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

TL;DR

This paper investigates the stability and isoperimetric properties of rotational constant mean curvature spheres in hyperbolic and spherical product spaces, establishing stability results, uniqueness, and bounds related to the isoperimetric problem.

Contribution

It proves stability of all rotational CMC spheres in hyperbolic spaces, characterizes stability in spherical spaces based on mean curvature, and fills gaps in the isoperimetric problem analysis.

Findings

All rotational CMC spheres in $\, ext{H}^n imes ext{R}$ are stable.

In $ ext{S}^n imes ext{R}$, small mean curvature spheres are unstable, large mean curvature spheres are stable.

Existence of stable, non-isoperimetric CMC spheres in $ ext{S}^n imes ext{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.