Stable Constant Mean Curvature Hypersurfaces in ${\mathbb{H}}^n$

Abstract

For every $n\ge 4$ and every $H$ in a neighborhood of $n-1$ (depending on $n$), we prove the existence of a properly embedded strongly stable CMC hypersurface in ${\mathbb{H}}^n$ with mean curvature $H$ and infinitely many ends, invariant under the action of a Schottky group. In particular, stable Bernstein-type rigidity fails in this range.

18 pages. Comments are welcome!