{
  "abstract": "We introduce and study a Yamabe-type variational problem associated with the curl operator on middle-degree forms on a closed oriented Riemannian manifold $(M^n,g)$ with $n\\equiv3\\pmod4$. The corresponding curl--Yamabe constant is defined by minimizing a conformally invariant, gauge-invariant quotient built from $\\|{\\rm curl}\\,α\\|_{\\frac{2n}{n+1}}$ and $\\int_M\\langle{\\rm curl}\\,α,α\\rangle$, and its Euler--Lagrange equation is the critical curl--Yamabe equation ${\\rm curl}\\,α=\\frac{n+1}{2}|α|^{\\frac{2}{n-1}}α$. Using sharp curl--Sobolev inequalities on the sphere and an Aubin-type cut-off construction, we prove that $0<Y_{\\rm curl}(M,[g])\\le Y_{\\rm curl}(\\mathbb{S}^n)$ for every such manifold, and that the infimum is attained whenever $Y_{\\rm curl}(M,[g])<Y_{\\rm curl}(\\mathbb{S}^n)$. Moreover, for $n>3$ we establish the strict inequality for manifolds that are not locally conformally flat, via a test-form expansion detecting the Weyl tensor, and deduce existence of a minimizer solving the curl--Yamabe equation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13998",
  "author_total": 2,
  "authors": [
    {
      "name": "Guofang Wang"
    },
    {
      "name": "Mingwei Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8rat7098qsyze.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13998",
  "fulltext_url": null,
  "id": "8rat7098qsyze",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13998",
  "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|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213292428",
  "pdf_url": "https://arxiv.org/pdf/2609.13998",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8rat7098qsyze.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13998",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13998"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The curl-Yamabe problem",
  "title_text": "The curl-Yamabe problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/8rat7098qsyze",
  "venue": "arXiv",
  "versions": []
}