{
  "abstract": "We introduce and prove a new conformally invariant Chern--Simons--Sobolev inequality, and classify all optimizers, which are the Hopf map $π:\\mathbb{S}^{2n+1}\\to\\mathbb{CP}^n$ composed with a symplectic diffeomorphism of $\\mathbb{CP}^n$ and an orientation-preserving conformal transformation of $\\mathbb{S}^{2n+1}$. The proof uses the sharp curl-Sobolev inequality and a rigidity argument for its equality case. Motivated by the main result, we also introduce the spherical Radu--Tchrakian--Yang functional, and prove the global minimality of the Hopf map for radii in a neighborhood of $((n!)^2/n)^{1/(4n-4)}$, and characterize all minimizers. We also prove local instability for sufficiently small and sufficiently large radii, with explicit lower bounds for the Morse index.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29769",
  "author_total": 1,
  "authors": [
    {
      "name": "Mingwei Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7tmfq196ajzq7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29769",
  "fulltext_url": null,
  "id": "7tmfq196ajzq7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29769",
  "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-ph"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7223228319",
  "pdf_url": "https://arxiv.org/pdf/2610.29769",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7tmfq196ajzq7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:19Z",
      "native_id": "2610.29769",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29769"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "The sharp Chern--Simons--Sobolev inequality and its variants",
  "title_text": "The sharp Chern--Simons--Sobolev inequality and its variants",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/7tmfq196ajzq7",
  "venue": "arXiv",
  "versions": []
}