{
  "abstract": "For a rotationally symmetric Ricci flow $g(t)=ds^2+f(s,t)^2g_{\\mathbb S^{n-1}}$, $n\\ge3$, we study the scale-invariant curvature anisotropy $A=1-f_s^2+ff_{ss}$ introduced by Angenent--Knopf. At the critical power $p_*=(n-1)/(n-2)$, with $Φ=sgn(A)|A|^{p_*/2}$, we prove the distributional identity \\[ (\\partial_t-Δ)|A|^{p_*} =-\\frac4{n-1} \\left|Φ_s+(n-1)\\frac{f_s}{f}Φ\\right|^2. \\] Integration against a conjugate heat kernel gives an anisotropic entropy formula. As an application, we obtain an entropy threshold for Type-I singularities on $\\mathbb S^n$: if the entropy associated with a singular-time kernel is at most $1$ at an earlier regular time, every tangent flow at the corresponding singular point is a round shrinking sphere. This criterion follows from the $0/1$ quantization of the singular-time anisotropic density.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37154",
  "author_total": 2,
  "authors": [
    {
      "name": "Hongxin Guo"
    },
    {
      "name": "Guozhi Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5xht3sh80x941.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37154",
  "fulltext_url": null,
  "id": "5xht3sh80x941",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37154",
  "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": "W7224121013",
  "pdf_url": "https://arxiv.org/pdf/2610.37154",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5xht3sh80x941.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37154",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37154"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow",
  "title_text": "An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/5xht3sh80x941",
  "venue": "arXiv",
  "versions": []
}