CMC hypersurface with finite index in hyperbolic space H4

Abstract

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 $H^4$. 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 $\mathrm{Ric}>1$ does not contain any complete noncompact minimal stable hypersurface with finite topology.

The proof relies on the μ-bubble initially introduced by Gromov and further developed by Chodosh-Li-Stryker in the context of stable minimal hypersurfaces.