{
  "id": "2001.00157",
  "version": "v1",
  "published": "2020-01-01T08:03:56.000Z",
  "updated": "2020-01-01T08:03:56.000Z",
  "title": "An inequality for length and volume in the complex projective plane",
  "authors": [
    "Mikhail G. Katz"
  ],
  "comment": "8 pages",
  "categories": [
    "math.DG",
    "math.AP",
    "math.GT"
  ],
  "abstract": "In the 1950s, Carl Loewner proved an inequality relating the shortest closed geodesics on a 2-torus to its area. Many generalisations have been developed since, by Gromov and others. We show that the shortest closed geodesic on a minimal surface S for a generic metric on CP^2 is controlled by the total volume, even though the area of S is not. We exploit the Croke-Rotman inequality, Gromov's systolic inequalities, the Kronheimer-Mrowka proof of the Thom conjecture, and White's regularity results for area minimizers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-01T08:03:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C23",
      "53C22"
    ],
    "keywords": [
      "complex projective plane",
      "inequality",
      "shortest closed geodesic",
      "whites regularity results",
      "gromovs systolic inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}