{
  "abstract": "Let $V$ be a nonempty relatively compact open subset of a smooth Riemannian manifold $(M^n,g)$ without boundary. We prove that $C_{\\mathrm N}(V,g)\\ge C_{\\mathrm N}(\\mathbb R^n)$ and $\\int_V \\mathrm{Sc}_g dμ_g\\ge0$ imply $\\mathrm{Rm}_g=0$ on $V$, where $C_{\\mathrm N}$ denotes the optimal constant in the Nash inequality and $\\mathrm{Sc}_g$ denotes the scalar curvature. This improves an earlier rigidity result of Druet, Hebey, and Vaugon [Int. Math. Res. Not. (1999), no. 14, 735--779].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32560",
  "author_total": 2,
  "authors": [
    {
      "name": "Liang Cheng"
    },
    {
      "name": "Zhenyu Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4r5hxqdrs43qf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32560",
  "fulltext_url": null,
  "id": "4r5hxqdrs43qf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32560",
  "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",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7223656947",
  "pdf_url": "https://arxiv.org/pdf/2610.32560",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4r5hxqdrs43qf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32560",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32560"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Local rigidity for the sharp Nash inequality",
  "title_text": "Local rigidity for the sharp Nash inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/4r5hxqdrs43qf",
  "venue": "arXiv",
  "versions": []
}