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

Erwann Aubry (1) , Jean-Francois Grosjean (2) , Julien Roth (3)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 IECN - Institut Élie Cartan de Nancy
3 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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 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 $q>\frac{n}{2}$, but not necessarily diffeomorphic to a sphere when $q<\frac{n}{2}$.

Mots clés: Mean curvature; Reilly inequality; Laplacian; Spectrum; pinching results; hypersurfaces

Domaines: Géométrie différentielle [math.DG]

Erwann Aubry, Jean-Francois Grosjean, Julien Roth. Hypersurfaces with small extrinsic radius or large $\lambda_1$ in Euclidean spaces. 2010. ⟨hal-00516633v1⟩