{
  "id": "2012.14041",
  "version": "v1",
  "published": "2020-12-28T00:36:11.000Z",
  "updated": "2020-12-28T00:36:11.000Z",
  "title": "Removable singularity of positive mass theorem with continuous metrics",
  "authors": [
    "Wenshuai Jiang",
    "Weimin Sheng",
    "Huaiyu Zhang"
  ],
  "comment": "35 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper, we consider asymptotically flat Riemannnian manifolds $(M^n,g)$ with $C^0$ metric $g$ and $g$ is smooth away from a closed bounded subset $\\Sigma$ and the scalar curvature $R_g\\ge 0$ on $M\\setminus \\Sigma$. For given $n\\le p\\le \\infty$, if $g\\in C^0\\cap W^{1,p}$ and the Hausdorff measure $\\mathcal{H}^{n-\\frac{p}{p-1}}(\\Sigma)<\\infty$ when $n\\le p<\\infty$ or $\\mathcal{H}^{n-1}(\\Sigma)=0$ when $p=\\infty$, then we prove that the ADM mass of each end is nonnegative. Furthermore, if the ADM mass of some end is zero, then we prove that $(M^n,g)$ is isometric to the Euclidean space by showing the manifold has nonnegative Ricci curvature in RCD sense. This extends the result of [Lee-LeFloch2015] from spin to non-spin, also improves the result of [Shi-Tam2018] and [Lee2013]. Moreover, for $p=\\infty$, this confirms a conjecture of Lee [Lee2013].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-28T00:36:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20",
      "83C99"
    ],
    "keywords": [
      "positive mass theorem",
      "removable singularity",
      "continuous metrics",
      "adm mass",
      "asymptotically flat riemannnian manifolds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}