{
  "abstract": "Let $K \\subset \\mathbb{R}^{m \\times n}$ be a compact $C^1$-submanifold with boundary, $p \\in (1,\\infty)$ and $Q := (0,1)^n$. We prove that $K$ satisfies a rigidity estimate of the form $\\|Du - (Du)_{Q}\\|_{L^p} \\leq C \\|\\mathrm{dist}_K(Du)\\|_{L^p}$, $u \\in W^{1,p}(Q,\\mathbb{R}^m)$, if and only if $K$ satisfies sequential rigidity and for each $A \\in K$, the tangent space to $K$ at $A$ satisfies exact rigidity. We further prove that this rigidity estimate is stable under small graphical perturbations of $K$. The key technical ingredient is proving that outside some small bad set where the maximal function of $\\mathrm{dist}_K^p(Du)$ is large, the size of the superlevel sets of $\\lvert Du - (Du)_Q\\rvert$ decays exponentially. This is achieved by an adaptation of the John-Nirenberg inequality for $BMO$-functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13218",
  "author_total": 1,
  "authors": [
    {
      "name": "Malte Borken"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9s1sw7fc28qv9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13218",
  "fulltext_url": null,
  "id": "9s1sw7fc28qv9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13218",
  "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"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7213230909",
  "pdf_url": "https://arxiv.org/pdf/2609.13218",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9s1sw7fc28qv9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13218",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13218"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A characterization of submanifolds of $\\mathbb{R}^{m \\times n}$ satisfying optimal rigidity estimates",
  "title_text": "A characterization of submanifolds of ℝ^m × n satisfying optimal rigidity estimates",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/9s1sw7fc28qv9",
  "venue": "arXiv",
  "versions": []
}