Tohoku Mathematical Journal
2008
March
SECOND SERIES VOL. 60, NO. 1

Tohoku Math. J.
60 (2008), 101-121

Title THE STRUCTURE OF WEAKLY STABLE CONSTANT MEAN CURVATURE HYPERSURFACES

Author Xu Cheng, Leung-fu Cheung and Detang Zhou

(Received June 6, 2006, revised July 2, 2007)
Abstract. We study the global behavior of weakly stable constant mean curvature hypersurfaces in a Riemannian manifold by using harmonic function theory. In particular, a complete oriented weakly stable minimal hypersurface in the Euclidean space must have only one end. Any complete noncompact weakly stable hypersurface with constant mean curvature $H$ in the 4 and 5 dimensional hyperbolic spaces has only one end under some restrictions on $H$.

2000 Mathematics Subject Classification. Primary 53C42; Secondary 53C21.
Key words and phrases. hypersurfaces, constant mean curvature, harmonic function.

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