{
  "id": "2003.00531",
  "version": "v1",
  "published": "2020-03-01T17:39:17.000Z",
  "updated": "2020-03-01T17:39:17.000Z",
  "title": "Nonexistence of radial optimal functions for the Sobolev inequality on Cartan-Hadamard manifolds",
  "authors": [
    "Tatsuki Kawakami",
    "Matteo Muratori"
  ],
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "It is well known that the Euclidean Sobolev inequality holds on any Cartan-Hadamard manifold of dimension $ n\\ge 3 $, i.e. any complete, simply connected Riemannian manifold with nonpositive sectional curvature. As a byproduct of the Cartan-Hadamard conjecture, a longstanding problem in the mathematical literature settled only very recently in a breakthrough paper by Ghomi and Spruck, we can now assert that the optimal constant is also Euclidean, namely it coincides with the one achieved in the Euclidean space $ \\mathbb{R}^n $ by the Aubin-Talenti functions. One may ask whether there exist at all optimal functions on a generic Cartan-Hadamard manifold $ \\mathbb{M}^n $. What we prove here, with ad hoc arguments that do not take advantage of the validity of the Cartan-Hadamard conjecture, is that this is false at least for functions that are radially symmetric with respect to the geodesic distance from a fixed pole. More precisely, we show that if the optimum in the Sobolev inequality is achieved by some radial function, then $\\mathbb{M}^n $ must be isometric to $ \\mathbb{R}^n $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-01T17:39:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J70"
    ],
    "keywords": [
      "radial optimal functions",
      "euclidean sobolev inequality holds",
      "cartan-hadamard conjecture",
      "nonexistence",
      "generic cartan-hadamard manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}