The structure of stable constant mean curvature hypersufaces

TL;DR

This paper investigates the global structure of stable constant mean curvature hypersurfaces in various Riemannian manifolds, establishing new one-end theorems using harmonic function techniques, with implications for minimal and CMC hypersurfaces.

Contribution

It introduces new one-end theorems for stable CMC hypersurfaces in general Riemannian manifolds, extending known results even in space forms, and applies harmonic function theory to prove these results.

Findings

Complete stable minimal hypersurfaces in Euclidean space have only one end.

Stable CMC hypersurfaces in hyperbolic space with certain mean curvature bounds have only one end.

Abstract

We study the global behavior of (weakly) stable constant mean curvature hypersurfaces in general Riemannian manifolds. By using harmonic function theory, we prove some one-end theorems which are new even for constant mean curvature hypersurfaces in space forms. In particular, a complete oriented weakly stable minimal hypersurface in $\mathbb{R}^{n+1}, n\geq 3,$ must have only one end. Any complete noncompact weakly stable CMC $H$-hypersurface in the hyperbolic space $\mathbb{H}^{n+1}, n=3,4,$ with $H^2\geq{10/9}, {7/4},$ respectively, has only one end.