{
  "id": "1608.05391",
  "version": "v1",
  "published": "2016-08-18T19:50:37.000Z",
  "updated": "2016-08-18T19:50:37.000Z",
  "title": "Liouville quantum gravity and the Brownian map III: the conformal structure is determined",
  "authors": [
    "Jason Miller",
    "Scott Sheffield"
  ],
  "comment": "32 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CV",
    "math.MP"
  ],
  "abstract": "Previous works in this series have shown that an instance of a $\\sqrt{8/3}$-Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure space (mm-space) agrees in law with the Brownian map (TBM). In this work, we show that given just the mm-space structure, one can a.s. recover the LQG sphere. This implies that there is a canonical way to parameterize an instance of TBM by the Euclidean sphere (up to M\\\"obius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the $\\sqrt{8/3}$-LQG sphere are equivalent. They ultimately encode the same structure (a topological sphere with a measure, a metric, and a conformal structure) and have the same law. From this point of view, the fact that the conformal structure a.s. determines the metric and vice-versa can be understood as a property of this unified law. The results of this work also imply that the analogous facts hold for Brownian and $\\sqrt{8/3}$-LQG surfaces with other topologies.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-18T19:50:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville quantum gravity",
      "brownian map",
      "lqg sphere",
      "resulting metric measure space",
      "canonical conformal structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}