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Tohoku Mathematical Journal
SECOND SERIES VOL. 60, NO. 1
|Tohoku Math. J.|
60 (2008), 101-121
THE STRUCTURE OF WEAKLY STABLE CONSTANT MEAN CURVATURE HYPERSURFACES
Xu Cheng, Leung-fu Cheung and Detang Zhou
(Received June 6, 2006, revised July 2, 2007)
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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