{
  "id": "1204.1695",
  "version": "v3",
  "published": "2012-04-08T01:09:27.000Z",
  "updated": "2017-08-24T10:00:34.000Z",
  "title": "A Sharp Comparison Theorem for Compact Manifolds with Mean Convex Boundary",
  "authors": [
    "Martin Li"
  ],
  "comment": "6 pages; published in Journal of Geometric Analysis",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary $\\partial M$. Assume that the mean curvature $H$ of the boundary $\\partial M$ satisfies $H \\geq (n-1) k >0$ for some positive constant $k$. In this paper, we prove that the distance function $d$ to the boundary $\\partial M$ is bounded from above by $\\frac{1}{k}$ and the upper bound is achieved if and only if $M$ is isometric to an $n$-dimensional Euclidean ball of radius $\\frac{1}{k}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-08-28T11:19:56.000Z",
      "comment": "revised version",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2017-08-24T10:00:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mean convex boundary",
      "sharp comparison theorem",
      "compact manifolds",
      "dimensional riemannian manifold",
      "dimensional euclidean ball"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.1695L"
    }
  }
}