2023/01/26 by Ronaldo F. de Lima, de Lima, Ronaldo F., Maria Fernanda Elbert +3
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2301.11038
openalex publication_date 2023/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We approach the one-parameter family of rotational constant mean curvature (CMC) spheres of \mathbb Hn×\mathbb R and \mathbb Sn×\mathbb R focusing on their stability and isoperimetry properties. We prove that all rotational CMC spheres of \mathbb Hn×\mathbb R are stable, and that the ones in \mathbb Sn×\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 Sn×\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 Hn×\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 Sn×\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 Hn×\mathbb R, and those of \mathbb Sn×\mathbb R with sufficiently large mean curvature, are nested.