{
  "id": "2211.06730",
  "version": "v1",
  "published": "2022-11-12T19:32:35.000Z",
  "updated": "2022-11-12T19:32:35.000Z",
  "title": "Stability of the positive mass theorem in dimension three",
  "authors": [
    "Conghan Dong"
  ],
  "comment": "22 pages",
  "categories": [
    "math.DG",
    "gr-qc"
  ],
  "abstract": "In this paper, we show that for a sequence of oriented complete pointed uniformly asymptotically Euclidean $3$-manifolds $(M_i, g_i, p_i)$ with non-negative integrable scalar curvature $R_{g_i}\\geq 0$, if their mass $m(g_i)\\to 0$, then by subtracting some subsets $Z_i\\subset M_i$ whose boundary area $|\\partial Z_i|\\leq C m(g_i)^{1/2}$, up to diffeomorphisms, $(M_i \\setminus Z_i, g_i, p_i)$ converge to the Euclidean space $\\mathbb{R}^3$ in the pointed metric topology. This confirms Huisken-Ilmanen's conjecture in terms of the flat metric topology. Moreover, if we assume the Ricci curvature bounded from below uniformly by $Ric_{g_i}\\geq -2 \\Lambda$, then $(M_i, g_i, p_i)$ converge to $\\mathbb{R}^3$ in the pointed Gromov-Hausdorff topology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-12T19:32:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive mass theorem",
      "confirms huisken-ilmanens conjecture",
      "flat metric topology",
      "gromov-hausdorff topology",
      "non-negative integrable scalar curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}