No abstract

Abstract. For odd-dimensional Poincaré-Einstein manifolds (X n+1 , g), we study the set of harmonic k-forms (for k < n/2) which are C m (with m ∈ N) on the conformal compactification X of X. This set 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 (X, ∂X) and the kernel of the Branson-Gover [3] differential operators (L k , G k ) on the conformal infinity (∂X, [h 0 ]). We also relate the set of C n−2k+1 ( k (X)) forms in the kernel of d + δ g to the conformal harmonics on the boundary in the sense of [3], 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.

Nous montrons qu'une variété riemannienne complète de dimension n, de courbure Ric n − 1 et dont la n-ième valeur propre est proche de n est Gromov-Hausdorff proche de (S n , can) et difféomorphe à S n. Ce résultat étend de manière optimale un résultat de P. Petersen [Invent. Math. 138 (1999) 1] (au passage nous comblons le trou annoncé par l'auteur dans l'erratum [Invent. Math. 155 (2004) 223]). Nous montrons également qu'une variété vérifiant l'inégalité Ric n − 1 et de volume proche de Vol S n #π 1 (M) est difféomorphe à un quotient isométrique S n /π1(M) et Gromov-Hausdorff proche de la métrique de courbure 1. Ceci améliore des résultats de T. Colding [Invent. Math. 124 (1996) 175] et T. Yamaguchi [Math. Ann. 284 (1989) 423].  2005 Elsevier SAS ABSTRACT.-We shall show that a complete Riemannian manifold of dimension n with Ric 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 [Invent. Math. 138 (1999) 1] (as a by-product, we fill a gap stated in the erratum [Invent. Math. 155 (2004) 223]). We shall also show that a manifold with Ric n − 1 and volume close to Vol S n #π 1 (M) is both Gromov-Hausdorff close and diffeomorphic to a space form S n /π1(M). This extends results of T.

Abstract. In this paper, we prove that Euclidean hypersurfaces with almost extremal extrinsic radius or λ1 have a spectrum that asymptotically contains the spectrum of the extremal sphere in the Reilly or Hasanis-Koutroufiotis Inequalities. We also consider almost extremal hypersurfaces which satisfy a supplementary bound on vM B n α and show that their spectral and topological properties depends on the position of α with respect to the critical value dim M . The study of the metric shape of these extremal hypersurfaces will be done in AG1[3], using estimates of the present paper. IntroductionThroughout the paper, X: M n → R n+1 is a closed, connected, immersed Euclidean hypersurface (with n 2). We set v M its volume, B its second fundamental form, H = 1 n tr B its mean curvature, r M its extrinsic radius (i.e. the least radius of the Euclidean balls containing M ), (λ M i ) i∈N the non-decreasing sequence of its eigenvalues labelled with multiplicities andThe Hasanis-Koutroufiotis inequality asserts that n H 2 2 , once again with equality if and only if M is the sphere S M (we give some short proof of these inequalities in section prel 2). Our aim is to study the spectral properties of the hypersurfaces that are almost extremal for each of this Inequalities. The results and estimates of this paper are used in AG1[3] to study the metric shape of the almost extremal hypersurfaces. We set µ2 the k-th eigenvalue of S M (labelled without multiplicities) and m k its multiplicity. Throughout the paper we shall adopt the notation that τ (ε|n, · · · ) is a positive function which depends on n, · · · and which converges to zero with ε → 0 when n, · · · are fixed.

No abstract