Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces

TL;DR

This paper establishes eigenvalue pinching theorems for compact hypersurfaces in spaces with bounded curvature, leading to rigidity results for stable constant mean curvature and almost umbilic hypersurfaces.

Contribution

It introduces new eigenvalue pinching theorems and applies them to prove rigidity and stability results for hypersurfaces with constant mean curvature or near-umbilicity.

Findings

Eigenvalue pinching theorems for hypersurfaces

Rigidity results for stable constant mean curvature hypersurfaces

Rigidity results for almost umbilic hypersurfaces

Abstract

In this paper we give pinching theorems for the first nonzero eigenvalue of the Laplacian on the compact hypersurfaces of ambient spaces with bounded sectional curvature. As application we deduce rigidity results for stable constant mean curvature hypersurfaces $M$ of these spaces $N$. Indeed, we prove that if $M$ is included in a ball of radius small enough then the Hausdorff-distance between $M$ and a geodesic sphere $S$ of $N$ is small. Moreover $M$ is diffeomorphic and quasi-isometric to $S$. As other application, we give rigidity results for almost umbilic hypersurfaces.