{
  "abstract": "We study the Gromov-Hausdorff stability of round spheres under perturbations of Perelman's nu-entropy. We show that in dimension two almost maximal entropy, together with an area normalisation, implies Gromov-Hausdorff closeness to the round two-sphere. In contrast, in dimensions at least three, we construct smooth metrics on the sphere whose entropies converge to that of the round sphere, but which remain a definite Gromov-Hausdorff distance away from it. These metrics can be chosen conformal and uniformly bi-Lipschitz equivalent to the round metric, with the same volume as the round sphere. Finally, in dimension three, we contrast this Gromov-Hausdorff instability with a dynamical stability result: metrics with nu-entropy above that of the round shrinking cylinder converge exponentially fast to round metrics under the volume-normalised Ricci flow.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06678",
  "author_total": 2,
  "authors": [
    {
      "name": "Reto Buzano"
    },
    {
      "name": "Shengwen Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dzvt0hp651br7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06678",
  "fulltext_url": null,
  "id": "dzvt0hp651br7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06678",
  "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.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06678",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dzvt0hp651br7.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06678",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06678"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Perelman's entropy and Gromov-Hausdorff versus dynamical stability of round spheres",
  "title_text": "Perelman's entropy and Gromov-Hausdorff versus dynamical stability of round spheres",
  "type": "preprint",
  "updated_at": "2026-11-07T18:20:08Z",
  "url": "https://research.draconitic.com/paper/dzvt0hp651br7",
  "venue": "arXiv",
  "versions": []
}