{
  "id": "1004.3188",
  "version": "v3",
  "published": "2010-04-19T13:30:15.000Z",
  "updated": "2010-12-23T12:26:27.000Z",
  "title": "Isoperimetric Inequalities for Minimal Submanifolds in Riemannian Manifolds: A Counterexample in Higher Codimension",
  "authors": [
    "Victor Bangert",
    "Nena Roettgen"
  ],
  "comment": "11 pages, We changed the title and added a section that exhibits the relation between our example and the question posed by Brian White concerning isoperimetric inequalities for minimal submanifolds",
  "journal": "Calculus of Variations and Partial Differential Equations: 45 (2012), no. 3, 455-466",
  "doi": "10.1007/s00526-011-0466-z",
  "categories": [
    "math.DG"
  ],
  "abstract": "For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B.White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The key step is the construction of a Riemannian metric on the closed four-dimensional ball B with the following properties: (1) B has strictly convex boundary. (2) There exists a complete nonconstant geodesic. (3) There does not exist a closed geodesic in B.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-12-23T12:26:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q20",
      "53C22",
      "49Q05",
      "53C42"
    ],
    "keywords": [
      "isoperimetric inequality",
      "higher codimension",
      "minimal submanifolds",
      "convex boundary",
      "counterexample"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.3188B"
    }
  }
}