{
  "abstract": "We establish sharp Sobolev-type geometric inequalities on $\\mathbb{S}^n$ involving the total $σ_k$-curvatures $\\int_{\\mathbb{S}^n}σ_k(g)\\,dv_g$. These results extend the optimal inequalities of Guan--Wang~\\cite{GWDuke} from the cone $\\mathcal{C}_k$ to the strictly larger cone $\\mathcal{C}_{k-1}$, thereby enlarging the range of admissible conformal metrics. Our approach is variational and is implemented through a fully nonlinear conformal flow. Working in $\\mathcal{C}_{k-1}$ introduces substantial analytic difficulties; in particular, one must obtain $C^2$ a priori estimates while simultaneously verifying that the flow remains parabolic. We resolve these issues via a carefully designed test function and by applying the maximum principle to the maximal eigenvalue of the Hessian matrix. As applications, we solve two open problems in dimensions 3 and 4. Finally, we give examples to show that these inequalities cannot be extended to $\\mathcal{C}_{k-2}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11421",
  "author_total": 3,
  "authors": [
    {
      "name": "Yuxin Ge"
    },
    {
      "name": "Guofang Wang"
    },
    {
      "name": "Wei Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fn3rz8vw6wcwr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11421",
  "fulltext_url": null,
  "id": "fn3rz8vw6wcwr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11421",
  "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": "W7212236350",
  "pdf_url": "https://arxiv.org/pdf/2609.11421",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fn3rz8vw6wcwr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11421",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11421"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Optimal geometric inequalities and fully nonlinear conformal flows",
  "title_text": "Optimal geometric inequalities and fully nonlinear conformal flows",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/fn3rz8vw6wcwr",
  "venue": "arXiv",
  "versions": []
}