  In this paper, we prove that there are no complete noncompact constant mean
curvature hypersurfaces with the mean curvature $H>1$ and finite index
satisfying universal subexponential end growth 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 stable minimal hypersurface with universal subexponential
end growth. The proof relies on the harmonic function theory developed by
Li-Tam-Wang and the $\mu$-bubble initially introduced by Gromov and further
developed by Chodosh-Li-Stryker in the context of stable minimal hypersurfaces.
