{
  "id": "1901.01173",
  "version": "v1",
  "published": "2019-01-04T15:28:33.000Z",
  "updated": "2019-01-04T15:28:33.000Z",
  "title": "On the Multiplicity One Conjecture in Min-max theory",
  "authors": [
    "Xin Zhou"
  ],
  "comment": "40 pages; comments welcome",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves. We prove that in a bumpy metric each volume spectrum is realized by the min-max value of certain relative homotopy class of sweepouts of boundaries of Caccioppoli sets. The main result follows by approximating such min-max value using the min-max theory for hypersurfaces with prescribed mean curvature established by the author with Zhu.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-04T15:28:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "min-max theory",
      "multiplicity",
      "conjecture",
      "bumpy metric",
      "volume spectrum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}