CMC hypersurface with finite index in hyperbolic space H⁴

In this paper, we prove that there are no complete noncompact constant mean curvature hypersurfaces with the mean curvature H>1 and finite index in hyperbolic space H⁴. This improves the previous best assumption, namely H>6463, to the optimal case. A more general nonexistence result can be proved in a 4-dimensional Riemannian manifold with certain curvature conditions. The proof relies on the harmonic function theory developed by Li-Tam-Wang and the -bubble initially introduced by Gromov and further developed by Chodosh-Li-Stryker in the context of stable minimal hypersurfaces.