The structure of stable constant mean curvature hypersufaces

Cheng, Xu; Cheung, Leung-fu; Zhou, Detang

The structure of stable constant mean curvature hypersufaces

Authors

Cheng, Xu

Cheung, Leung-fu

Zhou, Detang

Description

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.
27 pages