{
  "id": "1509.08061",
  "version": "v1",
  "published": "2015-09-27T07:40:14.000Z",
  "updated": "2015-09-27T07:40:14.000Z",
  "title": "Extendability of conformal structures on punctured surfaces",
  "authors": [
    "Jingyi Chen",
    "Yuxiang Li"
  ],
  "comment": "19 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "For a smooth immersion $f$ from the punctured disk $D\\backslash\\{0\\}$ into $\\mathbb{R}^n$ extendable continuously at the puncture, if its mean curvature is square integrable and the measure of $f(D)\\cap B_{r_k}=o(r_k)$ for a sequence $r_k\\to 0$, we show that the Riemannian surface $(D_r\\backslash\\{0\\},g)$ where $g$ is the induced metric is conformally equivalent to the unit Euclidean punctured disk, for any $r\\in(0,1)$. For a locally $W^{2,2}$ Lipschitz immersion $f$ from the punctured disk $D_2\\backslash\\{0\\}$ into $\\mathbb{R}^n$, if $\\|\\nabla f\\|_{L^\\infty}$ is finite and the second fundamental form of $f$ is in $L^2$, we show that there exists a homeomorphism $\\phi:D\\to D$ such that $f\\circ\\phi$ is a branched $W^{2,2}$-conformal immersion from the Euclidean unit disk $D$ into $\\mathbb{R}^n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-27T07:40:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "conformal structures",
      "punctured surfaces",
      "extendability",
      "second fundamental form",
      "euclidean unit disk"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150908061C"
    }
  }
}