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The structure of stable constant mean curvature hypersufaces

Xu Cheng·Leung-fu Cheung·Detang Zhou
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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.

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Xu Cheng, Leung-fu Cheung, Detang Zhou. 2006-02-01. The structure of stable constant mean curvature hypersufaces. https://arxiv.org/abs/math/0602007

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