Abstract
Let M n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an (n+1)-dimensional Riemannian manifold N n+1 whose (n− 1)th Ricci curvature satisfying Ric N(n−1) ≥(n−1)c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If |ϕ|2 − (n − 2)√n(n − 1)|H||ϕ| + n(2n − 1)(H 2 + c) ≥ 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c + H 2 > 0, then M must have only one end.
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Supported by the National Natural Science Foundation of China (10771187; 10671087); Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China; Jiangxi Province Natural Science Foundation (2008GZS0060)
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Fu, Hp., Xu, Hw. Weakly stable constant mean curvature hypersurfaces. Appl. Math. J. Chin. Univ. 24, 119–126 (2009). https://doi.org/10.1007/s11766-009-1815-y
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DOI: https://doi.org/10.1007/s11766-009-1815-y