Hypersurfaces with small extrinsic radius or large $λ_1$ in Euclidean spaces

Details Hypersurfaces with small extrinsic radius or large $λ_1$ in Euclidean spaces Hypersurfaces with small extrinsic radius or large $λ_1$ in Euclidean spaces

2010-09-10

arxiv.org/abs/1009.2010v2

Mathematics

Differential Geometry

24 pages

Scientific paper

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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