{
  "id": "math/0606675",
  "version": "v1",
  "published": "2006-06-27T10:10:45.000Z",
  "updated": "2006-06-27T10:10:45.000Z",
  "title": "Nonlinear evolution by mean curvature and isoperimetric inequalities",
  "authors": [
    "Felix Schulze"
  ],
  "comment": "42 pages",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "Evolving smooth, compact hypersurfaces in R^{n+1} with normal speed equal to a positive power k of the mean curvature improves a certain 'isoperimetric difference' for k >= n-1. As singularities may develop before the volume goes to zero, we develop a weak level-set formulation for such flows and show that the above monotonicity is still valid. This proves the isoperimetric inequality for n <= 7. Extending this to complete, simply connected 3-dimensional manifolds with nonpositive sectional curvature, we give a new proof for the Euclidean isoperimetric inequality on such manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-06-27T10:10:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "49Q20",
      "53C44"
    ],
    "keywords": [
      "mean curvature",
      "nonlinear evolution",
      "weak level-set formulation",
      "euclidean isoperimetric inequality",
      "normal speed equal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......6675S"
    }
  }
}