{
  "id": "1211.6210",
  "version": "v1",
  "published": "2012-11-27T04:25:45.000Z",
  "updated": "2012-11-27T04:25:45.000Z",
  "title": "A Compactness Theorem for Riemannian Manifolds with Boundary and Applications",
  "authors": [
    "Kenneth S. Knox"
  ],
  "comment": "17 pages; comments welcome",
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper we prove weak L^{1,p} (and thus C^{\\alpha}) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We obtain two stability theorems from the compactness result. The first theorem applies to 3-manifolds (contained in the aforementioned class) that have Ricci curvature close to 0 and whose boundaries are Gromov-Hausdorff close to a fixed metric on S^2 with positive curvature. Such manifolds are C^{\\alpha} close to the region enclosed by a Weyl embedding of the fixed metric into \\R^3. The second theorem shows that a 3-manifold with Ricci curvature close to 0 (resp. -2, 2) and mean curvature close to 2 (resp. 2\\sqrt 2, 0) is C^{\\alpha} close to a metric ball in the space form of constant curvature 0 (resp -1, 1), provided that the boundary is a topological sphere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-27T04:25:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20",
      "53C21"
    ],
    "keywords": [
      "riemannian manifolds",
      "compactness theorem",
      "ricci curvature close",
      "applications",
      "fixed metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.6210K"
    }
  }
}