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

Abstract : We 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.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-00338163
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Submitted on : Tuesday, November 11, 2008 - 6:31:04 PM
Last modification on : Thursday, April 29, 2021 - 12:06:08 PM

### Citation

Erwann Aubry. Pincement sur le spectre et le volume en courbure de Ricci positive. Annales Scientifiques de l'École Normale Supérieure, Société mathématique de France, 2005, 38, p 387--405. ⟨hal-00338163⟩

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