{
  "id": "1209.4601",
  "version": "v1",
  "published": "2012-09-20T18:37:52.000Z",
  "updated": "2012-09-20T18:37:52.000Z",
  "title": "Interior curvature estimates and the asymptotic plateau problem in hyperbolic space",
  "authors": [
    "Bo Guan",
    "Joel Spruck",
    "Ling Xiao"
  ],
  "comment": "20 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(\\kappa)=\\sigma\\in(0, 1)$ with a prescribed asymptotic boundary $\\Gamma$ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if $\\Gamma$ is (Euclidean) starshaped, the solution is unique and also (Euclidean) starshaped while if $\\Gamma$ is mean convex the solution is unique. We also show via a strong duality theorem that analogous results hold in De Sitter space. A novel feature of our approach is a \"global interior curvature estimate\".",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-09-20T18:37:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C21",
      "35J65",
      "58J32"
    ],
    "keywords": [
      "asymptotic plateau problem",
      "hyperbolic space",
      "bounded hyperbolic principal curvatures",
      "strictly locally convex hypersurface",
      "global interior curvature estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.4601G"
    }
  }
}