On Stable Constant Mean Curvature Spheres of $\mathbb H^n\times\mathbb R$ and $\mathbb S^n\times\mathbb R$ and their Uniqueness as Isoperimetric Hypersurfaces

Abstract

We consider the rotational constant mean curvature spheres of H n × R obtained by Hsiang and Hsiang and show that they are nested. We apply this property to prove the uniqueness of these spheres as isoperimetric hypersurfaces of H n × R, filling in a gap in the original proof given by Hsiang and Hsiang. We also give a direct proof of their stability, extending the result by Souam, who considered the case n = 2. Analogous results are obtained for the class of the rotational constant large mean curvature spheres of S n × R constructed by Pedrosa.