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

Abstract : 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.
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https://tel.archives-ouvertes.fr/tel-00004006
Contributor : Arlette Guttin-Lombard <>
Submitted on : Wednesday, December 17, 2003 - 3:15:32 PM
Last modification on : Tuesday, May 11, 2021 - 11:36:03 AM
Long-term archiving on: : Wednesday, September 12, 2012 - 12:05:25 PM

### Identifiers

• HAL Id : tel-00004006, version 1

### Citation

Erwann Aubry. Variétés de courbure de Ricci presque minorée: inégalités géométriques optimales et stabilité des variétés extrémales. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2003. Français. ⟨tel-00004006⟩

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