{
  "abstract": "Let $f:M^n\\to\\Sn^{n+q}(1)$ be a minimal immersion of a closed connected manifold with $n\\ge3$ and $q\\ge2$, and let $S$ be the squared length of its second fundamental form. We prove an integral inequality that gives $\\max_M S\\ge n$ if the immersion is not totally geodesic. If $n\\ne4$ or $q=2$, equality $\\max_M S=n$ holds if and only if the immersion is a Clifford minimal hypersurface; if $n=4$ and $q\\ge3$, equality holds if and only if the immersion is a Clifford minimal hypersurface or the standard minimal embedding of $\\mathbb{CP}^2$ in $\\Sn^7(1)$. Both classifications are understood up to an ambient isometry, a totally geodesic inclusion, and a finite covering. We construct the integral inequalities using polynomial weights and divergence identities. The required algebraic estimates are proved by sums of squared tensor norms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08126",
  "author_total": 3,
  "authors": [
    {
      "name": "Jianquan Ge"
    },
    {
      "name": "Kai Jia"
    },
    {
      "name": "Fagui Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5brgzve2v1s5k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08126",
  "fulltext_url": null,
  "id": "5brgzve2v1s5k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08126",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08126",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5brgzve2v1s5k.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08126",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08126"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The first gap for minimal submanifolds in spheres",
  "title_text": "The first gap for minimal submanifolds in spheres",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/5brgzve2v1s5k",
  "venue": "arXiv",
  "versions": []
}