Strongly stable CMC-one hypersurfaces in every hyperbolic space of dimension at least four

Abstract

For every integer $d\ge 4$, we prove strong stability for a subfamily of classical rotational hypersurfaces in ${\mathbb{H}}^d$ with normalized mean curvature one. The examples are complete, two-sided, properly embedded, and nowhere umbilic, with topology $\mathbb{R}\times {\mathbb{S}}^{d-2}$. An explicit positive supersolution yields a quantitative stability inequality for all compactly supported test functions. Consequently, endpoint horospherical rigidity fails in every ambient dimension at least four.