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We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\unicode[STIX]{x2202}M$ has dimension $n$ even. Its definition depends on the choice of metric $h_{0}$ on $\unicode[STIX]{x2202}M$ in the conformal class at infinity determined by $g$, we denote it by $\text{Vol}_{R}(M,g;h_{0})$. We show that $\text{Vol}_{R}(M,g;\cdot )$ is a functional admitting a ‘Polyakov type’ formula in the conformal class $[h_{0}]$ and we describe the critical points as solutions of some non-linear equation $v_{n}(h_{0})=\text{constant}$, satisfied in particular by Einstein metrics. When $n=2$, choosing extremizers in the conformal class amounts to uniformizing the surface, while if $n=4$ this amounts to solving the $\unicode[STIX]{x1D70E}_{2}$-Yamabe problem. Next, we consider the variation of $\text{Vol}_{R}(M,\cdot ;\cdot )$ along a curve of AHE metrics $g^{t}$ with boundary metric $h_{0}^{t}$ and we use this to show that, provided conformal classes can be (locally) parametrized by metrics $h$ solving $v_{n}(h)=\text{constant}$ and $\text{Vol}(\unicode[STIX]{x2202}M,h)=1$, the set of ends of AHE manifolds (up to diffeomorphisms isotopic to the identity) can be viewed as a Lagrangian submanifold in the cotangent space to the space ${\mathcal{T}}(\unicode[STIX]{x2202}M)$ of conformal structures on $\unicode[STIX]{x2202}M$. We obtain, as a consequence, a higher-dimensional version of McMullen’s quasi-Fuchsian reciprocity. We finally show that conformal classes admitting negatively curved Einstein metrics are local minima for the renormalized volume for a warped product type filling.

On any odd-dimensional oriented Riemannian manifold we define a volume form, which we call the odd Pfaffian, through a certain invariant polynomial with integral coefficients in the curvature tensor. We prove an intrinsic Chern-Gauss-Bonnet formula for incomplete edge singularities in terms of the odd Pfaffian on the fibers of the boundary fibration. The formula holds for product-type model edge metrics where the degeneration is of conical type in each fiber, but also for general classes of perturbations of the model metrics. The same method produces a Chern- Gauss-Bonnet formula for complete, non-compact manifolds with fibered boundaries in the sense of Mazzeo-Melrose and perturbations thereof, involving the odd Pfaffian of the base of the fibration. We deduce the rationality of the usual Pfaffian form on Riemannian orbifolds, and exhibit obstructions for certain metrics on a fibration to be realized as the model at infinity of a flat metric with conical, edge or fibered boundary singularities., Comment: This second version corrects a statement about the degenerate metric on the blow-up of a submanifold, a few typos and includes new references

We report on a particular case of the paper [7], joint with Raphael Ponge, showing that generically, the eta function of a first-order differential operator over a closed manifold of dimension n has first-order poles at all positive integers of the form n− 1, n− 3, n− 5, . . .. Version francaise abregee Soit D la classe des operateurs differentiels elliptiques symmetriques d’ordre 1 sur une variete Riemannienne fermee M , agissant sur les sections d’un fibre vectoriel Hermitien E. Il est connu que pour un operateur D ∈ D , le spectre de D en tant qu’operateur non-borne dans L(M,E) est discret. Les fonctions eta et zeta associees a D sont defines par les series

We describe the spectrum of the k k -form Laplacian on conformally cusp Riemannian manifolds. The essential spectrum is shown to vanish precisely when the k k and k − 1 k-1 de Rham cohomology groups of the boundary vanish. We give Weyl-type asymptotics for the eigenvalue-counting function in the purely discrete case. In the other case we analyze the essential spectrum via positive commutator methods and establish a limiting absorption principle. This implies the absence of the singular spectrum for a wide class of metrics. We also exhibit a class of potentials V V such that the Schrödinger operator has compact resolvent, although in most directions the potential V V tends to − ∞ -\infty . We correct a statement from the literature regarding the essential spectrum of the Laplacian on forms on hyperbolic manifolds of finite volume, and we propose a conjecture about the existence of such manifolds in dimension 4 whose cusps are rational homology spheres.

We find sufficient conditions for the absence of harmonic $L^2$ spinors on spin manifolds constructed as cone bundles over a compact K\"ahler base. These conditions are fulfilled for certain perturbations of the Euclidean metric, and also for the generalized Taub-NUT metrics of Iwai-Katayama, thus proving a conjecture of Vi\csinescu and the second author., Comment: Final version, 16 pages

We study the adiabatic limit of the eta invariant of the Dirac operator over cofinite quotient of PSL(2,R), which is a noncompact manifold with a nonexact fibred-cusp metric near the ends., 25 pages

We show that the Dirac operator on a spin manifold does not admit $L^2$ eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing., Comment: 12 pages

Under suitable invertibility hypothesis, the spectrum of the Dirac operator on certain open spin Riemannian manifolds is discrete, and obeys a growth law depending qualitatively on the (in)finiteness of the volume., 24 pages

The quantum permutation group of the set $X_n=\{1,..., n\}$ corresponds to the Hopf algebra $A_{aut}(X_n)$. This is an algebra constructed with generators and relations, known to be isomorphic to $\cc (S_n)$ for $n\leq 3$, and to be infinite dimensional for $n\geq 4$. In this paper we find an explicit representation of the algebra $A_{aut}(X_n)$, related to Clifford algebras. For $n=4$ the representation is faithful in the discrete quantum group sense., 9 pages

We give an analytic proof of the fact that the index of an elliptic operator on the boundary of a compact manifold vanishes when the principal symbol comes from the restriction of a K-theory class from the interior. The proof uses non-commutative residues inside the calculus of cusp pseudodifferential operators of Melrose.

We compute the axial anomaly for the Taub-NUT metric on $R^4$. We show that the axial anomaly for the generalized Taub-NUT metrics introduced by Iwai and Katayama is finite, although the Dirac operator is not Fredholm. We show that the essential spectrum of the Dirac operator is the whole real line.

We compute the index of the Dirac operator on a spin Riemannian manifold with conical singularities, acting from Lp(_+) to Lq(_-) with p, q > 1. When 1+n/p-n/q > 0 we obtain the usual Atiyah-Patodi-Singer formula, but with a spectral cut at (n + 1)/2 - n/q instead of 0 in the definition of the eta invariant. In particular we reprove Chou's formula for the L2 index. For 1+n/p-n/q _ 0 the index formula contains an extra term related to the Calderon projector.

The generalized Taub–NUT metrics exhibit in general gravitational anomalies. This is in contrast with the fact that the original Taub–NUT metric does not exhibit gravitational anomalies, which is a consequence of the fact that it admits Killing–Yano tensors forming Stackel–Killing tensors as products. We have found that for axial anomalies, interpreted as the index of the Dirac operator, the presence of Killing–Yano tensors is irrelevant. In order to evaluate the axial anomalies, we compute the index of the Dirac operator with the APS boundary condition on balls and on annular domains. The result is an explicit number-theoretic quantity depending on the radii of the domain. This quantity is 0 for metrics close to the original Taub–NUT metric but it does not vanish in general.

We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator $\delta$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values ${\eta}(\delta_t,0)$ and $t{\zeta}(\delta_t,0)$ are smooth functions of $t$ at $t=0$, and we identify their values at $t=0$ with the holonomy of the determinant bundle, respectively with a residue trace. For invertible families of operators, the functions ${\eta}(\delta_t,s)$ and $t{\zeta}(\delta_t,s)$ are shown to extend smoothly to $t=0$ for all values of $s$. After normalizing with a Gamma factor, the zeta function satisfies in the adiabatic limit an identity reminiscent of the Riemann zeta function, while the eta function converges to the volume of the Bismut-Freed meromorphic family of connection 1-forms., Comment: 32 pages, final version

We describe the heat kernel asymptotics for roots of a Laplace type operator Δ on a closed manifold. A previously known relation between the Wodzicki residue of Δ and heat trace asymptotics is shown to hold pointwise for the corresponding densities.

The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals that generate the 0 0 -dimensional Hochschild cohomology groups, the index of a fully elliptic fibered cusp operator is expressed as the sum of a local contribution of Atiyah-Singer type and a global term on the boundary. We announce a result relating this boundary term to the adiabatic limit of the eta invariant in a particular case.

We recall a Fredholm criterion for fully elliptic cusp(pseudo)differential operators on a compact manifold with corners ofarbitrary codimension, acting on suitable Sobolev spaces. Then we give aformula for the index in terms of regularized `trace' functionalssimilar to the residue trace of Wodzicki and Guillemin.

Resume Dans cette Note, on demontre l'existence de la limite adiabatique de la fonction η ( s ) d'un operateur sur l'espace total d'une fibration au dessus de S 1 , construit a partir d'une famille d'operateurs differentiels inversibles d'ordre 1. Nous identifions cette limite a l'holonomie d'une famille meromorphe de connexions dans le fibre trivial. Dans le meme contexte, la fonction ζ diverge. On donne une formule pour les deux premiers coefficients du developpement asymptotique. Le premier resultat reste vrai pour une famille non-inversible si on se restreint a s =0. Dans le cas d'une famille d'operateurs de Dirac, on retrouve la formule d'holonomie de Bismut–Freed. Pour citer cet article : S. Moroianu, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 131–134

We consider the calculus Ψ*,* de(X, deΩ½) of double-edge pseudodifferential operators naturally associated to a compact manifold X whose boundary is the total space of a fibration. This fits into the setting of boundary fibration structures, and we discuss the corresponding geometric objects. We construct a scale of weighted double-edge Sobolev spaces on which double-edge pseudodifferential operators act as bounded operators, characterize the Fredholm elements in Ψ*,* de(X) by means of the invertibility of an appropriate symbol map, and describe a K-theoretical formula for the Fredholm index extending the Atiyah–Singer formula for closed manifolds. The algebra of operators of order (0, 0) is shown to be a Ψ*-algebra, hence its K-theory coincides with that of its C *-closure, and we give a description of the corresponding cyclic 6-term exact sequence. We define a Wodzicki-type residue trace on an ideal in Ψ*,* de(X, deΩ½), and we show that it coincides with Dixmier's trace for operators of order –dim X in ...

We give a short proof of the fact that compact pluricanonical locally conformally K\"ahler manifolds have parallel Lee form., Comment: 6 pages, to appear in IMRN

A Ricci surface is a Riemannian 2-manifold $(M,g)$ whose Gaussian curvature $K$ satisfies $K\Delta K+g(dK,dK)+4K^3=0$. Every minimal surface isometrically embedded in $\mathbb{R}^3$ is a Ricci surface of non-positive curvature. At the end of the 19th century Ricci-Curbastro has proved that conversely, every point $x$ of a Ricci surface has a neighborhood which embeds isometrically in $\mathbb{R}^3$ as a minimal surface, provided $K(x), Comment: 27 pages; final version, to appear in Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

On a spin manifold with conformal cusps, we prove under an invertibility condition at infinity that the eta function of the twisted Dirac operator has at most simple poles and is regular at the origin. For hyperbolic manifolds of finite volume, the eta function of the Dirac operator twisted by any homogeneous vector bundle is shown to be entire., Comment: 22 pages, 2 figures

We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{\Gamma,\Sigma}^{\rm o}(\lambda)$ associated to the spinor bundle $\Sigma$ on odd dimensional convex co-compact hyperbolic manifolds $X:=\Gamma\backslash\hh^{2n+1}$. We define a natural eta invariant $\eta(D)$ associated to the Dirac operator $D$ on $X$ and prove that $\eta(D)=\frac{1}{\pi i}\log Z_{\Gamma,\Sigma}^{\rm o}(0)$, thus extending Millson's formula to this setting. As a byproduct, we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We also define an eta invariant for the odd signature operator and, under some conditions, we describe it on the Schottky space of 3-dimensional Schottky hyperbolic manifolds and relate it to Zograf factorization formula., Comment: 36 pages

We consider a compact manifold whose boundary is a locally trivial fiber bundle and an associated pseudodifferential algebra that models fibered cusps at infinity. Using trace-like functionals that generate the 0-dimensional Hochschild cohomology groups, we express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior and a term that comes from the boundary. This answers the index problem formulated by Mazzeo and Melrose. We give a more precise answer in the case where the base of the boundary fiber bundle is the circle. In particular, for Dirac operators associated to a "product fibered cusp metric", the index is given by the integral of the Atiyah-Singer form in the interior minus the adiabatic limit of the eta invariant of the restriction of the operator to the boundary., Comment: 22 pages. Prepublication du Laboratoire Emile Picard n.253. See also http://picard.ups-tlse.fr