Tohoku Mathematical Journal
2010
September
SECOND SERIES VOL. 62, NO. 3

Tohoku Math. J.
62 (2010), 383-392

Title ON STABLE CONSTANT MEAN CURVATURE HYPERSURFACES

Author Hai-Ping Fu and Zhen-Qi Li

(Received March 2, 2009, revised March 24, 2010)
Abstract. We study complete non-compact stable constant mean curvature hypersurfaces in a Riemannian manifold of bounded geometry, and prove that there are no nontrivial $L^2$ harmonic 1-forms on such hypersurfaces. We also show that any smooth map with finite energy from such a hypersurface to a compact manifold with non-positive sectional curvature is homotopic to constant on each compact set. In particular, we obtain some one-end theorems of complete non-compact weakly stable constant mean curvature hypersurfaces in the space forms.

2000 Mathematics Subject Classification. Primary 53C40; Secondary 58E20.
Key words and phrases. Stable hypersurface, $L^2$ harmonic forms, constant mean curvature, harmonic map, ends.

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