CMC hypersurface with finite index in hyperbolic space $\mathbb{H}^4$

In this paper, we prove that there are no complete noncompact constant mean curvature hypersurfaces with the mean curvature $H > 1$, finite index and finite topology in hyperbolic space $\mathbb{H}^4$. A more general nonexistence result can be proved in a $4$-dimensional Riemannian manifold with certain curvature conditions. We also show that $4$-manifold with $\operatorname{Ric} > 1$ does not contain any complete noncompact minimal stable hypersurface with finite topology. The proof relies on the $\mu$-bubble initially introduced by Gromov and further developed by Chodosh-Li-Stryker in the context of stable minimal hypersurfaces.