On stable hypersurfaces with constant mean curvature in Euclidean spaces

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 $\partial M=\varnothing$. 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.

12 pages