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 $H$.
Xu Cheng. Leung-fu Cheung. Detang Zhou. "The structure of weakly stable constant mean curvature hypersurfaces." Tohoku Math. J. (2) 60 (1) 101 - 121, 2008. https://doi.org/10.2748/tmj/1206734408