Weakly stable constant mean curvature hypersurfaces

Let Mn be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an (n+1)-dimensional Riemannian manifold Nn+1 whose (n-1)th Ricci curvature satisfying RicN(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)(H2 + c) ≥ 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c + H2 > 0, then M must have only one end.