{
  "abstract": "The Euclidean paradigm that spheres optimize mean curvature variational problems breaks down in the sub-Riemannian Heisenberg group: neither the Pansu sphere nor the Korányi sphere is optimal for the variational problems associated with the Minkowski and Heintze-Karcher inequalities. Motivated by this phenomenon, we develop a variational theory for geometric problems driven by the horizontal mean curvature, focusing on the total mean curvature functional and the related Minkowski inequality, introducing suitable notions of non-characteristic stationarity and stability. We fully classify critical surfaces, identifying a new one-parameter family of rotationally invariant surfaces, which we call Pansu-Minkowski spheres. We show that a distinguished member, the optimal Pansu-Minkowski sphere, emerges as the unique critical point of the Minkowski quotient, and uniquely minimizes it among rotationally invariant competitors. We prove non-characteristic stability and local minimality of Pansu-Minkowski spheres under rotationally invariant perturbations, while showing their instability under unrestricted perturbations. Finally, we show that the Minkowski quotient does not admit any positive lower bound: in the first sub-Riemannian Heisenberg group, no Minkowski inequality holds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.29912",
  "author_total": 3,
  "authors": [
    {
      "name": "Mattia Fogagnolo"
    },
    {
      "name": "Andrea Pinamonti"
    },
    {
      "name": "Simone Verzellesi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/24ttm0q1hmzfx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.29912",
  "fulltext_url": null,
  "id": "24ttm0q1hmzfx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.29912",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7162808904",
  "pdf_url": "https://arxiv.org/pdf/2605.29912",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/24ttm0q1hmzfx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2605.29912",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.29912"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Unexpected phenomena for mean curvature functionals in the Heisenberg group",
  "title_text": "Unexpected phenomena for mean curvature functionals in the Heisenberg group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/24ttm0q1hmzfx",
  "venue": "arXiv",
  "versions": []
}