{
  "abstract": "We prove that every closed weakly stable constant mean curvature surface immersed in the Riemannian Heisenberg space ${\\rm Nil}_3$ is an embedded rotational sphere. As a consequence, we solve the isoperimetric problem in ${\\rm Nil}_3$: the isoperimetric domains of ${\\rm Nil}_3$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08309",
  "author_total": 2,
  "authors": [
    {
      "name": "José M. Espinar"
    },
    {
      "name": "Pablo Mira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29hv9x14sfapv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08309",
  "fulltext_url": null,
  "id": "29hv9x14sfapv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08309",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08309",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29hv9x14sfapv.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08309",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08309"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The isoperimetric problem in the Riemannian Heisenberg $3$-space",
  "title_text": "The isoperimetric problem in the Riemannian Heisenberg $3$-space",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:49Z",
  "url": "https://research.draconitic.com/paper/29hv9x14sfapv",
  "venue": "arXiv",
  "versions": []
}