{
  "id": "1502.04859",
  "version": "v1",
  "published": "2015-02-17T10:55:48.000Z",
  "updated": "2015-02-17T10:55:48.000Z",
  "title": "A holographic principle for the existence of parallel spinor fields and an inequality of Shi-Tam type",
  "authors": [
    "Oussama Hijazi",
    "Sebastián Montiel"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1502.04087",
  "journal": "Asian Journal of Mathematics, International Press, 2014, 18, pp.489 - 506",
  "doi": "10.4310/AJM.2014.v18.n3.a6",
  "categories": [
    "math.DG"
  ],
  "abstract": "Suppose that $\\Sigma=\\partial M$ is the $n$-dimensional boundary of a connected compact Riemannian spin manifold $( M,\\langle\\;,\\;\\rangle)$ with non-negative scalar curvature, and that the (inward) mean curvature $H$ of $\\Sigma$ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to the conformal metric $\\langle\\;,\\;\\rangle_H=H^2\\langle\\;,\\;\\rangle$ is at least $n/2$ and equality holds if and only if there exists a parallel spinor field on $ M$. As a consequence, if $\\Sigma$ admits an isometric and isospin immersion $\\phi$ with mean curvature $H_0$ as a hypersurface into another spin Riemannian manifold $ M_0$ admitting a parallel spinor field, then \\begin{equation} \\label{HoloIneq} \\int_\\Sigma H\\,d\\Sigma\\le \\int_\\Sigma \\frac{H^2_0}{H}\\, d\\Sigma \\end{equation} and equality holds if and only if both immersions have the same shape operator. In this case, $\\Sigma$ has to be also connected. In the special case where $M_0=\\R^{n+1}$, equality in (\\ref{HoloIneq}) implies that $M$ is an Euclidean domain and $\\phi$ is congruent to the embedding of $\\Sigma$ in $M$ as its boundary. We also prove that Inequality (\\ref{HoloIneq}) implies the Positive Mass Theorem (PMT). Note that, using the PMT and the additional assumption that $\\phi$ is a strictly convex embedding into the Euclidean space, Shi and Tam \\cite{ST1} proved the integral inequality \\begin{equation}\\label{shi-tam-Ineq} \\int_\\Sigma H\\,d\\Sigma\\le \\int_\\Sigma H_0\\, d\\Sigma, \\end{equation} which is stronger than (\\ref{HoloIneq}) .",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-17T10:55:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C27",
      "53C40",
      "53C80",
      "58G25"
    ],
    "keywords": [
      "parallel spinor field",
      "holographic principle",
      "shi-tam type",
      "inequality",
      "equality holds"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}