{
  "abstract": "We prove a Pólya--Szegö inequality for functions defined on an $n$-dimensional submanifold $Σ$ of a complete noncompact Riemannian manifold with nonnegative sectional curvature. The associated rearrangement is a Schwarz rearrangement on $\\mathbb R^n$, and the constant depends on the $L^n$-norm of the mean curvature of $Σ$ and an isoperimetric quantity obtained by Brendle. As applications, we derive Sobolev, Log-Sobolev, Hardy, and Gagliardo--Nirenberg inequalities on submanifolds of arbitrary codimension under a small total mean curvature assumption. In the critical Sobolev case, we obtain Moser--Trudinger inequalities on finite-volume submanifolds and exact growth inequalities on submanifolds with infinite volume. Under suitable assumptions, the Pólya--Szegö constant equals one; in this case, the critical constants in the inequalities coincide with the sharp Euclidean ones.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22676",
  "author_total": 4,
  "authors": [
    {
      "name": "Anh Xuan Do"
    },
    {
      "name": "Nguyen Lam"
    },
    {
      "name": "Guozhen Lu"
    },
    {
      "name": "Raoní Ponciano"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bddh2nasvvwfy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22676",
  "fulltext_url": null,
  "id": "bddh2nasvvwfy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22676",
  "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.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214031673",
  "pdf_url": "https://arxiv.org/pdf/2609.22676",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bddh2nasvvwfy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22676",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22676"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications",
  "title_text": "Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/bddh2nasvvwfy",
  "venue": "arXiv",
  "versions": []
}