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

Abstract : 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 $q\leqslant n$.
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https://hal.archives-ouvertes.fr/hal-00516633
Contributor : Jean-Francois Grosjean <>
Submitted on : Thursday, November 25, 2010 - 4:29:43 PM
Last modification on : Friday, July 9, 2021 - 11:30:07 AM
Long-term archiving on: : Monday, February 28, 2011 - 8:58:27 AM

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### Identifiers

• HAL Id : hal-00516633, version 2
• ARXIV : 1009.2010

### Citation

Erwann Aubry, Jean-Francois Grosjean, Julien Roth. Hypersurfaces with small extrinsic radius or large $\lambda_1$ in Euclidean spaces. 2010. ⟨hal-00516633v2⟩

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