{
  "id": "1110.5498",
  "version": "v7",
  "published": "2011-10-25T13:35:37.000Z",
  "updated": "2015-06-23T05:15:46.000Z",
  "title": "Lipschitz-Volume rigidity in Alexandrov geometry",
  "authors": [
    "Nan Li"
  ],
  "comment": "This is the published version on AIM",
  "journal": "Adv. Math., 275, (30 April 2015), 114--146",
  "categories": [
    "math.DG",
    "math.MG"
  ],
  "abstract": "We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map $f\\colon X=\\amalg X_\\ell\\to Y$ between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of $X$. We furthermore characterize the metric structure on $Y$ with respect to $X$ when $f$ is also onto. This implies the converse of Petrunin's Gluing Theorem: if a gluing of two Alexandrov spaces via a bijection between their boundaries produces an Alexandrov space, then the bijection must be an isometry.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2013-10-27T05:37:09.000Z",
      "title": "Volume and gluing rigidity in Alexandrov geometry",
      "abstract": "We develop a new technique to approximate the length of curves in Alexandrov space subject to a dimension comtrol. Using this technique, we show that a 1-Lipschitz map $f: \\amalg X_\\alpha\\to Y$ between Alexandrov spaces preserves volume if and only if it preserves the length of paths. We furthermore characterize the metric on $Y$ when $f$ is also onto. This implies the converse of Petrunin's Gluing Theorem: if the gluing of two Alexandrov spaces is an Alexandrov space, then the gluing is along the boundary and via an isometry.",
      "comment": "25 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v7",
      "updated": "2015-06-23T05:15:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "alexandrov geometry",
      "gluing rigidity",
      "alexandrov spaces preserves volume",
      "alexandrov space subject",
      "petrunins gluing theorem"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.5498L"
    }
  }
}