On Stability and Isoperimetry of Constant Mean Curvature Spheres of ${\mathbb{H}}^n\times \mathbb{R}$ and ${\mathbb{S}}^n\times \mathbb{R}.$

Abstract

We approach the one-parameter family of rotational constant mean curvature (CMC) spheres of ${\mathbb{H}}^n\times \mathbb{R}$ and ${\mathbb{S}}^n\times \mathbb{R}$ focusing on their stability and isoperimetry properties. We prove that all rotational CMC spheres of ${\mathbb{H}}^n\times \mathbb{R}$ are stable, and that the ones in ${\mathbb{S}}^n\times \mathbb{R}$ with sufficiently small (resp.~large) mean curvature are unstable (resp.~stable). We also show that there exists a one-parameter family of stable CMC rotational spheres in ${\mathbb{S}}^n\times \mathbb{R}$ which are not isoperimetric (i.e., they do not bound isoperimetric regions). We establish the uniqueness of the regions enclosed by the rotational CMC spheres of ${\mathbb{H}}^n\times \mathbb{R}$ as solutions to the isoperimetric problem, filling in a gap in the original proof given by Hsiang and Hsiang. We establish, as well, a sharp upper bound for the volume of the spherical regions of ${\mathbb{S}}^n\times \mathbb{R}$ which are unique solutions to the isoperimetric problem. In essence, all these results come from the fact that the rotational CMC spheres of ${\mathbb{H}}^n\times \mathbb{R}$, and those of ${\mathbb{S}}^n\times \mathbb{R}$ with sufficiently large mean curvature, are nested.

20 pages, 7 figures. To appear in Mathematische Zeitschrift