{
  "abstract": "We establish a quantitative stability estimate for codimension-one immersions into a round sphere. Let $(M,g)$ be an oriented Riemannian manifold and suppose that a prescribed shape operator is realised by a smooth isometric immersion $θ:(M,g)\\to\\mathbb S^{n+1}$. We prove that, on every relatively compact strongly Lipschitz domain and for every $1<p<\\infty$, any Sobolev immersion $ϕ$ is close to $θ$ modulo an ambient rotation, with the $W^{1,p}$-distances between both the immersions and their Gauss maps controlled by the $L^p$ stretching-plus-bending energies of $ϕ$. No {\\it a priori} bounds on the fundamental forms of $ϕ$ are required. The proof proceeds by extending the immersions along normal geodesics and reducing the problem to an equidimensional geometric rigidity estimate on the sphere. A finite localisation and patching argument handles the possible non-injectivity of the normal extension of the reference immersion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37665",
  "author_total": 2,
  "authors": [
    {
      "name": "Siran Li"
    },
    {
      "name": "Isaac Newell"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/61qxffnv1neq3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37665",
  "fulltext_url": null,
  "id": "61qxffnv1neq3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37665",
  "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.CA"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "physics.class-ph"
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      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
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  "openalex": "W7214896642",
  "pdf_url": "https://arxiv.org/pdf/2609.37665",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/61qxffnv1neq3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:11Z",
      "native_id": "2609.37665",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37665"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.class-ph",
      "subfield": "general"
    }
  ],
  "title": "A geometric rigidity estimate for codimension-1 immersions into spheres",
  "title_text": "A geometric rigidity estimate for codimension-1 immersions into spheres",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/61qxffnv1neq3",
  "venue": "arXiv",
  "versions": []
}