Institute of Mathematics Simion Stoilow of the [610088]

Institute of Mathematics \Simion Stoilow" of the
Romanian Academy
HABILITATION THESIS
Spectral analysis of elliptic operators
in hyperbolic geometry
Sergiu Moroianu
Specialization: Mathematics
Bucharest, 2012

"[…] Forma f ar a fond nu numai c a nu aduce nici
un folos, dar este de-a dreptul stric acioas a, indc a
nimiceste un mijloc puternic de cultur a. […] Mai
bine s a nu facem de loc academii, cu sectiunile lor,
cu sedintele solemne, cu discursurile de receptiune, cu
analele pentru elaborate dec^ at s a le facem toate aceste
f ar a maturitatea stiintic a ce singur a le d a ratiunea
de a .
Titu Maiorescu, 1868.

Contents
I Abstract 3
1 Abstract of the thesis 5
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Rezumat 7
2.1 Motivatie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Rezumat stiintic . . . . . . . . . . . . . . . . . . . . . . . . . 7
II Works on elliptic operators in hyperbolic geom-
etry 9
2.3 Hyperbolic surfaces . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Hyperbolic 3-manifolds . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 Quasi-fuchsian manifolds . . . . . . . . . . . . . . . . . 17
2.4.2 Schottky manifolds . . . . . . . . . . . . . . . . . . . . 18
2.5 Hyperbolic cone manifolds . . . . . . . . . . . . . . . . . . . . 19
2.6 Odd Selberg zeta function . . . . . . . . . . . . . . . . . . . . 23
2.6.1 The Selberg trace formula . . . . . . . . . . . . . . . . 24
1

2
III Research perspectives 33
IV Bibliography 39

Part I
Abstract
3

Chapter 1
Abstract of the thesis
1.1 Motivation
I have defended my Habilitation thesis more than 8 years ago, in June 2004
at the Universit e Paul Sabatier in Toulouse, France. Due to the demanding
standards of the bureaucracy in the ministry of research, that thesis cannot
be recognized in our country. The Habilitation procedure was introduced
only in 2011, together with guidelines on how to recognize automatically
Habilitation degrees obtained in the European Union. Unfortunately these
guidelines (art. 3 (2) from the ministry of education and research order nr.
5.690/13.10.2011) require the candidate to have already directed at least
one PhD student: [anonimizat]. In my case, since I work in Bucharest, such a
requirement cannot possibly be fullled. Knowing the well-deserved place our
country holds in Europe in terms of scientic prestige and of research output
per capita, it is perhaps justied that Habilitation theses from perypheral
countries like France are not recognized automatically. It appears therefore
necessary for me to gather a new Habilitation thesis in order to nally earn
the right to direct PhD theses in Romania.
1.2 Abstract
Hyperbolic 3-manifolds may have conical singularities along geodesic curves.
We consider such hyperbolic cone-manifolds with innite-length singular
curves. We prove that when all the angles around the singularities are
5

6 CHAPTER 1. ABSTRACT OF THE THESIS
less than, the hyperbolic cone-manifold is inifnitesimally rigid when the
cone angles are xed and when the natural conformal structure with marked
points, corresponding to the end of the singular lines, is also xed. Further-
more, small deformations of the cone angles and of the marked conformal
structure at innity correspond to unique small deformations of the cone-
manifold structure. Under our assumption on the angles, the singular locus
consists of closed geodesics, complete geodesics, and of a graph with vertices
of valence at most 3, with possibly half-innite edges.
On non-compact smooth hyperbolic 3 manifolds of nite geometry without
cusps, or more generally on convex co-compact odd-dimensional hyperbolic
manifolds, we analyze the geometric Selberg zeta function Zo
;() corre-
sponding to the spinor bundle. We prove the meromorphic extension to the
complex plane and describe the zeros and the poles of Zo
;(). In the proof,
we construct the resolvent of the Dirac operator on convex co-compact hy-
perbolic manifolds, prove its meromorphic extension and derive the existence
and properties of Eisenstein and scattering operators as corollary. We re-
mark that there exists a naturally-dened spectral eta invariant of the Dirac
operator on convex cocompact hyperbolic manifolds obtained as quotients
nH2n+1. We prove that this invariant is related to the Selberg zeta func-
tion: exp(i(D)) =Zo
;(0). Formally this extends a result of Millson from
the case of closed manifolds. Assuming that the exponent of convergence of
the fundamental group is small enough, the same identities hold for the od
signature operator. In the particular case of Schottky 3-manifolds, the eta
invariant, which depends just on the point in the Schottky moduli space cor-
responding to the conformal structure on the underlying Riemann surface, is
the argument of a holomorphic function appearing in the Zograf factorization
formula as the ratio of two k ahler potentials for the Weil-Petersson metric.

Chapter 2
Rezumat
2.1 Motivatie
Mi-am sustinut teza de Abilitare (Habilitation  a Diriger des Recherches) cu
8 ani inaintea prezentei lucrari, la Universitatea Paul Sabatier din Toulouse,
Franta in iunie 2004. Datorita standardelor ridicate ale birocratiei din min-
isterul educatiei, teza mea nu poate  insa, din pacate, recunoscuta pe plan
local. Procedura de Abilitare a fost introdusa in 2011, urmata de modal-
itati practice de recunoastere automata a diplomelor de Abilitare obtinute
in alte tari UE. Din nefericire, articolul 3 (2) din ordinul 5.690/13.10.2011
cere candidatului sa  condus deja cel putin un doctorant in strainatate.
In cazul meu, intrucat lucrez in Romania, aceasta cerinta nu poate  in-
deplinita. Tinand cont de locul bine-meritat pe care il ocupa tara noasta
in Europa prin prestigiul stiintic si prin rezultatele concrete ale cercetarii,
e fara indoiala pe deplin justicat ca teze de Abilitare din tari mai putin
dezvoltate matematic precum Franta sa nu e recunoscute automat. Sunt
astfel nevoit sa redactez prezenta teza pentru a obtine in nal dreptul de a
conduce doctorate in Romania.
2.2 Rezumat stiintic
Varietatile hiperbolice de dimensiune 3 pot avea singularitati conice de-a
lungul unor curbe geodezice. Consideram varietati de dimensiune 3 hyper-
bolice care sunt convex co-compacte cu singularitati conice de-a lungul unor
7

8 CHAPTER 2. REZUMAT
geodezice posibil innite sau de-a lungul unui graf geodezic. Astfel de singu-
laritati sunt folosite de zicieni ca model pentru particule masive fara spin.
Demonstram un rezultat de rigiditate innitezimala cand unghiurile in ju-
rul curbelor singulare sunt mai mici decat : orice deformare innitezimala
schimba sau aceste unghiuri, sau structura conforma la innit cu puncte
marcate corespunzand directiilor singulare. Mai mult, orice variatie sucient
de mica a structurii conforme marcate la innit sau a unghiurilor singulare
poate  realizata printr-o unica deformare a structurii de varietate conica
hyperbolica.
Aratam ca functia zeta Selberg de tip impar Zo
;() pe o varietate de di-
mensiune impara convex co-compacta X= nH2n+1asociata bratului de
spinori admite o extensie meromorfa la planul complex si descriem structura
polilor si a zerourilor. Ca unealta de abordare analizam spectrul operatorului
Dirac si dezvoltam teoria difuziei (scattering) pe varietati asimptotic hiper-
bolice. Aratam ca exista un invariant eta (D) asociat natural operatorului
Dirac peste varietati hyperbolice convex co-compacte si demonstram iden-
titatea exp( i(D)) =Zo
;(0), extinzand astfel formula Millson din cazul
compact. Sub ipoteza ca exponentul grupului convex co-compact este su-
cient de mic, denim un invariant eta pentru operatorul de signatura impar
si aratam ca pe varietati Schottky de dimensiune 3 invariantul eta este ar-
gumentul unei functii olomorfe care apare in formula Zograf de factorizare
care leaga doua potentiale K ahler naturale pentru metrica Weil-Petersson pe
spatiul de moduli de varietati Schottky.

Part II
Works on elliptic operators in
hyperbolic geometry
9

11
In my thesis Residue functionals on the algebra of adiabatic pseudo-
dierential operators , defended in 1999 at MIT (and recognized in Romania
in 2004) with professor Richard Melrose, I studied the Hochschild homology
of the algebra of adiabatic pseudo-dierential operators. In the subsequent
papers [55], [59] and [62] I extended this study to other classes of pseudo-
dierential operators, namely bered-cusp and double-edge operators. The
study of Hochschild homology of pseudo-dierential algebra is essentially
motivated by index theory. I studied Fredholm index problems in [54], [56]
and [63]. Together with Robert Lauter we nd an index formula for pseudo-
dierential operators on manifolds with corners and on manifolds with bered
boundary. A related subject of study was the K-theory of the algebra of sus-
pended operators [58], which allowed me to prove a cobordism invariance
statement for the index of elliptic operators on manifolds with corners in
[64], and also to give an analytic proof of the cobordism invariance for ellip-
tic operators on closed manifolds [67].
I gradually shifted from index theory towards more general studies of the
spectrum of linear elliptic operators. My actual Habilitation thesis [53], de-
fended in 2004, originated from my works on the spectrum of elliptic opera-
tors on noncompact manifolds, namely [72], [57], [61]. In [72] I nd that the
spin Dirac operator on a hyperbolic manifold of nite volume continues to
obey a sort of Weyl asymptotic formula for eigenvalues classically known on
compact manifolds. In [57] I prove that the eta and zeta functions of elliptic
rst-order dierential operators on the total space of a bration with com-
pact base have an adiabatic limit, generalizing works by Bismut-Fried and
Cheeger. Together with Christian B ar we described in [60] the small-time
heat kernel asymptotics for roots of Laplace type operators on closed man-
ifolds, displaying some non-local terms in the asymptotic expansion. With
Andr e Legrand [66] we found an Lpindex formula for conical manifolds,
retrieving and extending the Chou index formula in the L2case.
In this current Habilitation thesis I will only detail some of the directions in
my recent research activity dealing with the interaction between hyperbolic
geometry and the analysis of elliptic operators. I will therefore not explain
in detail my other subjects of research. Below I will brie
y list some projects
I have carried out in dierent areas revolving around elliptic dierential op-
erators on Riemannian manifolds.
In [74] and [77], together with Sylvain Gol enia, I have analyzed the spectrum
of magnetic Laplacians and of the Laplacians on dierential forms. We nd

12
very interesting geometric examples of noncompact Riemannain manifolds
where the spectrum of the Hodge Laplacian on forms is pure-point and obeys
the Weyl law. The examples include certain complete hyperbolic manifolds
of nite volume. We thus correct a result from the literature concerning the
spectrum of the Hodge laplacian on geometrically nite hyperbolic manifolds.
I have studied the generalized Taub-NUT metrics together with Ion Cotaescu,
Andrei Moroianu and Mihai Visinescu [65], [68], [80]. We compute the eta
invariant of the Dirac operator on dierent domains, thus determining the
index of the Dirac operator with the Atyiah-Patodi-Singer boundary condi-
tion. On the whole space we prove that the index is nite although the Dirac
operator itself is not Fredholm. Finally on a wider class of spaces we show
that there do not exist L2harmonic spinors.
I have studied the Dirac spectrum on manifolds with gradient conformal
vector elds in the joint paper [71] with Andrei Moroianu, generalizing a
result of J. Lott about zero modes. We prove that the there do not exist L2
eigenspinors, which is the opposite case when compared to [74]. Interestingly,
the setting in [71] includes innite volume convex co-compact hyperbolic
manifolds, while [74] includes nite volume hyperbolic manifolds with cusps.
The index formulae from [70] and [73], the last one in collaboration with
Victor Nistor, are by now quite old. I have more or less turned away from
index theory in recent years.
The results on the generic presence of singularities of spectral eta and zeta
functions on closed manifolds from [82] close an interesting question, that
of invariants appearing as values of zeta-type spectral functions at integer
points. Such a value is meaningful whenever the associated zeta function is
regular, in particular it then becomes conformally invariant.
The new conformally covariant series of dierential operators starting with
the Dirac operator discovered in [81] are very new and surprising. Even the
third-order operator we get is entirely new: the operator
L:=D3cl(d(scal))
2(n1)2 clRicr
n2+scal
(n1)(n2)D;
dened in terms of the metric g, is conformally covariant, in the sense that
^L1=en+3
2!L1en3
2
if^L1is dened in terms of the conformal metric ^h=e2!h.

2.3. HYPERBOLIC SURFACES 13
In the rest of this presentation I will focus on my current research revolving
around hyperbolic geometry. Already the paper [74] was in the setting of
nite-volume hyperbolic manifolds. Together with Paul Loya and Jinsung
Park we examined in [75] the adiabatic limit of the eta invariant on circle
brations over noncompact hyperbolic surfaces. The regularity of the eta
function on nite-volume hyperbolic manifolds was investigated in [78] jointly
with the same co-authors as above. All these works set the stage for the
two papers I dene as the core of this Habilitation thesis, namely [76] by
J.-M. Schlenker, Sergiu Moroianu, Quasi-fuchsian manifolds with particles,
published in the Journal of Dierential Geometry 83(2009), 75{129, and [79]
by C. Guillarmou, Sergiu Moroianu, J. Park, Eta invariant and Selberg Zeta
function of odd type over convex co-compact hyperbolic manifolds, published
in Advances in Mathematics 225(2010), no. 5, 2464{2516.
2.3 Hyperbolic surfaces
I should start this presentation with the most elementary and perhaps most
fascinating objects of geometry, namely surfaces. A surface is a topological
space  locally dieomorphic to R2. Every surface admits a smooth structure ,
namely an atlas of such dieomorphisms whose change of charts consists
of smooth local automorphisms of R2. Every two such atlases dier by a
homeomorphism of . Every smooth atlas contains a conformal atlas, in
the sense that the changes of charts are conformal maps with respect to
the canonical
at metric on R2, however these conformal structures are no
longer unique up to dieomorphism. In fact, their equivalence classes modulo
isotopy gives rise to one of the most intriguing objects in mathematics, the
celebrated Teichm uller space.
For simplicity we consider here only the case of orientable connected surfaces,
since the language of Riemann surfaces comes handy in that setting. In
general one needs to include anti-holomorphic maps in atlases of generalized
holomorphic structure. Every orientable surface  admits a holomorphic
atlas. By the Riemann mapping theorem (proved in a nal form in 1907 by
Poincar e and Koebe) its universal cover is therefore bi-holomorphic to one of
the three simply connected Riemann surfaces { the sphere, the plane Cor the
hyperbolic plane H2. Only the last deserves some explanation. By denition,

14
(H2;gH2) is the upper half-plane endowed with the Poincar e metric:
H2=fz2C;y==(z)>0)g; g H2=y2jdzj2:
It is a complete metric of Gaussian curvature 1. Of course, its conformal
structure is the same as the one dened by the euclidean metric jdzj2on the
upper half-plane. Clearly, a self-isometry of ( H2;gH2) is either holomorphic or
anti-holomorphic. The group of orientation-preserving isometries of ( H2;gH2)
can be identied with PSL 2(R) by the action
a b
c d
:z7!az+b
cz+d:
However, it is true conversely that every bi-holomorphic automorphism of H2
is a hyperbolic isometry , in other words it belongs to PSL 2(R).
The surface  can be identied with the quotient of its universal cover by
the action of the fundamental group, which must act without xed points.
The only bi-holomorphism of CP1without xed points is the identity, hence
in the rst case  itself must be CP1. In the second case when ~ =C, the
bi-holomorphisms are given by ane transformations of the form
z7!az+b; a;b 2C; a6= 0:
The only such maps without xed points are when a= 1, hence translations,
which have the pleasant property of being also isometries for the Euclidean
metricjdzj2. It follows that  is the quotient of Cby a discrete group of
translations, hence (according to the rank of the group) it must be C, a
cylinder or a torus, and it inherits a
at metric from C. The most interesting
case is the remaining case when ~ =H2. In this case  is the quotient of H2
by a discrete subgroup in PSL 2(R) acting properly discontinuously (without
xed points). Therefore  inherits a complete hyperbolic metric.
Theorem 2.1 (Riemann-Poincar e-Koebe Uniformization theorem) .Every
connected Riemann surface other than CP1,C, a cylinder or a torus,
admits in its conformal class a unique complete metric of constant Gaussian
curvature1.
Even for compact surfaces, there exists a non-trivial deformation space of
hyperbolic metrics (or equivalently of Riemann surface structures). Assume

2.3. HYPERBOLIC SURFACES 15
therefore that  is a compact orientable surface of genus g2 with a xed
smooth structure. The Riemann moduli space Mgis the set of equivalence
classes of holomorphic structures modulo the group Di of dieomorphisms
of . The Teichm uller space Tgis the quotient of the set of holomorphic struc-
tures by the smaller group Di 0ofisotopies , namely dieomorphisms homo-
topic (inside Di) to the identity map of . The quotient G:= Di=Di 0is
the mapping class group of , and clearly M=T=G. We can view Tas the
symplectic reduction of the space of metrics on  by the semi-direct product
of Di 0and of C1(;R), the last one acting by conformal transformations
on metrics, i.e.,
f
g:=efg:
Hence Tinherits a manifold structure. Alternately, Tis the space of equiv-
alence classes of hyperbolic metrics on  modulo isotopy. Its tangent space
at a given holomorphic structure on  is the space of holomorphic quadratic
dierentials, i.e., of holomorphic sections in the square of the line bundle
(T)1;0. By the Riemann-Roch formula this space has dimension equal to
3g3, hence Tis a smooth complex manifold of dimension 3 g3. By us-
ing Fenchel-Nielsen coordinates given by any pants decompositions, we can
see that Tis topologically a ball of dimension 6 g6. Moreover Tcarries
a natural K ahler metric, the Weil-Petersson metric dened by the L2inner
product of two holomorphic quadratic dierentials using the unique under-
lying hyperbolic metric on .
In conclusion, the Riemann moduli space is the quotient of a contractible
K ahler manifold by a discrete group of isometries, the mapping class group,
acting properly discontinuously but with some xed points at those complex
structures admitting non-trivial automorphisms (for instance, hyperelliptic
curves). The isotropy subgrup of any point in Tgis nite, hence Mis a K ahler
orbifold. Its properties are sometimes studied under the guise of G-invariant
objects on T.
Let us point out that the mapping class group is a very interesting object in
its own right. Representations of Gcan be obtained from equivariant objects
onTg, like the holomorphic section in the Chern-Simons line bundle over Tg
recently constructed in a preprint by C. Guillarmou and this author.

16
2.4 Hyperbolic 3-manifolds
Hyperbolic 3-manifolds are a central object of current research in mathe-
matics. Although it is not the focus of my work in this thesis, I should pay
at least lip service to the geometrization program, featuring compact and
nite-volume hyperbolic manifolds. According to Thurston's hyperbolization
conjecture, every closed aspherical and atoroidal 3-manifold with innite fun-
damental group admits a hyperbolic metric. The hyperbolization conjecture
was recently proved based on Perelman's ideas on the Ricci
ow, completing
a program started by Hamilton in the 1980's. Many leading mathemati-
cians have contributed to this achievement, both before and after Perelman's
work. The hyperbolization conjecture is a part of the geometrization con-
jecture, which states that every 3-manifold can be decomposed in essentially
unique \geometric" pieces by cutting along embedded spheres and tori. Each
of the geometric pieces admits a complete geometry (in the sense of Cartan)
of nite volume from a list of eight maximal geometries. Such a geometry is
an atlas modeled on ( X;G ) whereXis a model simply-connected 3-manifold
andGis a Lie group acting transitively on X, maximal with this property
and with compact stabilizers.
One instance of such a geometry is the spherical geometry modeled on
(S3;O(4)). The geometrization conjecture (or rather theorem) implies the
celebrated Poincar e conjecture, stating that every compact simply connected
smooth 3-manifold is dieomorphic to S3.
However by far most examples of geometrizable 3-manifolds are hyperbolic.
Oriented hyperbolic manifolds are modelled on ( H3;PSL 2(C)). Here we con-
sider the upper half-space model for hyperbolic space as embedded inside
quaternions orthogonal on k:
H3=f(z+jt);z2C;t> 0g
and the M obius group PSL 2(C) acts by quaternion linear fractional transfor-
mations: a b
c d
:q=z+jt7!(aq+b)(cq+d)1:
One can check that this action preserves the metric t2jdqj2of constant sec-
tional curvature1.
In dimension 3 a nite-volume hyperbolic metric on a given manifold, when it
exists, is unique by Mostow rigidity. Therefore there cannot be any analog of

2.4. HYPERBOLIC 3-MANIFOLDS 17
the Teichm uller space and its rich K ahler structure. Nevertheless something
stronger than an analogy – an identication – does occur between Teichm uller
space and the space of deformations of certain innite-volume hyperbolic 3-
manifold of nite geometry.
In this thesis we are interested in two classes of manifolds which are locally
modelled on ( H3;PSL 2(C)) but which are neither necessarily complete nor
of nite volume (in fact, they are always of innite volume). The typical
examples are convex co-compact hyperbolic manifolds, which we describe
below.
Let PSL 2(C) be a discrete group acting properly discontinuously and
freely on H3which admits a hyperbolic polyhedron (with a nite number
of vertices, edges and faces) as fundamental domain. Then the quotient
X:= nH3is a complete hyperbolic manifold of nite geometry. We assume
thatXis non-compact and that every vertex at innity (i.e., on the Riemann
sphere viewed as compactication of Cf0g) lies in a 2-dimensional face at
innity. Such a manifold X(and the group ) are called convex co-compact ,
and then the volume of Xis necessarily innite. An alternate description is
requiringXto admit a compact geodesically convex subset (a subset CX
is called geodesically convex if every geodesic arc with end-points in Cis
entirely contained in C).
A convex co-compact manifold Xis dieomorphic to the interior of a compact
manifold Xwith boundary . To describe the metric near the boundary we
choose a boundary-dening function x:X![0;1), i.e., a smooth function
vanishing precisely on the boundary  = fx= 0gand such that dx6= 0 in
. Nearx= 0 the hyperbolic metric on Xtakes the form
g=x2(dx2+h(x)); h (x) =h0+x2h2+x4h4
whereh0is a metric on , and h2;h4are symmetric two tensors on .
There exist two fundamental examples of convex co-compact hyperbolic 3-
manifolds: quasi-fuchsian manifolds, and Schottky manifolds.
2.4.1 Quasi-fuchsian manifolds
A discrete subgroup PSL 2(R) is called a fuchsian group . If we assume
that does not contain elliptic or parabolic elements (i.e., elements
with
jtr(
)j2) then the quotient  := nH2is a smooth complete hyperbolic

18
surface. The hyperbolic metric from H2is -invariant so it descends to a
metricgon  of gaussian curvature 1. Since PSL 2(R)PSL 2(C), we can
also consider the quotient X:= nH3. It is isometric to a cylinder
(R;g); g =dr2+ cosh(r)2g:
If  is compact then Xis called a Fuchsian 3-manifold . It is clearly complete
and non-compact. Moreover, it has a natural compactication by adding two
copies of  at r=1, corresponding to the set of directions of geodesics
escaping to innity (two such geodesics are identied if they get closer to
each other in time, which implies that they become exponentially closer to
each other). The two faces can also be realized as the quotient of the
upper and of the lower half-planes by the action of . This action is in fact an
action on CP1preserving the real circle Rtf1g . The fuchsian group has
therefore the property that its action on CP1is properly discontinuous on
two simply connected domains separated by a Jordan curve which is precisely
the limit set of .
By a fundamental result of Alhfors and Bers, the group can be deformed
inside PSL 2(C) as a convex co-compact subgroup. More precisely, given two
Riemann surface structures (or equivalently two conformal structures) con a
xed smooth surface , there exists a faithful representation of in PSL 2(C)
without elliptic or parabolic elements, unique up to conjugacy, such that the
resulting 3-manifold X:=()nH3induces the conformal structures cat
its two ideal boundary components . Let us explain this geometrically.
Choose boundary-dening functions xnear each boundary component so that
the induced metrics h
0:= limx!0x2gon the boundaries are hyperbolic
(this is always possible by the uniformization theorem, and xis unique up to
second order errors at the boundary). A quasi-fuchsian manifold turns out
to be a hyperbolic manifold dieomorphic to R, such that near t=1
the metric takes the form
g=x2(dx2+h(x)); h (x) =h
0+x2h
2+x4h
4
whereh
0are hyperbolic metrics on f1g , andh
2are divergence-free
symmetric two tensors on  of constant trace 1.
2.4.2 Schottky manifolds
Every Riemann surface  of genus gcan be realized as the ideal boundary of
a convex cocompact handlebody. To see this, choose disjoint simple curves

2.5. HYPERBOLIC CONE MANIFOLDS 19
1;gon  such that the cut along them leads to a topological sphere with 2 g
disks removed. The Schottky reverse cut theorem states that this surface is
biholomorphic to the complement of gpairs of mutually disjoint disks Dj;D0
j,
j= 1;:::;g inCP1. The interior of Djcan be mapped onto the interior of
D0
jby an element
j2PSL 2(C), and the elements
1;:::;
ggenerate a free
group of rank g. The quotient of H3by this group is called a Schottky
manifold, and its boundary at innity can be identied with the initial surface
.
In both cases, Schottky and quasi-fuchsian, the conformal structure at in-
nity determines uniquely the hyperbolic convec co-compact metric in the
interior. Therefore, the Schottky, respectively the quasi-fuchsian spaces are
identied with Tg, respectively T2
g. Actually this fact remains true for every
geometrically nite hyperbolic 3-manifold without rank 3 cusps: the space of
its deformations is in bijection with the Teichm uller space of its ideal bound-
ary. This correspondence may be interpreted as a rigidity statement: convex
co-compact manifolds are rigid unde deformations preserving the conformal
structure at innity. One of the results emphasized in this thesis deals with
the extension of this phenomenon to convex co-compact manifolds with con-
ical singularities.
2.5 Hyperbolic cone manifolds
A general notion of hyperbolic cone-manifolds was introduced by Thurston in
[95]. They are stratied spaces whose open startum is a hyperbolic manifold.
In dimension 2 the sigularities are easy to describe, as they are isolated
points. In this case, a conical point has a neighborhood isometric to a model
neighborhood Vdened as follows: consider the universal cover Uof the
complement of the point iinH2. The group Racts by isometries on this
cover, lifting the action of the isotropy group S1PSL 2(R) ofi. The
model conical manifold Vis dened as the quotient of Uby translation with
2[0;1). This set is incomplete but we can dene a one-point completion
by adding the tip of the cone. If = 2we obtainV2=H2, for other angles
the metric is singular at the tip of the cone.
In the case of 3-dimensional manifolds, the singular set can be a disjoint
union of geodesic curves, dened as in the previous case. Then the metric
in a neighborhood of each singular point is determined by the angle, a real

20
number which is locally constant along the singular locus. The metric takes
the explicit form
V= (SR>0;dt2+ sinh2(t)h);
where (S;h) is the spherical metric surface with two conical points of angle
. However the singularities can also occur along a geodesic graph . An
important remark due to Weiss [99] is that when all conical angles are less
than, the singular graph has all vertices of valence 3.
As explained above, convex co-compact manifods without conical singular-
ities are rigid relative to the conformal structure at the ideal boundary, so
they are parametrized (on a given smooth manifold) by those conformal
structures, i.e., by Teichm uller space. A natural question is to see to what
extent conical hyperbolic manifolds are rigid in that sense.
Compact conical hyperbolic manifolds were considered by Hodgson and Ker-
ckho [31]. In their setting the singular set is a disjoint union of simple closed
geodesic curves and each of these singularities is determined by four real pa-
rameters: the angle and the translation component of holonomy around the
singular curve (which form together the so-called complex angle), and the
lenght and the twist along the curve (which determine the complex length).
Their result states that for real angles less than 2 the cone manifold is in-
nitesimally rigid by deformations preserving the complex angle. Moreover,
they showed that changes in the complex angles parametrize small deforma-
tions of the hyperbolic cone-manifold structure. Weiss [99] proved a similar
result for cone manifolds with graph singularities, however he must assume
that the angles are less than .
The innitesimal rigidity result of Hodgson and Kerkho was generalized
by Bromberg [8] to the setting of convex co-compact cone manifolds, with
singularities again along closed curves. His results states that any innitesi-
mal deformation of such a cone-manifold structure must imply a deformation
of the angles (supposed to be less than or of the conformal structure at
innity.
In my paper [76] with Jean-Marc Schlenker we examine hyperbolic 3-
manifolds with singularities along graphs with possibly innite-length edges,
i.e., half-innite geodesics stretching in the funnels. We must assume that
the angles around the half-innite geodescs are bounded above by for sev-
eral technical resons. Although possible in principle, we have prefered not to
allow cusps of rank 2 in the manifold, mainly because their presence would

2.5. HYPERBOLIC CONE MANIFOLDS 21
make the presentation more cumbersome.
As mentioned above, a subset Cin a hyperbolic cone-manifold Mis called
geodesically convex if it is non-empty and any geodesic segment in Mwith
endpoints in Cis contained in C. This notion is global and thus stronger
than the usual condition of convexity on the boundary, for instance points are
not geodesically convex unless Mis contactible. Directly from the denition,
any nonempty intersection of geodesically convex sets is again geodesically
convex. We prove that when the conical angles are less than , every closed
geodesic is contained in every geodesically convex set. This allows us to
dene a smallest geodesically convex set, containing all closed geodesics,
playing the role of the convex core in convex co-compact manifolds. We just
need to assume that Mcontains a compact convex set. If Mis a complete,
hyperbolic cone-manifold of dimension 3 with cone singularities of angles at
mostalong a nite graph with possibly innite-length edges, we will call it
aconvex co-compact manifold with particles if it contains a compact subset
Cwhich is convex.
Such a manifold is always homeomorphic to the interior of a compact mani-
fold with boundary, which can be chosen to be the smalles geodesically convex
subset ofM. The vertices of valence 1 in the singular graph correspond to
half-innite edges going to innity.
It follows that Mis then homeomorphic to the interior of a compact man-
ifold with boundary which we will call N(Nis actually homeomorphic to
the compact convex subset Cin the denition). The singular set of Mcor-
responds under the homeomorphism with a graph embedded in N, such
that vertices of adjacent to only one edge are in the boundary of N.
Innitesimal deformations of possibly incomplete hyperbolic manifolds are
parametrized by the 1-cohomology group of a
at bundle of \innitesimal
Killing vector elds". This idea goes back to Weil [97] in this setting and is
an instance of the so-called Kodaira-Spencer deformation theory. Basically,
an innitesimal deformation gives rise to a closed 1-form on Mtwisted by
the bundle Eof innitesimal Killing vector elds, well dened up to addition
of an exact form. The bundle Eis the bundle associated to the holonomy
representation of 1(M) into PSL 2(C), unique up to conjugacy, and to the
adjoint representation of PSL 2(C) on its lie algebra. Geometrically Ecan
also be identied with the complexication of the tangent bundle, endowed
with an explicit
at connection. Any trivial deformation, i.e., the innites-
imal action of a vector eld, leads to an exact 1-form, hence the twisted

22
de Rham cohomology class of the deformation 1-form is well dened. This
construction was recently used for cone-manifolds by Hodgson and Kerckho
[31], with a special emphasis on the square-integrability of representatives for
the deformation cohomology class.
We prove two related results about the rigidy of hyperbolic convex co-
compact manifolds with particles.
Theorem 2.2. The metric on a convex co-compact manifold with particles
(M;g)is innitesimally rigid under deformations preserving the cone angles
and the marked conformal structure at innity.
To state the second main result, let R(Mr) be the representation variety
of1(Mr) intoPSL 2(C) and:1(Mr)!PSL 2(C) the holonomy repre-
sentation of the regular part of M. We call Rcone(Mr) the subset of those
representations for which the holonomy of meridians of the singular curves
have no translation component, that is, these holonomies are pure rotations.
Thus2Rcone(Mr), and, in the neighborhood of , the points of Rcone(Mr)
are precisely the holonomies of cone-manifolds in the sense of our denition.
Theorem 2.3. Let(M;g)be a convex co-compact manifold with particles, c
the induced marked conformal structure at innity, and 1;


;N2(0;)
the conical angles. In the neighborhood of the holonomy representation ,
the moduli space of convex co-compact manifolds with particles structures on
M, i.e., the quotient of Rcone(Mr)byPSL (2;C), is parameterized by the
parameters c;1;


;n.
The proof of the rigidity theorem is based, as we mentioned above, on the
analysis of the deformation class in H1(M;E ). We rst show that the defor-
mation can be normalized, modulo trivial deformations, into a form where
the 1-form!is square integrable. The main body of work is essentially show-
ing that the L2cohomology group H1
L2(M;E ) vanishes. For this we use again
in a crucial way the hypothesis that the cone angles are less than to deduce
the essential self-adjointness of a twisted Dirac operator. Thus the crux of
the articles lies in the ne analysis of certain elliptic operators appearing
naturally in the problem. In general, rigidity would follow from the vanish-
ing ofH1(M;E ). In the case of a closed manifold, this vanishing reduces,
by Hodge theory, to the vanishing of the space of harmonic 1-forms twisted
byE. This Hodge cohomology space turns out to be zero since the Hodge
laplacian twisted by Eis strictly positive on 1-forms, using a Weitzenb ock

2.6. ODD SELBERG ZETA FUNCTION 23
formula due to Matsushima and Murakami. This formal positivity is typical
for negative curvature. When the manifold Mis no longer compact, it is
no longer true that H1(M;E ) is zero since as we have seen there do exist
deformations changing the conformal structure at innity. One works in-
stead inside the Hilbert space of L2forms twisted by E. First we prove that
modulo trivial deformations, any deformation preserving the angles and the
conformal structure at innity marked by the endpoints of outgoing conical
geodesics can be represented by a L2form. Then we show that this form
must be exact, with an L2primitive.
The second theorem is a consequence of the rst. Its proof consists in a
computation of dimensions for the deformation space (without restrictions
on the angles) and the fact that the global innitesimal deformations (i.e.,
deformations of the whole manifold) form a Lagrangian subspace in the space
of germs of deformations near the singular locus. This idea of a Lagrangian
space inside the moduli space appears also in the second work featured in
this thesis.
2.6 Odd Selberg zeta function of the Dirac
operator on hyperbolic manifolds
We place ourselves in the same setting as in the previous section but without
\particles", namely we study convex co-compact hyperbolic manifolds and
their spectral zeta functions.
First let us recall a few basic things about the spectrum of elliptic operators
on a closed manifold. Let Dbe a symmetric rst-order elliptic dierential
operator on a closed manifold M, acting on sections of a vector bundle E. We
can viewDas an unbounded linear operator acting in the Hilbert space of
square-integrable section in E, with domain C1(M;E ). ThenDis essentially
self-adjoint, and its spectrum is formed of real eigenvalues accumulating in
absolute value to innity. Moreover, if dim( M) =nand Rank(E) = k, then
the counting function Nfor the eigenvalues of D,
N(x) := #f2Spec(D);jj<xg;
satises the Weyl asymptotic formula
lim
x!1N(x)
xn=1
(2)nZ
SMj(D)j

24
where(D) is the principal simbol of DandSMthe sphere inside the
tangent bundle to M. This asymptotic law implies that the series dening
the spectral zeta function
(D;S) :=X
06=2Spec(D)jjs
is absolutely convergent for <(s)>n. Similarly, the eta function
(D;S) :=X
2Spec(D)jjs1
is absolutely convergent for <(s)> n. Both these functions extend mero-
morphically to the complex plane with a regular point in the origin. The eta
invariant is the value of the eta function at s= 0. It is the boundary term
in the index formula for compact manifolds with boundary [4] in the case
whereDis the tangential component of an elliptic operator on a compact
manifoldXwith boundary M. By the denition of the Gamma function (or
more sophisticatedly using the Mellin transform) one can write
(D;s) = Tr
D(D2)s+1
2
=1
((s+ 1)=2)Z1
0ts1
2Tr
DetD2
dt;
One can think of (D) :=(D;0) as computing the asymmetry \Tr( DjDj1)"
of the spectrum, or the diference between the number of positive and of
negative eigenvalues.
On non-compact manifolds the situation is drastically dierent in general.
What makes things dicult is that symmetric elliptic operators may admit
several self-adjoint extensions, or none. The spectrum may be pure-point
and obey the Weyl law, as in [72], or it can be purely continuous without
any eigenvalue as in [81]. The eta function may exist, but it can have a pole
ats= 0. Even in the case of hyperbolic surfaces of nite volume, it is not
known, with the notable exception of arithmetic surfaces, whether there exist
eigenvalues at all other than 0.
2.6.1 The Selberg trace formula
The original Selberg trace formula is a fundamental and surprising relation-
ship between the spectrum of the Laplacian on a hyperbolic surface and the

2.6. ODD SELBERG ZETA FUNCTION 25
so called length spectrum, i.e., the set of lengths of closed geodesics. Selberg
proved this formula in particular as a way to show that on arithmetic sur-
faces there exist enough eigenvalues to satisfy the growth rate predicted by
the Weyl law. However the formula applies to locally symmetric spaces in
quite wide generality.
Before applying it to Dirac operators on manifolds of innite volume, let us
review at least the simplest instance of the Selberg trace formula, that of
compact hyperbolic surfaces. Let Sbe a closed hyperbolic surface,
= dd
the associated scalar Laplacian, and fan even holomorphic function, which in
practice will be f(z) = exp(t(s2+1
4)) andRs0(s) = (s2
0s2)1fort>0 and
=(s0)>0. LetFbe the holomorphic function dened by F(s2+1
4) =f(s)
and dene the trace of F(
) as
tr(
) =X
2Spec(
)F():
The operator F(
) makes sense for every F, is bounded when Fis bounded
on the positive real line, and is trace-class when F() decreases faster than
jj1, again on the real line towards + 1. Thus tr(
) is under these hy-
potheses the actual operator trace of F(
). Let now
H2be the Laplacian
on the universal cover H2ofS, andF(
H2) the coresponding operator on
L2(H2). This operator has a Schwartz kernel A(z;z0), possibly singular on
the diagonal z=z0and decreasing away from it, which is PSL 2(R)-bivariant
and therefore depends only on the distance between zandz0. Thus there
exists a convolution kernel Gsuch thatA(z;z0) =G(2 coshd(z;z0)2), with
Gdecreasing at innity. This function Gcan be computed explicitly in terms
offand vice-versa , using the so-called Abel transform.
The operator F(
) also has a Schwartz kernel AS(m;m0) dened on SS
with singularities on the diagonal. In fact, one can write the lift of ASto
H2H2as a sum over the fundamental group PSL 2(R):
As(z;z0) =X

2A(z;
z):
This can be re-written in terms of the convolution kernel GasP

2G(2 coshd(z;
z0)2). Now the trace of a pseudodierential opera-
tor of order strictly less than 2 = dim( S) with kernel ASis computed by
tr(F(
)) =Z
SAS(m;m )dm:

26
Thus by choosing a fundamental domain
H2, we get
tr(F(
)) =Z

X

2G(2 coshd(z;
z)2):
Under suitable hypothesis on fthe above series is absolutely convergent in
L1so we reverse the order of integration and summation. Moreover, we group
the terms according to conjugacy classes inside :
tr(F(
)) =X
[
]20I[
]; I [
]:=X

2[
]Z

G(2 coshd(z;
z)2)dz:
Each conjugacy class is either the class of the identity, or a hyperbolic class
since we assume Shas no cusps. The conjugacy class of the identity contains
just itself. The centralizer of an elliptic element
6= 1 in a Fuchsian group
is a cyclic group containing
, so
=
n
0for somen1 and
0a primitive
element of . Thus the conjugacy class of
is in bijection with =h
0i, hence
by the bi-invariance of the distance function, we can rewrite
I[
]=Z
h
0inH2G(2 coshd(z;
z)2)dz:
In conclusion, we have written
tr(F(
)) =I1+X
16=[
0]20X
n1Z
h
0inH2G(2 coshd(z;
n
0z)2)dz:
Recall now that conjugacy classes of hyperbolic elements in are in bijection
with oriented closed geodesics in S= nH2, and that the trace of a group
element is cosh(d
2) wheredis le length of the corresponding oriented closed
geodesic. We let Ldenote the set of length (with multiplicities) of the closed
geodesics, also called the geometric spectrum. After some manipulations
with Abel transforms, from the above we get the Trace formula:
tr(F(
)) = (g1)Z
Rf()tanh()d+X
d2Ld^f(nd)
2 sinh(nd
2)
where ^fis the Fourier transform of f.

2.6. ODD SELBERG ZETA FUNCTION 27
One remarkable oshot of the trace formula is the analytic extension of the
Selberg zeta function. This is a geometrically dened holomorphic function
in terms of the length spectrum of S:
(z) :=Y
d2L(d;z);  (d;z) =1Y
n=0(1e(n+z)d):
The product is absolutely convergent for Re( z)>1 and hence the zeta
function is holomorphic there. The Selberg zeta function for the resolvent
function proves that extends holomorphically to Cwith zeros at points
determined by the spectrum of
. Explicitly, except for some \trivial" zeros
at the non-negative integers, has zeros precisely at the points1
2+iz2Csuch
that1
4+z22Spec(
), with multiplicity equal to that of the corresponding
eignevalue. Moreover the Selberg zeta function satises a functional equation
making it symmetric around1
2. This provides a hyperbolic counterpart of
the Riemann hypothesis, satised by the Selberg zeta function, with the
exception of a nite number of real zeros corresponding to the eigenvalues
of
which are less than1
4(the so-called small eigenvalues). Note also the
identity
(z) = exp
1X
m=1X
d2Lezmd
m(1emd)!
:
There is no conceptual diculty in extending the above reasoning to closed
hyperbolic surfaces of higher dimensions, or more generally to locally sym-
metric spaces [89]. Also, the trace formula can be applied to operators acting
in homogeneous bundles, like the Dirac operator on spinors. This has been
carried out by Millson [49], who showed that the fractional part of the eta
invariant of the odd signature operator Aon odd forms odd=2m
p=02p1
on a hyperbolic 4 k1-dimensional manifold X:= nH4m1is determined
by the central value of the corresponding zeta function of odd type. To de-
ne the zeta function, clearly more information than the length of the closed
geodesics is needed. In dimension 2 the trace is the unique invariant of a hy-
perbolic group element, in higher dimensions and when we deal with vector
bundles various holonomies appear naturally. Millson denes
Zo
;() := exp0
@X

2P1X
k=1+(R(
)k)(R(
)k)
jdet(IdP(
)k)j1
2ekd(
)
k1
A

28
where Pis the set of primitive closed geodesics in X, or equivalently the set
of conjugacy classes in
;R(
)2SO(4m2) is the holonomy in the form
bundle along a geodesic
,denotes the character associated to the two
irreducible representations of SO(4 m2) corresponding to the ieigenspace
of the Hodge operator ?acting on 2m1,P(
) is the linear Poincar e map
along
, andd(
) is the length of the closed geodesic
. The function Zo
;()
extends meromorphically to 2C, its zeros and poles occur on the line
<() = 0 with order given in terms of the multiplicity of the eigenvalues of
A, and the following remarkable identity holds:
ei(A)=Zo
;(0):
Moscovici and Stanton [85] extended this result to closed locally symmetric
spaces of higher rank.
The geometric Selberg zeta function makes perfect sense on certain non-
compact hyperbolic manifolds, at least in the domain of absolute conver-
gence, since closed geodesics continue to be related to the traces of hyper-
bolic elements on the fundamental group. One example of this is the case of
nite-volume surfaces, treated by Selberg himslef. Another example is the
analysis of the odd-type zeta function for the spin Dirac operator carried out
by J. Park in [86]. There the functional equation of the zeta function involves
the determinant of the scattering matrix. Together with C. Guillarmou and
J. Park I have studied the Selberg zeta function on convex co-compact hyper-
bolic manifolds associated to the spin Dirac operator. We then specialize to
the case of dimension 3 and we extend our results, basically without change,
to the odd signature operator. For Schottky manifolds we get an interesting
relation to Teichm uller theory, which is developed further in a joint paper
with C. Guillarmou.
We start our investigation by studying the meromorphic extension of the
resolvent of the Dirac operator on a general asymptotically hyperbolic man-
ifold. These are complete Riemannian manifolds ( X;g) such that Xis the
interior of a smooth manifold with boundary X, and the metric gis of the
formg=x2gwhere gis a smooth Riemannian metric on X, andxis a
boundary-dening function, x2C1(X;[0;1)) such that x1(0) =@Xand
dxdoes not vanish on @X, and nally dxhas length 1 with respect to  g. Then
the sectional curvatures of gtend to1 near@X, ie., Xis indeed asymtot-
ically hyperbolic. Clearly convex co-compact manifolds are particular cases
of asymptotically hyperbolic manifolds.

2.6. ODD SELBERG ZETA FUNCTION 29
The Dirac operator Dacting on the spinor bundle  on a spin asymptotically
hyperbolic manifold Xhas real spectrum (it is essentially self-adjoint since
the manifold is complete), and one can dene its resolvent for <()>0 in
two ways
R+() := (D+i)1; R () := (Di)1
as analytic families of bounded operators acting on L2(X; ). Our rst task
is to prove meromorphic extension of these analytic families: the resolvents
R() extend meromorphically to 2Cas operators from C1
0(X;) to the
dual space C1(X;), and moreover the polar part of the meromorphic
extensions is of nite rank. This is proved by actually constructing the
resolvent inside the calculus of \zero" pseudodierential operators of Mazzeo-
Melrose [46].
The rst step in the Melrose construction of resolvents in a wide class of
pseudo-dierential calculi is, besides, the obvious principal symbol, the re-
solvent of the so-called normal operator, which is a model operator at the
boundary obtained by freezing coecients. In our case this normal operator
turns out to be precisely the Dirac operator on the hyperbolic space Hd+1.
The explicit computation of Camporesi [13] shows that the Schwartz kernel
RHd+1(;m;m0) of the resolvent ( D2+2)1onHd+1, dened for<()>0,
is given essentially by a hypergeometric function times the parallel transport
fromm0tom. We remark that the parallel transport lifts smoothly on the
so-called zero double space, which is the (real) blow-up of the boundary diag-
onal in XX. ThusRHd+1(;m;m0) is a conormal distribution on the zero
double space. Using this, we can show that R(;m;m0) extends analyti-
cally to2Cas a conormal distribution in 1;+d
2;+d
2
0 (X;0), where  0
is the spinor bundle on Xwith respect to  g. More precisely, the asymptotic
structure is as follows: let ;0be dening functions for the left and right
hyperfaces of X2
0. Then (0)d
2R(;m;m0) has a conormal singularity
at the lifted diagonal corresponding to the pseudodierential order 1 and
is smooth to the three boundary faces, in the sense that is has Taylor se-
ries there. We now remark that the formula is analytic in 2C, in other
wordsRHd+1(;m;m0) extends analytically to C. Clearly when<()0 the
associated operator is no longer bounded on L2, but its Schwartz kernel is
analytic as a family of conormal distributions.
Once the rst step of the machinery is established, the second step con-
sists in constructing the resolvent modulo residual (smoothing) terms.

30
This relies on the composition rules of 0(X). We obtain a parametrix
in 1;+d
2;+d
2
0 (X;0) for the resolvent R(;m;m0), with an error in
 1;1;+d
2
0 . This error term is already compact, we use then a Neumann
series argument to solve away the Taylor series at the front face. This may
in principle introduce logarithmic terms in ;0;but they are shown not
to occur by the indicial equation of Di.
This structure theorem for the resolvent implies that the spectrum of Don
asymptotically hyperbolic manifolds is absolutely continuous and given by
R. The Eisenstein series are dened directly from the resolvent, by con-
structing generalized eigenspinors with prescribed asymptotic behaviour at
innity. The scattering operator S() :C1(@X;)!C1(@X;) is just the
leading \outgoing" asymptotic term in the Eisenstein series. It is a meromor-
phic family of pseudodierential operators on @Xof order 2, with principal
symbol essentially the same as that of the Dirac operator. In [81] we show
that the intrinsic conformal covariance of the scattering operator leads to
the construction of conformally covariant dierential operators on any spin
manifoldM, of any odd order 2 k+1dim(M), coniciding with the 2 k+1-th
power of the Dirac operator up to lower order terms. The fact that the scat-
tering operator is invertible outside some isolated points where it has zeros
or poles gives a new proof of the cobordism invariance of the index. Also
1
2(IdS(0)) turns out to be the Calderon projector for the Dirac operator,
i.e., the orthogonal L2projector on the Cauchy data space, or the boundary
values of harmonic spinors on X. The scattering operator plays an important
role in the study of the odd-type Selberg zeta function.
The exponent of a convex co-compact group is dened using an arbitrary
m2Hd+1by
inf(
2RjX

2ed(m;
m )<1)
:
For> n, we dene the Selberg zeta function of odd type Zo
;() asso-
ciated to the spinor bundle  exactly like Millson did for the odd signature
operator, except that R(
) denotes now the holonomy in the spinor bundle 
along
, anddenotes the character of the two irreducible representations
of Spin(2n) corresponding to the ieigenspaces of the Cliord multiplication
cl(T
) with the tangent vector eld T
to
.
One of our two main results in [81] is the pole structure of the zeta function:

2.6. ODD SELBERG ZETA FUNCTION 31
Theorem 2.4. The odd-type Selberg zeta function Zo
;()on an odd di-
mensional spin convex co-compact hyperbolic manifold X= nH2n+1has a
meromorphic extension to C, and it is analytic near f<()0g. A point0
with<(0)<0gis a zero or a pole of Zo
;()if and only if the meromor-
phic extension of R+()or ofR()has a pole at 0, and in that case the
multiplicity of 0is
rank Res 0R()rank Res 0R+():
The proof of this theorem uses of course the Selberg trace formula, for the
particular case of the heat kernel. The main problem is perhaps that the
odd heat operator Dexp(tD2) on a asymptotically hyperbolic manifold
is not of trace class. We can try to regularize the pointwise trace using the
Hadamard regularization procedure, but in fact we remark that for convex co-
compact hyperbolic manifolds this pointwise trace turns out to be absolutely
integrable, as a consequence of the explicit formula of Camporesi-Pedon [14]
of the odd heat kernel on hyperbolic space (which in particular has vanishing
pointwise trace), and a second construction of a parametric for the resolvent
when the sectional curvatures are 1 near innity showing that the error
from the model case is rapidly decreasing at innity.
Let us comment about the novelty element here. Freed [20] proved the mero-
morphic extension of a large class of dynamical zeta functions on compact
hyperbolic manifolds. His approach applies also to the convex co-compact
setting, as noted by Patterson-Perry [87], but without giving any information
about the structure of the poles and zeros. Since the original work of Selberg
one suspects that these are related to spectral and topological invariants of
the manifold. In the scalar case a description of the poles exists by the re-
cent work of Patterson-Perry [87] and Bunke-Olbrich [10], but for general
homogeneous bundles the zeta function is not well understood.
Our second main result proves that Millson's formula linking the eta invariant
and the Selberg zeta function holds on convex co-compact manifolds for the
Dirac operator, and also for the odd signature operator under the assumption
that the exponent is small enough. The eta invariant can no longer be
dened in terms of eigenvalues (as there are none in our setting, the spectrum
being purely absolutely continuous!) but the integral
(D) :=1pZ1
0t1
2Z
Xtr(DetD2)(m) dv(m)
dt;

32
is still absolutely continuous, again from the cancellations in the formula for
the odd heat kernel on hyperbolic space due to Camporesi and Pedon [14].
Theorem 2.5. LetX= nH2n+1be an odd dimensional spin convex co-
compact hyperbolic manifold. Then
ei(D)=Zo
;(0): (2.1)
IfXis of dimension 4 kthe eta invariant on the boundary can also be dened
for the odd signature operator, under the technical condition on the exponent
<2k1. Then the identity ei(A)=Zo
;(0) continues to hold. We expect
this identity to be valid irrespective of . The technical problems appearing
in our proof for this last case appear since the continuous spectrum of the
signature operator has two layers, coresponding to closed and co-closed forms.
Such a phenomenon occurs also in the analysis of the Laplacian on forms on
hyperbolic manifolds of nite volume [77].

Part III
Research perspectives
33

35
After two years at the University of Bucharest, where I had graduated the
rst three years of study, I have obtained my DEA in the Ecole Polytechnique
in Paris, France and my PhD in MIT, USA. I had the chance to study in
the best Romanian university and in two leading research institutions in the
world, with some of the best Romanian and foreign mathematicians of our
times: Kostake Teleman, Paul Gauduchon, Richard Melrose. I am indebted
to them for teaching me some great pieces of mathematics. After my thesis
I have spent two years at IMAR between 1999{2001, then I have taken post-
doc positions in Hamburg in the research group of Christian B ar, the current
president of the German Mathematical Society, and in Toulouse, where I
have worked with Andr e Legrand and Jean-Marc Schlenker. The Graduate
School in Toulouse accepted my Habilitation project, I have defended my
Habilitation  a Diriger des Recherches thesis in June 2004. My jury consisted
of A. Legrand, E. Leichtnam, V. Nistor, M. Hilsum and J.-M. Schlenker.
My activity in the last eight years has been mainly carried out while a re-
searcher (CS II) at the Institute of Mathematics "Simion Stoilow" of the
Romanian Academy. Between 2006{2012 I have been a Invited Professor in
several top research institutions in France: University of Toulouse, Ecole Nor-
male Superieure, Ecole Polytechnique, and a visiting scientist in the Institut
des Hautes Etudes Scientiques and Institut Henri Poincar e. My teach-
ing activity has consisted in two courses at the Scoala Normala Superioara
Bucharest on \Hyperbolic Geometry" and on \Uniformization of Surfaces",
and a mini-course at a summer school in KIAS – Korea Institute for Ad-
vanced Studies. I have co-directed (together with Colin Guillarmou) in 2011
and 2012 two groups of two students each from the Ecole Normale Superieure
(Ulm) for their Licence-3 memoirs. I had a post-doc on my Marie Curie ERG
grant (Sylvain Gol enia, presently Ma^ tre de Conferences in Bordeaux) with
whom I wrote two papers published in Ann H Poincar e and Trans Am Math
Soc. I directed a BS thesis of a student in Bucharest University. We had
to invoke a dierent nominal advisor for bureaucratic resons, a fact which I
disliked.
Since 2011 I am a member of the Mathematics committee in charge of vali-
dating university degrees (CNATDCU), including Habilitation theses. I am
currently on two Habilitation thesis committees nominated by CNATDCU.
I have obtained two national grants in the last ve years as Principal Inves-
tigator, and three international grants since 2001 as PI for the Romanian
side.

36
1. Grant PNII-TE-0053/2011 "Quantum invariants in hyperbolic geome-
try", 2012-2014.
2. Grant PNII-ID-1188/2009 "Geometric and quantum invariants of 3-
manifolds and applications", 2009-2011.
3. CNRS { Romanian Academy grant, 2006{2007.
4. Marie Curie European reintegration grant MERG-006375, 2004-2005.
5. DFG grant 436 RUM 17/7/01, 2001.
The works on hyperbolic geometry included in this thesis were partially de-
veloped in the framework of the grant PNII-ID-1188/2009. They open new
perspectives on this important objects by furthering the knowledge on the
deformation theory of hyperbolic cone-manifolds, and on the spectral the-
ory of the Dirac operator on convex co-compact hyperblic manifolds. Some
of the resulting directions of research are covered by my grant PNII-TE-
0053/2011, and they deal notably with Chern-Simons theory on Teichm uller
space, which is the subject of a current joint project with Colin Guillarmou.
We essentially believe that complex Chern-Simons invariants will include si-
multaneously the information encoded in the eta invariant and in the renor-
malized volume. Our research program could result in a representation of
the mapping class group in a Hilbert space of holomorphic section in the
Hermitian Chern-Simons line bundle over the Teichm uller space. Such a rep-
resentation would be of substantial interest for TQFT, since up-to-date it is
not known whether the mapping class group of genus gsurfaces admits any
faithful nite-dimensional representations.
Outside hyperbolic geometry, I am currently interested in the more general
notion of Einstein metrics. Together with Bernd Ammann and Andrei Mo-
roianu we have proved that the Cauchy problem for the Einstein equation in
Riemannian geometry, i.e., the problem of constructing an Einstein metric
starting from a hypersurface and a symmetric tensor playing the role of a
second fundamental form, always has solutions if the initial data are real
analytic. We also give examples when the solution exists if and only if the
initial metric is real analytic. This problem, or at least its Lorentzian coun-
terpart, has been at the center of mathematical relativity since the work
of Choquet-Bruhat some 50 years ago. In the semi-Riemannian framework
the Cauchy problem always has solutions, since it behaves essentially like a

37
wave equation. In the Riemannian case we discovered that this is not the
case. Further work is needed to nd weaker conditions than real-analyticity
to ensure existence of the solutions.
In another project with Andrei Moroianu we examine Ricci surfaces, that
is, surfaces whose Gaussian curvature satises the nonlinear elliptic partial
dierential equation
K
K+jdKj2+ 4K3= 0:
We show that every Ricci surface can be locally embedded either in the
at
Euclidean space R3as a minimal surface, or in the
at Lorentz space R2;1
as a maximal surface. This completes the answer to a question raised and
partially answered by Ricci-Curbastro in 1895, who solved the easy case, that
of non-vanishing curvature. An intersting question which we do not solve is
what genus gsurfaces admit Ricci metrics. We show that when gis odd,
every hyperelliptic surface admits a Ricci metric. This is one of my projects
for future research, that I could share with a prospective student.
In a fourth ongoing project, this time with Colin Guillarmou and Jean-Marc
Schlenker, we look at the renormalized volume for asymptotically hyperbolic
Einstein manifolds. We discovered that in a given conformal class at in-
nity the renormalized volume is maximized when a certain quantity vn, a
higher-dimensional analog of the Gauss curvature and of the Q-curvature,
is constant. This renormalized volume plays therefore an analog role to the
renormalized volume from convex co-compact manifolds, which is a K ahler
potential on the Teichm uller space of the ideal boundary. The study of con-
formal manifolds admitting metrics with vnconstant could be a very fruitful
thesis subject.
All the above projects could lead to reasonably dicult and meaningful PhD
thesis subjects. The virtue of these subjects, as compared to the sort of
thesis subjects I encounter in my activity on the CNATDCU mathematics
committee, is being actually connected to the main stream of mathematical
research, following a denite goal to prove an actual mathematical result of
interest to the scientic community, as opposed to the prevailing attitude in
our country in the last 10-20 years, which means focusing on some oral exams,
some conspects from the literature, culminating in a lengthy manuscript
which too often does not go beyond the depth level of a MS thesis.

38
Closing remarks
I chose to return to Romania after my PhD at MIT (1999) in the hope of
contributing towards a world-level mathematical atmosphere in Romania.
My PhD thesis took 5 years to be recognized by the bureaucracy in the
ministry. I was available to direct PhD theses in 2004, after nishing my post-
doctoral fellowships in Germany and France and returning to Bucharest with
an Habilitation degree. Most of my Romanian colleagues who held similar
positions to mine (associate professor) in the US or France could, and did,
direct PhD students. Unfortunately, myself I could not get the \right to
direct PhD's" since I was not a full professor. My application for a full
professor position was turned down because I had not had written 2 books, as
the twisted standards of that time required. I preferred instead to concentrate
on my research, hoping for better times. Then the notion of Habilitation was
nally introduced in Romania in 2011. In my grant application PNII-TE-
0053/2011 I had anticipated hiring a PhD student… only to realize that
my Habiliation from France (2004) was after all still not good enough to
be allowed to direct theses in Romania! I am therefore presenting this new
thesis, although I nd it irresponsible to ask someone with a PhD from MIT
and HDR from Toulouse to write yet another thesis for bureaucratic reasons,
even more in light of the abysmal level of research in post-1989 Romania as
compared to the US or France, the leading mathematical countries of our
times.
Although my French Habilitation is not recognized by our authorities, I am
currently on two Habilitation committees in Romania. Indeed, foreign mem-
bers on these committees cannot be reasonably required to possess a Roma-
nian Habilitation, but just any foreign Habilitation or equivalent diploma.
The criteria are stated uniformly for the whole committee, not just for its
foreign members. I thus fulll the criteria thanks to my French Habilitation!
The Habilitation process in the hands of Romanian bureaucracy has good
chances of becoming yet another spectacular illustration of Titu Maiorescu's
celebrated \forme f ar a fond" theory, an unfortunate mainstay of Romanian
culture in the past two centuries.
Despite these diculties I am convinced that doing mathematical research
in Romania is still possible today. My intention is to develop my career here
and I aim to create a strong group in geometric analysis. Being allowed to
direct PhD theses will be a step in the right direction.

Part IV
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39

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[53] Sergiu Moroianu, Hablitation  a Diriger des Recherches, Universit e Paul
Sabatier, Toulouse, June 2004.
[54] R. Lauter, Sergiu Moroianu, Fredholm Theory for degenerate pseudodif-
ferential operators on manifolds with bered boundaries, Comm. Partial
Di. Equat. 26(2001), 233{285.
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[56] R. Lauter, Sergiu Moroianu, The index of cusp operators on manifolds
with corners, Ann. Global Anal. Geom. 21(2002), 31{49.
[57] Sergiu Moroianu, Sur la limite adiabatique des fonctions ^ eta et z^ eta,
Comptes Rendus Math. 334(2002), 131{134.
[58] Sergiu Moroianu, K-Theory of suspended pseudo-dierential operators,
K-Theory 28(2003), 167{181.
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manifolds with bered cusps, T. Am. Mat. Soc. 355(2003), 3009{3046.
[60] C. B ar, Sergiu Moroianu, Heat Kernel Asymptotics for Roots of Gener-
alized Laplacians, Int. J. Math. 14(2003), 397{412.

46 BIBLIOGRAPHY
[61] Sergiu Moroianu, Adiabatic limits of eta and zeta functions of elliptic
operators, Math. Z. 246(2004), 441{471.
[62] Sergiu Moroianu, Homology of adiabatic pseudo-dierential operators,
Nagoya Math. J. 175(2004), 171{221.
[63] R. Lauter, Sergiu Moroianu, An index formula on manifolds with bered
cusp ends, J. Geom. Analysis 15(2005), 261{283.
[64] Sergiu Moroianu, Cusp geometry and the cobordism invariance of the
index, Adv. Math. 194(2005), 504{519.
[65] I. Cotaescu, Sergiu Moroianu, M. Visinescu, Gravitational and axial
anomalies for generalized Euclidean Taub-NUT metrics, J. Phys. A
{ Math. Gen. 38(2005), 7005{7019.
[66] A. Legrand, Sergiu Moroianu, On theLpindex of spin Dirac operators
on conical manifolds, Studia Math. 177(2006), 97{112.
[67] Sergiu Moroianu, On Carvalho's K-theoretic formulation of the cobor-
dism invariance of the index, Proc. Amer. Math. Soc. 134 (2006),
3395{3404.
[68] Sergiu Moroianu, M. Visinescu, L2-index of the Dirac operator of gener-
alized Euclidean Taub-NUT metrics, J. Phys. A – Math. Gen. 39(2006),
6575{6581.
[69] T. Banica, Sergiu Moroianu, On the structure of quantum permutation
groups, Proc. Amer. Mat. Soc. 135(2007), 21{29.
[70] Sergiu Moroianu, Fibered cusp versus d-index theory, Rendiconti Semin.
Math. Padova. 117(2007), 193{203.
[71] A. Moroianu, Sergiu Moroianu, The Dirac spectrum on manifolds with
gradient conformal vector elds, (with Andrei Moroianu), J. Funct.
Analysis. 253nr. 1 (2007), 207{219.
[72] Sergiu Moroianu, Weyl laws on open manifolds, Math. Annalen 340,
nr. 1 (2008), 1{21.

BIBLIOGRAPHY 47
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Phys. (2008), 123{148.
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on conformally cusp manifolds, Ann. H. Poincar e 9(2008), 131{179.
[75] P. Loya, Sergiu Moroianu, J. Park, Adiabatic limit of the eta invari-
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1593{1616.
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Eta functions of an Elliptic Operator, Int. J. Math. 23, nr. 6. (2012).
[83] C. Guillarmou, Sergiu Moroianu, Chern-Simons line bundle on Te-
ichm uller space, preprint arXiv:1102.1981, submitted.
[84] B. Ammann, A. Moroianu, Sergiu Moroianu, The Cauchy problems for
Einstein metrics and parallel spinors, preprint arXiv:1106.2066, sub-
mitted.

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