{
  "id": "1908.09814",
  "version": "v1",
  "published": "2019-08-26T17:36:07.000Z",
  "updated": "2019-08-26T17:36:07.000Z",
  "title": "Total curvature and the isoperimetric inequality in Cartan-Hadamard manifolds",
  "authors": [
    "Mohammad Ghomi",
    "Joel Spruck"
  ],
  "comment": "81 pages, 7 figures",
  "categories": [
    "math.DG",
    "math-ph",
    "math.AP",
    "math.MG",
    "math.MP"
  ],
  "abstract": "We prove that the total positive Gauss-Kronecker curvature of any closed hypersurface embedded in a complete simply connected manifold of nonpositive curvature $M^n$, $n\\geq 2$, is bounded below by the volume of the unit sphere in Euclidean space $\\mathbf{R}^n$. This yields the optimal isoperimetric inequality for bounded regions of finite perimeter in $M$, via Kleiner's variational approach, and thus settles the Cartan-Hadamard conjecture. The proof employs a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for smooth approximation of the signed distance function. Immediate applications include sharp extensions of the Faber-Krahn and Sobolev inequalities to manifolds of nonpositive curvature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-26T17:36:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20",
      "58J05",
      "52A38",
      "49Q15"
    ],
    "keywords": [
      "total curvature",
      "cartan-hadamard manifolds",
      "total positive gauss-kronecker curvature",
      "nonpositive curvature",
      "kleiners variational approach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 81,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}