Weakly stable constant mean curvature hypersurfaces

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-1))(N) >= (n - 1)c. Denote by H and phi the mean curvature and the trace-free second fundamental form of M respectively. If vertical bar phi vertical bar(2) - (n - 2)root n(n - 1)vertical bar H vertical bar vertical bar phi vertical bar + 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.