{
  "abstract": "We establish curvature-based bounds for the $p$-capacity of convex domains and related capacity--area inequalities. Let $K$ denote the Gauss--Kronecker curvature of $\\partialΩ$. In Euclidean space, if $ 0<K_-\\leq K\\leq K_+, $ then, for every $1<p\\leq \\frac{n+1}{2}$, $$ \\operatorname{Cap}_p\\left(B(K_+^{\\frac{1}{1-n}})\\right) \\leq \\operatorname{Cap}_p(Ω) \\leq \\operatorname{Cap}_p\\left(B(K_-^{\\frac{1}{1-n}})\\right). $$ The lower comparison remains valid for the full range $1<p<n$. Moreover, the exponent range in the upper comparison is optimal in general: when $n\\geq 4$ and $\\frac{n+1}{2}<p<n$, the upper comparison may fail. Thus $p=\\frac{n+1}{2}$ is the critical exponent for the curvature-radius upper comparison. Under the pinching condition $ \\frac{K_-}{K_+}=ρ\\in (0, 1], $ we further obtain an explicit scale-invariant capacity--area lower bound for every $1<p<n$. In the special case $n=3$ and $p=2$, $$ \\frac{\\operatorname{Cap}_2(Ω)} {\\sqrt{|\\partialΩ|}} \\geq \\frac{4\\sqrtπ\\,ρ^{1/4}} {1+\\sqrtρ}, $$ which implies the Pólya--Szegö inequality whenever $$ ρ\\geq ρ_*:= \\left( \\frac{π^2-4-π\\sqrt{π^2-8}}{4} \\right)^2 \\simeq 0.15484. $$ On a Cartan--Hadamard manifold with $\\operatorname{Sec}\\leq -k^2<0$, we obtain a $p$-capacity lower bound retaining all principal curvatures of the boundary, together with corresponding Gauss--Kronecker curvature estimates and a rigidity characterization of the equality case by a warped-product exterior geometry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07564",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiaoshang Jin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cc8m4c0sa5acf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07564",
  "fulltext_url": null,
  "id": "cc8m4c0sa5acf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07564",
  "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": "W7220999690",
  "pdf_url": "https://arxiv.org/pdf/2610.07564",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cc8m4c0sa5acf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07564",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07564"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Curvature-Based Bounds for $p$-Capacity and Capacity--Area Inequalities",
  "title_text": "Curvature-Based Bounds for p-Capacity and Capacity--Area Inequalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/cc8m4c0sa5acf",
  "venue": "arXiv",
  "versions": []
}