{
  "id": "1611.04056",
  "version": "v1",
  "published": "2016-11-13T00:02:26.000Z",
  "updated": "2016-11-13T00:02:26.000Z",
  "title": "Scalar curvature and singular metrics",
  "authors": [
    "Yuguang Shi",
    "Luen-Fai Tam"
  ],
  "comment": "47pages, All comments are welcomed",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $M^n$, $n\\ge3$, be a compact differentiable manifold with nonpositive Yamabe invariant $\\sigma(M)$. Suppose $g_0$ is a continuous metric, smooth outside a compact set $\\Sigma$, and is in $W^{1,p}_{loc}$ for some $p>n$. Suppose the scalar curvature of $g_0$ is at least $\\sigma(M)$ outside $\\Sigma$. We prove that $g_0$ is Einstein outside $\\Sigma$ if the codimension of $\\Sigma$ is at least $2$. If in addition, $g_0$ is Lipschitz then $g_0$ is smooth and Einstein after a change the smooth structure. If $\\Sigma$ is a compact embedded hypersurface, and $g_0$ is smooth up to $\\Sigma$ from two sides of $\\Sigma$, and if the difference of the mean curvatures along $\\Sigma$ at two sides of $\\Sigma$ has a fixed appropriate sign. Then $g_0$ is also Einstein outside $\\Sigma$. For manifolds with dimension between $3$ and $7$, we obtain a positive mass theorem on an asymptotically flat manifold without spin assumption for metrics with a compact singular set of codimension at least $2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-13T00:02:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20",
      "83C99"
    ],
    "keywords": [
      "scalar curvature",
      "singular metrics",
      "einstein outside",
      "compact singular set",
      "asymptotically flat manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}