PreprintarXiv licencearXiv
The isoperimetric problem in the Riemannian Heisenberg $3$-space
Abstract
We prove that every closed weakly stable constant mean curvature surface immersed in the Riemannian Heisenberg space Nil3 is an embedded rotational sphere. As a consequence, we solve the isoperimetric problem in Nil3: the isoperimetric domains of Nil3 are exactly the regions enclosed by rotational constant mean curvature spheres, and for each prescribed volume there is a unique such domain, up to ambient isometry. For small volumes, the isoperimetric result was proved by Tomter in 1993.
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BibTeX
@misc{espinar2026isoperimetric,
title = {{The isoperimetric problem in the Riemannian Heisenberg $3$-space}},
author = {José M. Espinar and Pablo Mira},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08309},
eprint = {2611.08309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08309},
}
RIS
TY - UNPB
TI - The isoperimetric problem in the Riemannian Heisenberg $3$-space
AU - José M. Espinar
AU - Pablo Mira
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08309
UR - https://arxiv.org/abs/2611.08309
AB - We prove that every closed weakly stable constant mean curvature surface immersed in the Riemannian Heisenberg space Nil_3 is an embedded rotational sphere. As a consequence, we solve the isoperimetric problem in Nil_3: the isoperimetric domains of Nil_3 are exactly the regions enclosed by rotational constant mean curvature spheres, and for each prescribed volume there is a unique such domain, up to ambient isometry. For small volumes, the isoperimetric result was proved by Tomter in 1993.
ER -