{
  "id": "1712.02540",
  "version": "v1",
  "published": "2017-12-07T09:05:16.000Z",
  "updated": "2017-12-07T09:05:16.000Z",
  "title": "Escobar-Yamabe compactifications for Poincare-Einstein manifolds and rigidity theorems",
  "authors": [
    "Xuezhang Chen",
    "Mijia Lai",
    "Fang Wang"
  ],
  "comment": "12",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $(X^{n},g_+) $ $(n\\geq 3)$ be a Poincar\\'{e}-Einstein manifold which is $C^{3,\\alpha}$ conformally compact with conformal infinity $(\\partial X, [\\hat{g}])$. On the conformal compactification $(\\overline{X}, \\bar g=\\rho^2g_+)$ via some boundary defining function $\\rho$, there are two types of Yamabe constants: $Y(\\overline{X},\\partial X,[\\bar g])$ and $Q(\\overline{X},\\partial X,[\\bar g])$. (See definitions (\\ref{def.type1}) and (\\ref{def.type2})). In \\cite{GH}, Gursky and Han gave an inequality between $Y(\\overline{X},\\partial X,[\\bar g])$ and $Y(\\partial X,[\\hat{g}])$. In this paper, we first show that the equality holds in Gursky-Han's theorem if and only if $(X^{n},g_+)$ is isometric to the standard hyperbolic space $(\\mathbb{H}^{n}, g_{\\mathbb{H}})$. Secondly, we derive an inequality between $Q(\\overline{X},\\partial X,[\\bar g])$ and $Y(\\partial X, [\\hat g])$, and show that the equality holds if and only if $(X^{n},g_+)$ is isometric to $(\\mathbb{H}^{n}, g_{\\mathbb{H}})$. Based on this, we give a simple proof of the rigidity theorem for Poincar\\'{e}-Einstein manifolds with conformal infinity being conformally equivalent to the standard sphere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-07T09:05:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C25"
    ],
    "keywords": [
      "rigidity theorem",
      "escobar-yamabe compactifications",
      "poincare-einstein manifolds",
      "conformal infinity",
      "equality holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}