  We study the global behavior of (weakly) stable constant mean curvature
hypersurfaces in general Riemannian manifolds. By using harmonic function
theory, we prove some one-end theorems which are new even for constant mean
curvature hypersurfaces in space forms.
  In particular, a complete oriented weakly stable minimal hypersurface in
$\mathbb{R}^{n+1}, n\geq 3,$ must have only one end. Any complete noncompact
weakly stable CMC $H$-hypersurface in the hyperbolic space $\mathbb{H}^{n+1},
n=3,4,$ with $H^2\geq{10/9}, {7/4},$ respectively, has only one end.
