{
  "id": "1205.5474",
  "version": "v2",
  "published": "2012-05-24T14:57:17.000Z",
  "updated": "2014-12-03T02:21:14.000Z",
  "title": "Contracting the boundary of a Riemannian 2-disc",
  "authors": [
    "Yevgeny Liokumovich",
    "Alexander Nabutovsky",
    "Regina Rotman"
  ],
  "comment": "34 pages, 10 figures",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $D$ be a Riemannian 2-disc of area $A$, diameter $d$ and length of the boundary $L$. We prove that it is possible to contract the boundary of $D$ through curves of length $\\leq L + 200d\\max\\{1,\\ln {\\sqrt{A}\\over d} \\}$. This answers a twenty-year old question of S. Frankel and M. Katz, a version of which was asked earlier by M.Gromov. We also prove that a Riemannian $2$-sphere $M$ of diameter $d$ and area $A$ can be swept out by loops based at any prescribed point $p\\in M$ of length $\\leq 200 d\\max\\{1,\\ln{\\sqrt{A}\\over d} \\}$. This estimate is optimal up to a constant factor. In addition, we provide much better (and nearly optimal) estimates for these problems in the case, when $A<<d^2$. Finally, we describe the applications of our estimates for study of lengths of various geodesics between a fixed pair of points on \"thin\" Riemannian $2$-spheres.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-24T14:57:17.000Z",
      "abstract": "Let D be a Riemannian 2-disc of area A, diameter d and length of the boundary L. We prove that it is possible to contract the boundary of D through curves of length \\leq L+200d max{1, ln(sqrt(A)/d)}. This answers a twenty-year old question of S.Frankel and M.Katz, a version of which was asked earlier by M.Gromov. We also prove that a Riemannian 2-sphere M of diameter d and area A can be swept out by loops based at any prescribed point p in M of length \\leq 200 d max{1, ln(sqrt(A)/d)}. This estimate is optimal up to a constant factor. We apply this estimate to obtain curvature-free upper bounds for lengths of two shortest simple periodic geodesics and of various geodesics connecting a fixed pair of points on Riemannian 2-spheres.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-03T02:21:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C23"
    ],
    "keywords": [
      "riemannian",
      "twenty-year old question",
      "constant factor",
      "asked earlier",
      "prescribed point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.5474L"
    }
  }
}