{
  "abstract": "Let $x:M^n\\to\\mathbb{S}^{n+q}$ be a minimal immersion of a complete connected manifold with flat normal bundle, where $n\\ge3$ and $q\\ge2$. We prove that if the squared norm of the second fundamental form $S$ satisfies $S<2n$, then $M^n$ lies in $\\mathbb{S}^{n+1}$. The standard minimal products of three spheres show that the upper bound $2n$ is optimal. We also prove a codimension reduction theorem under the condition $S+λ_2\\le 2n+\\frac{2n}{3n-2}$, where $λ_2$ is the second largest eigenvalue of the Gram matrix of the shape operators, implying that $M^n$ lies in $\\mathbb{S}^{n+1}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07789",
  "author_total": 2,
  "authors": [
    {
      "name": "Juan Li"
    },
    {
      "name": "Zhiyuan Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7vjb36v23dneg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07789",
  "fulltext_url": null,
  "id": "7vjb36v23dneg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07789",
  "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",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7221044623",
  "pdf_url": "https://arxiv.org/pdf/2610.07789",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7vjb36v23dneg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07789",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07789"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Codimension Reduction of Minimal Submanifolds with Flat Normal Bundle in Spheres",
  "title_text": "Codimension Reduction of Minimal Submanifolds with Flat Normal Bundle in Spheres",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/7vjb36v23dneg",
  "venue": "arXiv",
  "versions": []
}