On Stability and Isoperimetry of Constant Mean Curvature Spheres of $\mathbb H^n\times\mathbb R$ and $\mathbb S^n\times\mathbb R.$
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909391921348608 |
|---|---|
| author | de Lima, Ronaldo F. Elbert, Maria F. Nelli, Barbara |
| author_facet | de Lima, Ronaldo F. Elbert, Maria F. Nelli, Barbara |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_11038 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Stability and Isoperimetry of Constant Mean Curvature Spheres of $\mathbb H^n\times\mathbb R$ and $\mathbb S^n\times\mathbb R.$ de Lima, Ronaldo F. Elbert, Maria F. Nelli, Barbara Differential Geometry 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. |
| title | On Stability and Isoperimetry of Constant Mean Curvature Spheres of $\mathbb H^n\times\mathbb R$ and $\mathbb S^n\times\mathbb R.$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2301.11038 |