{
  "abstract": "We give an elementary proof of a tangent-cone identity which might be known to specialists in geometric measure theory. It concerns Lipschitz mappings $g$ from the closed unit ball $B\\subset\\mathbb{R}^2$ to $\\mathbb{R}^2$. Let $x$ be an interior point of differentiability of $g$ at which $Dg(x)$ has rank one. If the upper one-dimensional Hausdorff density of $g(B)$ at $g(x)$ is at most one, then the tangent cone to $g(B)$ at $g(x)$ coincides with the image of $Dg(x)$. This property is used in the proof of rigidity for the four-gradient problem. We provide a detailed, self-contained proof based on differentiability, connectedness and elementary estimates for one-dimensional Hausdorff measure, without using advanced methods from geometric measure theory.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05161",
  "author_total": 1,
  "authors": [
    {
      "name": "Igor Buchowiec"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1pdgdvrgcj2py.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05161",
  "fulltext_url": null,
  "id": "1pdgdvrgcj2py",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05161",
  "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.MG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05161",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1pdgdvrgcj2py.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.05161",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05161"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "An elementary proof of a tangent-cone identity for rank-one planar Lipschitz maps",
  "title_text": "An elementary proof of a tangent-cone identity for rank-one planar Lipschitz maps",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:34Z",
  "url": "https://research.draconitic.com/paper/1pdgdvrgcj2py",
  "venue": "arXiv",
  "versions": []
}