{
  "id": "1610.07036",
  "version": "v1",
  "published": "2016-10-22T11:28:45.000Z",
  "updated": "2016-10-22T11:28:45.000Z",
  "title": "On the stability for Alexandrov's Soap Bubble theorem",
  "authors": [
    "Rolando Magnanini",
    "Giorgio Poggesi"
  ],
  "comment": "20 pages, dedicated to prof. Shigeru Sakaguchi on the occasion of his $60^{th}$ birthday",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "Alexandrov's Soap Bubble theorem dates back to $1958$ and states that a compact embedded hypersurface in $\\mathbb{R}^N$ with constant mean curvature must be a sphere. For its proof, A.D. Alexandrov invented his reflection priciple. In $1982$, R. Reilly gave an alternative proof, based on integral identities and inequalities, connected with the torsional rigidity of a bar. In this article we study the stability of the spherical symmetry: the question is how much a hypersurface is near to a sphere, when its mean curvature is near to a constant in some norm. We present a stability estimate that states that a compact hypersurface $\\Gamma\\subset\\mathbb{R}^N$ can be contained in a spherical annulus whose interior and exterior radii, say $\\rho_i$ and $\\rho_e$, satisfy the inequality $$ \\rho_e - \\rho_i \\le C \\Vert H - H_0 \\Vert^{\\tau_N}_{L^1 (\\Gamma)}, $$ where $\\tau_N=1/2$ if $N=2, 3$, and $\\tau_N=1/(N+2)$ if $N\\ge 4$. Here, $H$ is the mean curvature of $\\Gamma$, $H_0$ is some reference constant and $C$ is a constant that depends on some geometrical and spectral parameters associated with $\\Gamma$. This estimate improves previous results in the literature under various aspects. We also present similar estimates for some related overdetermined problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-22T11:28:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "35N25",
      "35B35",
      "35A23"
    ],
    "keywords": [
      "alexandrovs soap bubble theorem dates",
      "constant mean curvature",
      "compact embedded hypersurface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}