On constant mean curvature hypersurfaces with finite index

Abstract.

We discuss the non-existence of complete noncompact constant mean curvature hypersurfaces with finite index in a 4- or 5-dimensional manifold. As a consequence, we obtain that a complete noncompact constant mean curvature hypersurface in $\mathbb{R}^{m}(m=4, 5 )$ with finite index must be minimal.