1) + σ(M2).
The following example is well-known.
Example A.11. Via the Fourier transform, the Laplacian ∆ of Rm corresponds to multipli-
cation with |x|2 on L2(Rm). Therefore, the spectrum of Rm is absolutely continuous with
σ(Rm) = [0,∞).
Together with this example, Corollary A.9 (2) yields
Corollary A.12. For any complete and connected Riemannian manifold M , the spectrum of
M × Rk is absolutely continuous, for any k ≥ 1.
Acknowledgements
We would like to thank the Max Planck Institute for Mathematics in Bonn for its support
and hospitality. The second author’s research was also partially supported by SEED Grant
RD/0519-IRCCSH0-024. We would like to thank our institutions for providing us with ideal
working conditions. We are also grateful to the referees for their helpful comments.
References
[1] Azencott R., Wilson E.N., Homogeneous manifolds with negative curvature. I, Trans. Amer. Math. Soc.
215 (1976), 323–362.
[2] Azencott R., Wilson E.N., Homogeneous manifolds with negative curvature. II, Mem. Amer. Math. Soc. 8
(1976), iii+102 pages.
[3] Ballmann W., Lectures on spaces of nonpositive curvature, DMV Seminar, Vol. 25, Birkhäuser, Basel, 1995.
[4] Ballmann W., Matthiesen H., Mondal S., Small eigenvalues of surfaces of finite type, Compos. Math. 153
(2017), 1747–1768, arXiv:1506.06541.
https://doi.org/10.2307/1999731
https://doi.org/10.1090/memo/0178
https://doi.org/10.1007/978-3-0348-9240-7
https://doi.org/10.1112/S0010437X17007291
https://arxiv.org/abs/1506.06541
On the Spectrum of Certain Hadamard Manifolds 19
[5] Ballmann W., Polymerakis P., On the essential spectrum of differential operators over geometrically finite
orbifolds, J. Differential Geom., to appear, arXiv:2103.13704.
[6] Bessa G.P., Jorge L.P., Montenegro J.F., The spectrum of the Martin–Morales–Nadirashvili minimal surfaces
is discrete, J. Geom. Anal. 20 (2010), 63–71, arXiv:0809.1173.
[7] Donnelly H., Garofalo N., Riemannian manifolds whose Laplacians have purely continuous spectrum, Math.
Ann. 293 (1992), 143–161.
[8] Donnelly H., Garofalo N., Schrödinger operators on manifolds, essential self-adjointness, and absence of
eigenvalues, J. Geom. Anal. 7 (1997), 241–257.
[9] Donnelly H., Xavier F., On the differential form spectrum of negatively curved Riemannian manifolds,
Amer. J. Math. 106 (1984), 169–185.
[10] Heber J., On the geometric rank of homogeneous spaces of nonpositive curvature, Invent. Math. 112 (1993),
151–170.
[11] Ichinose T., Operators on tensor products of Banach spaces, Trans. Amer. Math. Soc. 170 (1972), 197–219.
[12] Kasue A., A note on L2 harmonic forms on a complete manifold, Tokyo J. Math. 17 (1994), 455–465.
[13] Kazdan J.L., Warner F.W., Curvature functions for compact 2-manifolds, Ann. of Math. 99 (1974), 14–47.
[14] Mazzeo R., Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic mani-
folds, Amer. J. Math. 113 (1991), 25–45.
[15] Polymerakis P., Spectral estimates for Riemannian submersions with fibers of basic mean curvature, J. Geom.
Anal. 31 (2021), 9951–9980, arXiv:2003.09843.
[16] Reed M., Simon B., Methods of modern mathematical physics. IV. Analysis of operators, Academic Press,
New York, 1978.
[17] Rellich F., Über das asymptotische Verhalten der Lösungen von ∆u + λu = 0 in unendlichen Gebieten,
Jahresber. Dtsch. Math.-Ver. 53 (1943), 57–65.
[18] Wolter T.H., Geometry of homogeneous Hadamard manifolds, Internat. J. Math. 2 (1991), 223–234.
[19] Xavier F., Convexity and absolute continuity of the Laplace–Beltrami operator, Math. Ann. 282 (1988),
579–585.
https://arxiv.org/abs/2103.13704
https://doi.org/10.1007/s12220-009-9101-z
https://arxiv.org/abs/0809.1173
https://doi.org/10.1007/BF01444709
https://doi.org/10.1007/BF01444709
https://doi.org/10.1007/BF02921722
https://doi.org/10.2307/2374434
https://doi.org/10.1007/BF01232428
https://doi.org/10.2307/1996304
https://doi.org/10.3836/tjm/1270127966
https://doi.org/10.2307/1971012
https://doi.org/10.2307/2374820
https://doi.org/10.1007/s12220-021-00634-z
https://doi.org/10.1007/s12220-021-00634-z
https://arxiv.org/abs/2003.09843
https://doi.org/10.1142/S0129167X91000430
https://doi.org/10.1007/BF01462884
1 Introduction
2 Preliminaries
3 Vanishing of point spectrum
4 Absolutely continuous spectrum
5 Homogeneous Hadamard manifolds
6 Determination of the spectrum
A Some spectral theory
A.1 Convolution of measures
A.2 Decomposition of spectra
A.3 Spectra of Riemannian manifolds
References
|
| id | nasplib_isofts_kiev_ua-123456789-211981 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T09:05:59Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ballmann, Werner Mukherjee, Mayukh Polymerakis, Panagiotis 2026-01-20T16:14:09Z 2023 On the Spectrum of Certain Hadamard Manifolds. Werner Ballmann, Mayukh Mukherjee and Panagiotis Polymerakis. SIGMA 19 (2023), 050, 19 pages 1815-0659 2020 Mathematics Subject Classification: 58J50; 53C20 arXiv:2301.10986 https://nasplib.isofts.kiev.ua/handle/123456789/211981 https://doi.org/10.3842/SIGMA.2023.050 We demonstrate the absolute continuity of the spectrum and determine the spectrum as a set for two classes of Hadamard manifolds, as well as for specific domains and quotients of one of these classes. We would like to thank the Max Planck Institute for Mathematics in Bonn for its support and hospitality. The second author’s research was also partially supported by SEED Grant RD/0519-IRCCSH0-024. We would like to thank our institutions for providing us with ideal working conditions. We are also grateful to the referees for their helpful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Spectrum of Certain Hadamard Manifolds Article published earlier |
| spellingShingle | On the Spectrum of Certain Hadamard Manifolds Ballmann, Werner Mukherjee, Mayukh Polymerakis, Panagiotis |
| title | On the Spectrum of Certain Hadamard Manifolds |
| title_full | On the Spectrum of Certain Hadamard Manifolds |
| title_fullStr | On the Spectrum of Certain Hadamard Manifolds |
| title_full_unstemmed | On the Spectrum of Certain Hadamard Manifolds |
| title_short | On the Spectrum of Certain Hadamard Manifolds |
| title_sort | on the spectrum of certain hadamard manifolds |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211981 |
| work_keys_str_mv | AT ballmannwerner onthespectrumofcertainhadamardmanifolds AT mukherjeemayukh onthespectrumofcertainhadamardmanifolds AT polymerakispanagiotis onthespectrumofcertainhadamardmanifolds |