Hypersurfaces with small extrinsic radius or large ${\lambda }_1$ in Euclidean spaces

Erwann Aubry, Jean-Francois Grosjean, Julien Roth

We prove that hypersurfaces of ${\mathbb{R}}^{n+1}$ 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…