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We prove in a simple and coordinate-free way the equivalence between the classical definitions of the mass or of the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asymptotically hyperbolic case.

This note computes the "renormalized volume" and a renormalizedGauss-Bonnet-Chern formula for the Euler characteristic ofasymptotically complex hyperbolic Einstein (in short: ACHE)4-manifolds., Comment: revised version ; reference to math.DG/0404455 added

International audience; Nous démontrons que toute métrique d'Einstein asymptotiquement hyperbolique réelle ou complexe possède un développement polyhomogène au voisinage de son bord à l'infini. La preuve s'étend également au cas dit local, c'est-à-dire quand le bord à l'infini est un ouvert de R^n . Ces résultats sont nouveaux en hyperbolique réel dans le cas local en dimension impaire et en hyperbolique complexe dans tous les cas.

In this paper we prove the Spinc analog of the Hijazi inequality on the first eigenvalue of the Dirac operator on compact Riemannian manifolds and study its equality case. During this study, we are naturally led to consider generalized Killing spinors on Spinc manifolds and we prove that such objects can only exist on low-dimensional manifolds (up to dimension three). This allows us to give a nice geometrical description of the manifolds satisfying the equality case of the above-mentioned inequality and to classify them in dimension three and four.

An argument in our paper The Huber theorem for non-compact conformally flat manifolds [Comment. Math. Helv. 77 (2002), 192?220] was not justified. Using recent work by G. Tian and J. Viaclovsky, we show that our result holds true

International audience

In this paper, we develop a general study of contributions at infinity of Bochner-Weitzenb\"ock-type formulas on asymptotically flat manifolds, inspired by Witten's proof of the positive mass theorem. As an application, we show that similar proofs can be obtained in a much more general setting as any choice of an irreducible natural bundle and a very large choice of first-order operators may lead to a positive mass theorem along the same lines if the necessary curvature conditions are satisfied., Comment: Communications in Mathematical Physics, Springer Verlag, 2016, to appear

We prove an optimal Penrose-like inequality for the mass of any asymptotically flat Riemannian 3-manifold having an inner minimal 2-sphere and nonnegative scalar curvature. Our result shows that the mass is bounded from below by an expression involving the area of the minimal sphere (as in the original Penrose conjecture) and some nomalized Sobolev ratio. As expected, the equality case is achieved if and only if the metric is that of a standard spacelike slice in the Schwarzschild space.

Le theme de cet article est la recherche de compactifications conformes compactes et suffisamment regulieres de varietes riemanniennes asymptotiquement plates: une telle compactification existe si les tenseurs de Weyl et de Cotton-York decroissent a l'infini plus vite que r -4 et r -5 . Nous etudions egalement le cas critique ou ces tenseurs ont exactement la decroissance citee.

International audience; We give a simple characterization of the parabolic geodesics introduced by Cap, Slovak and Zadnik for all parabolic geometries. This goes through the definition of a natural connection on the space of Weyl structures. We then show that parabolic geodesics can be characterized as the following data: a curve on the manifold and a Weyl structure along the curve, so that the curve is a geodesic for its companion Weyl structure and the Weyl structure is parallel along the curve and in the direction of the tangent vector of the curve.

Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an isometry extension property is proved: continuous groups of isometries at conformal infinity extend into the bulk of any complete conformally compact Einstein metric. Relations of this property with the invariance of the Gauss-Codazzi constraint equations under deformations are also discussed., 32 pages, supercedes math.DG/0501067; final published version

We give a differential geometric description of the Cartan (or tractor) bundle and its canonical connection in CR geometry, thus offering a direct, alternative, definition to the usual abstract approach., minor changes ; wrong author in reference [7] corrected

We define a renormalized characteristic class for Einstein asymptotically complex hyperbolic (ACHE) manifolds of dimension 4: for any such manifold, the polynomial in the curvature associated to the characteristic class euler-3signature is shown to converge. This extends a work of Burns and Epstein in the Kahler-Einstein case. This extends a work of Burns and Epstein in the Kahler-Einstein case. We also define a new global invariant for any 3-dimensional pseudoconvex CR manifold, by a renormalization procedure of the eta invariant of a sequence of metrics which approximate the CR structure. Finally, we get a formula relating the renormalized characteristic class to the topological number euler-3signature and the invariant of the CR structure arising at infinity., Lemma 2.6 changed because of a mistake. Section 5 (using lemma 2.6) rewritten

International audience; We relate a recently introduced non-local geometric invariant of compact strictly pseudoconvex Cauchy-Riemann (CR) manifolds of dimension 3 to various eta-invariants in CR geometry: on the one hand a renormalized eta-invariant appearing when considering a sequence of metrics converging to the CR structure by expanding the size of the Reeb field; on the other hand the eta-invariant of the middle degree operator of the contact complex. We then provide explicit computations for a class of examples: transverse circle invariant CR structures on Seifert manifolds. Applications are given to the problem of filling a CR manifold by a complex hyperbolic manifold, and more generally by a Kahler-Einstein or an Einstein metric.

We show that complete conformally flat manifolds of dimension n>2 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line, or are globally conformally equivalent to flat space or to a spherical spaceform. This extends previous works by Q.-M. Cheng, M.H. Noronha, B.-L. Chen and X.-P. Zhu, and S. Zhu., revised version, added reference to previous paper by S. Zhu on the same subject

We prove that the round metric on the sphere has the largest first eigenvalue of the Dirac operator among all metrics that are larger than it. As a corollary, this gives an alternative proof of an extremality result for scalar curvature due to M. Llarull., Comment: to appear in G.A.F.A

It was proved in 1957 by Huber that any complete surface with integrable Gauss curvature is conformally equivalent to a compact surface with a finite number of points removed. Counterexamples show that the curvature assumption must necessarily be strengthened in order to get an analogous conclusion in higher dimensions. We show in this paper that any non compact Riemannian manifold with finite $ L^{n/2} $ -norm of the Ricci curvature satisfies Huber-type conclusions if either it is a conformal domain with volume growth controlled from above in a compact Riemannian manifold or if it is conformally flat of dimension 4 and a natural Sobolev inequality together with a mild scalar curvature decay assumption hold. We also get partial results in other dimensions.

International audience

We present a set of global invariants, called "mass integrals", which can be defined for a large class of asymptotically hyperbolic Riemannian manifolds. When the "boundary at infinity" has spherical topology one single invariant is obtained, called the mass; we show positivity thereof. We apply the definition to conformally compactifiable manifolds, and show that the mass is completion-independent. We also prove the result, closely related to the problem at hand, that conformal completions of conformally compactifiable manifolds are unique., Comment: 27 pages, Latex2e with several style files; various misprints corrected, positivity theorem for black holes considerably strengthened, to appear in Pacific Jour. of Mathematics

International audience

In this thesis we give an overview of the volume entropy rigidity problem. A conjecture by Gromov and Katok states that, on a locally symmetric space (M; g0), the symmetric metric g0 has minimal volume entropy among metrices with the same total volume. The text is self-contained, assuming a basic knowledge in differential geometry. Therefore we discuss in the first chapter some background material used in the sequel. The case of compact quotients of H2 _ H2 was unknown before this work ; we give a fully detailled proof. The key-point is to build a calibrating form as in [BCG95]. As a by-product, we present some applications provided by the proof of the volume entropy rigidity conjecture. We conclude by an informal section explaining the motivations of the problem to a non-mathematical reader.; Nous donnons dans cette thèse une preuve du problème de l’entropie volumique minimale dans les quotients compacts de H2_H2. Une conjecture de Gromov et Katok prétend en effet que, sur un espace localement symétrique (M; g0), la métrique de plus petite entropie volumique parmi les métriques de volume fixé est la métrique g0. Le texte se veut relativement abordable. C’est pourquoi nous avons intégré un premier chapitre qui contient une bonne partie du matériel qui sera utilisé par la suite. Puis nous passons en revue les preuves des différents cas du problème déjà traités. Le cas des quotients compacts de H2_H2 n’était pas connu avant ce travail ; nous en détaillons minutieusement la preuve. Notre démarche consiste à faire fonctionner la méthode de calibration imaginée dans [BCG95]. Nous présentons aussi les principales applications qui découlent de la preuve de la conjecture de Gromov et Katok. Nous concluons par une discussion heuristique qui explique les enjeux du problème que nous étudions.