Conformal harmonic forms, Branson-Gover operators and Dirichlet problem at infinity

For odd dimensional Poincar\'e-Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for k<\ndemi) which are $C^m$ (with m\in\nn) on the conformal compactification $\bar{X}$ of $X$. This is infinite dimensional for small $m$ but it becomes finite dimensional if $m$ is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology H^k(\bar{X},\pl\bar{X}) and the kernel of the Branson-Gover \cite{BG} differential operators $(L_k,G_k)$ on the conformal infinity (\pl\bar{X},[h_0]). In a second time we relate the set of $C^{n-2k+1}(\Lambda^k(\bar{X}))$ forms in the kernel of $d+\delta_g$ to the conformal harmonics on the boundary in the sense of \cite{BG}, providing some sort of long exact sequence adapted to this setting. This study also provides another construction of Branson-Gover differential operators, including a parallel construction of the generalization of $Q$ curvature for forms.Comment: 35 page

Metric shape of hypersurfaces with small extrinsic radius or large λ1 \lambda_1

We determine the Hausdorff limit-set of the Euclidean hypersurfaces with large $\lambda_1$ or small extrinsic radius. The result depends on the $L^p$ norm of the curvature that is assumed to be bounded a priori, with a critical behaviour for $p$ equal to the dimension minus 1

Finiteness of π1\pi_1 and geometric inequalities in almost positive Ricci curvature.

International audienceWe show that complete $n$-manifolds whose part of Ricci curvature less than a positive number is small in $L^p$ norm (for $p>n/2$) have bounded diameter and finite fundamental group. On the contrary, complete metrics with small $L^{n/2}$-norm of the same part of the Ricci curvature are dense in the set of metrics of any compact differentiable manifold

Pincement sur le spectre et le volume en courbure de Ricci positive

Paru dans Ann. Sci. Ec. Norm. sup (2005)International audienceWe show that a complete Riemannian manifold of dimension $n$ with \Ric\geq n{-}1 and its $n$-st eigenvalue close to $n$ is both Gromov-Hausdorff close and diffeomorphic to the standard sphere. This extends, in an optimal way, a result of P. Petersen. We also show that a manifold with \Ric\geq n{-}1 and volume close to \frac{\Vol\sn}{#\pi_1(M)} is both Gromov-Hausdorff close and diffeomorphic to the space form \frac{\sn}{\pi_1(M)}. This extends results of T. Colding and T. Yamaguchi

Hypersurfaces with small extrinsic radius or large λ1\lambda_1 in Euclidean spaces

We prove that hypersurfaces of $\R^{n+1}$ which are almost extremal for the Reilly inequality on $\lambda_1$ and have $L^p$-bounded mean curvature ($p>n$) are Hausdorff close to a sphere, have almost constant mean curvature and have a spectrum which asymptotically contains the spectrum of the sphere. We prove the same result for the Hasanis-Koutroufiotis inequality on extrinsic radius. We also prove that when a supplementary $L^q$ bound on the second fundamental is assumed, the almost extremal manifolds are Lipschitz close to a sphere when $q>n$, but not necessarily diffeomorphic to a sphere when $q\leqslant n$.Comment: 24 page

APPROXIMATION OF THE SPECTRUM OF A MANIFOLD BY DISCRETIZATION

Abstract. We approximate the spectral data (eigenvalues and eigenfunctions) of compact Riemannian manifold by the spectral data of a sequence of (computable) discrete Laplace operators associated to some graphs immersed in the manifold. We give an upper bound on the error that depends on upper bounds on the diameter and the sectional curvature and on a lower bound on the injectivity radius. hal-00776850, version 1- 16 Jan 2013 1

Variétés de courbure de Ricci presque minorée: inégalités géométriques optimales et stabilité des variétés extrémales

Jury: Yves COLIN DE VERDIÈRE (Professeur, Université de Grenoble), Président; Sylvestre GALLOT (Professeur, Université de Grenoble), Directeur de thèse; Étienne GHYS (Directeur de Recherche CNRS, ENS Lyon); Jacques LAFONTAINE (Professeur, Université de Montpellier);We study the geometry of manifolds whose Ricci curvature is almost bounded from below (i.e. such that, for a fixed real $k$, a $L^p$ norm of the function $(\underline(\rm Ric)-k)^-$ is small, where $\underline(\rm Ric)(x)$ is the smallest eigenvalue of the Ricci curvature at $x$). We prove, under this assumption, the extensions of the classical geometric inequalities of Myers, Bishop-Gromov, Lichnerowicz,...; we then characterize the almost extremal manifolds (extending some results of T.~Colding and P.~Petersen). For closed Riemannian manifolds $M^n$ with almost positive Ricci curvature, the Hodge-Laplacian on 1-forms admits at most $n$ small eigenvalues. If there are exactly $n$ small eigenvalues ($n-1$ are sufficient if $M$ is orientable) then $M$ is diffeomorphic to a nilmanifold, and the metric is almost left-invariant. These results are applications of analytic estimates established in the first part of the thesis.On s'intéresse à la géométrie des variétés de courbure de Ricci presque supérieure à $k$ (i.e. telle qu'une norme $L^p$---locale ou globale---de la fonction $(\underline(\rm Ric)-k)^-$ soit petite, où $\underline(\rm Ric)(x)$ est la plus petite valeur propre de la courbure de Ricci en $x$). On démontre sous cette hypothèse les équivalents des inégalités géométriques classiques de Myers, de Bishop-Gromov, de Lichnerowicz,... puis on caractérise les variétés qui réalisent presque les cas d'égalité (généralisant des travaux de T.~Colding et de P.~Petersen). Sur une variété compacte $M^n$ de courbure presque positive, le laplacien sur les 1-formes a au plus $n$ petites valeurs propres. S'il a exactement $n$ petites valeurs propres ($n-1$ suffisent si $M$ est orientable) alors $M$ est difféomorphe à une Nilvariété et la métrique est presque invariante à gauche. Ces résultats découlent d'estimées analytiques établies dans la première partie de la thèse