Complete hypersurfaces with constant mean curvature and finite index in hyperbolic spaces

In this article, we prove that any complete finite index hypersurface in the hyperbolic space ℍ4(−1) ℍ5(−1)) with constant mean curvature H satisfying H2 (H2 > respectively) must be compact. Specially, we verify that any complete and stable hypersurface in the hyperbolic space ℍ4(−1) (resp. ℍ5(−1)) with constant mean curvature H satisfying H2 > (resp. H2) must be compact. It shows that there is no manifold satisfying the conditions of some theorems in [7, 9].