{
  "id": "1306.4233",
  "version": "v1",
  "published": "2013-06-18T15:09:07.000Z",
  "updated": "2013-06-18T15:09:07.000Z",
  "title": "The GBC mass for asymptotically hyperbolic manifolds",
  "authors": [
    "Yuxin Ge",
    "Guofang Wang",
    "Jie Wu"
  ],
  "comment": "41 pages",
  "categories": [
    "math.DG",
    "gr-qc"
  ],
  "abstract": "The paper consists of two parts. In the first part, by using the Gauss-Bonnet curvature, which is a natural generalization of the scalar curvature, we introduce a higher order mass, the Gauss-Bonnet-Chern mass $m^{\\H}_k$, for asymptotically hyperbolic manifolds and show that it is a geometric invariant. Moreover, we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs and establish a relationship between the corresponding Penrose type inequality for this mass and weighted Alexandrov-Fenchel inequalities in the hyperbolic space $\\H^n$. In the second part, we establish these weighted Alexandrov-Fenchel inequalities in $\\H^n$ for any horospherical convex hypersurface $\\Sigma$. As an application, we obtain an optimal Penrose type inequality for the new mass defined in the first part for asymptotically hyperbolic graphs with a horizon type boundary $\\Sigma$, provided that a dominant energy condition $\\tilde L_k\\ge0$ holds. Both inequalities are optimal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-18T15:09:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotically hyperbolic manifolds",
      "gbc mass",
      "weighted alexandrov-fenchel inequalities",
      "asymptotically hyperbolic graphs",
      "first part"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1238938,
      "adsabs": "2013arXiv1306.4233G"
    }
  }
}