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

TL;DR

This paper shows that hypersurfaces in Euclidean space nearly extremal for certain geometric inequalities are geometrically close to spheres, with spectral properties similar to spheres, under specific curvature bounds.

Contribution

It establishes quantitative closeness to spheres for hypersurfaces nearly extremal for Reilly and Hasanis-Koutroufiotis inequalities, considering various curvature bounds and regularity conditions.

Findings

Hypersurfaces nearly extremal for Reilly inequality are Hausdorff close to spheres.

Under additional $L^q$ bounds, hypersurfaces are Lipschitz close to spheres.

Spectral properties of these hypersurfaces asymptotically match those of spheres.

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 $q\leqslant n$.