Stable complete noncompact hypersurfaces with constant mean curvature

This article concerns the structure of complete noncompact stable hypersurfaces Mⁿ with constant mean curvature H > 0 in a complete noncompact oriented Riemannian manifold Nⁿ⁺¹. In particular, we show that a complete noncompact stable constant mean curvature hypersurface Mⁿ, n = 5, 6, in the Euclidean space must have only one end. Any such hypersurface in the hyperbolic space with, respectively, has only one end.