{
  "abstract": "We establish local Hamilton-type gradient estimates for positive $C^{2,1}$ solutions of $u_t=Δu^m$ on complete Riemannian manifolds whose Ricci curvature is bounded from below. For every fixed $m>1$ and every fixed $0<m<1$, there exist $β=β(m,n)>0$ and $C=C(m,n)>0$ such that a solution $0<u\\leqslant A$ in $B_{2R}(x_0)\\times(t_0-T,t_0]$ satisfies the following local gradient estimate \\begin{equation*} \\sup_{B_R(x_0)\\times(t_0-T/2,t_0]}|\\nabla u^β| \\leqslant C A^β\\left(\\frac1R+\\sqrt{k}+\\frac{A^{(1-m)/2}}{\\sqrt T}\\right), \\end{equation*} where $\\mathrm{Ric}_M\\geqslant-k$. The proof uses an intrinsic quantitative alternative to locate, around each prescribed positive point, a cylinder on which the solution has a controlled upper-to-lower ratio. A local gradient estimate on this cylinder is combined with a stopping argument. As a consequence, every uniformly bounded positive ancient solution on a connected complete manifold with nonnegative Ricci curvature is constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31373",
  "author_total": 2,
  "authors": [
    {
      "name": "Jun Sun"
    },
    {
      "name": "Jiaming Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4kt6h102zfxj9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31373",
  "fulltext_url": null,
  "id": "4kt6h102zfxj9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31373",
  "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.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214641341",
  "pdf_url": "https://arxiv.org/pdf/2609.31373",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4kt6h102zfxj9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31373",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31373"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Hamilton-Type Gradient Estimates for Porous Medium and Fast Diffusion Equations on Riemannian Manifolds for all exponents",
  "title_text": "Hamilton-Type Gradient Estimates for Porous Medium and Fast Diffusion Equations on Riemannian Manifolds for all exponents",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/4kt6h102zfxj9",
  "venue": "arXiv",
  "versions": []
}