{
  "abstract": "Let $E\\subset\\mathbf R^n$ be a set of dimension $2$ with an isolated singular point $0$, such that $E-\\{0\\}$ is an orientable $\\mathcal C^2$ submanifold of $\\mathbf R^n$ of dimension $2$, and hence a Riemann surface with the conformal structure induced by the ambient metric. We study the extension of its conformal structure across $0$, under hypotheses that are metric rather than analytic. Denoting by $S(0,r)$ the sphere of radius $r$ centred at $0$, we call the singularity $0$ $\\mathit{M\\text{-}regular}$ if (i) the pair $(E-\\{0\\},\\{0\\})$ satisfies a Whitney condition (the radial direction becomes asymptotically tangent to $E-\\{0\\}$ at $0$) and (ii) the length of the spherical slice $S(0,r)\\cap E$ is bounded above by a linear function of $r$ for small $r$ ($\\mathit{metric\\ decay}$). The main result (1989) is that an $M$-regular singularity is $\\mathit{conformally\\ point\\text{-}like}$, that is, a punctured neighbourhood of $0$ in $E$ is conformal to the punctured unit disc $Δ^*$, so that the conformal structure extends across $0$. The proof uses the modulus of rings (extremal length), which generalizes the classical planar technique to $2$-dimensional submanifolds of $\\mathbb{R}^n$. Two classes of surfaces are shown to be conformally point-like: all surfaces of revolution generated by a single $C^1$ curve, and orientable subanalytic sets of dimension $2$ with isolated singularities, for which $M$-regularity is proven directly from Hironaka's structure theory. In this case the metric decay condition follows from a direct estimate of the volume of the slices $S(0,r)\\cap E$ (Lemma 7.5), based on Hironaka's Whitney theorem and Stasica's decomposition into $L$-analytic pieces. A further original result is a $C^\\infty$ counterexample showing that a strict Whitney condition alone does $\\mathit{not}$ imply the linear length bound: the two hypotheses defining $M$-regularity are independent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.19452",
  "author_total": 1,
  "authors": [
    {
      "name": "Flavio Mosca"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/em9vw868t3tn5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.19452",
  "fulltext_url": null,
  "id": "em9vw868t3tn5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.19452",
  "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.CV"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7203857105",
  "pdf_url": "https://arxiv.org/pdf/2608.19452",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/em9vw868t3tn5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2608.19452",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.19452"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A problem for isolated singularities of surfaces",
  "title_text": "A problem for isolated singularities of surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/em9vw868t3tn5",
  "venue": "arXiv",
  "versions": []
}