On stable hypersurfaces with constant mean curvature in Euclidean spaces

TL;DR

This paper establishes curvature estimates and a Bernstein-type theorem for stable hypersurfaces with constant mean curvature in Euclidean spaces, linking local volume growth to global properties under certain curvature conditions.

Contribution

It introduces new curvature bounds and a Bernstein-type theorem for stable hypersurfaces with constant mean curvature, extending previous results to arbitrary dimensions.

Findings

Curvature estimates relate local and global volume growth.

A Bernstein-type theorem for complete stable hypersurfaces with finite $L^p$-norm curvature.

Results apply to hypersurfaces in Euclidean spaces of any dimension.

Abstract

In this paper, we derive curvature estimates for strongly stable hypersurfaces with constant mean curvature immersed in $\mathbb{R}^{n+1}$, which show that the locally controlled volume growth yields a globally controlled volume growth if $\partial M=\emptyset$. Moreover, we deduce a Bernstein-type theorem for complete stable hypersurfaces with constant mean curvature of arbitrary dimension, given a finite $L^p$-norm curvature condition.