Finite Index and Do Carmo Question for Constant Mean Curvature Hypersurfaces

We prove that any finite $δ$-index hypersurface $M$ in ${\mathbb R}^{n+1}$ with constant mean curvature must be minimal, provided either of the following conditions holds: - the volume growth of $M$ is sub-exponential; - the Ricci curvature of $M$ satisfies $\operatorname{Ric}_M\geq -\frac{3(1-δ)}{n-1}|A|^2g,$ where $A$ is the second fundamental form and $g$ is the metric on $M.$ In the second case, our result further implies that, in addition to being minimal, such an $M$ must be a hyperplane. We emphasize that no restriction on the dimension is imposed. Moreover, the statement in the second case is new even for finite index hypersurfaces ($δ=0$).