{
  "id": "2307.05277",
  "version": "v1",
  "published": "2023-07-11T14:12:19.000Z",
  "updated": "2023-07-11T14:12:19.000Z",
  "title": "Positive mass theorems for spin initial data sets with arbitrary ends and dominant energy shields",
  "authors": [
    "Simone Cecchini",
    "Martin Lesourd",
    "Rudolf Zeidler"
  ],
  "comment": "18 pages",
  "categories": [
    "math.DG",
    "gr-qc"
  ],
  "abstract": "We prove a positive mass theorem for spin initial data sets $(M,g,k)$ that contain an asymptotically flat end and a shield of dominant energy (a subset of $M$ on which the dominant energy scalar $\\mu-|J|$ has a positive lower bound). In a similar vein, we show that for an asymptotically flat end $\\mathcal{E}$ that violates the positive mass theorem (i.e. $\\mathrm{E} < |\\mathrm{P}|$), there exists a constant $R>0$, depending only on $\\mathcal{E}$, such that any initial data set containing $\\mathcal{E}$ must violate the hypotheses of Witten's proof of the positive mass theorem in an $R$-neighborhood of $\\mathcal{E}$. This implies the positive mass theorem for spin initial data sets with arbitrary ends, and we also prove a rigidity statement. Our proofs are based on a modification of Witten's approach to the positive mass theorem involving an additional independent timelike direction in the spinor bundle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-11T14:12:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C21",
      "53C24",
      "53C27"
    ],
    "keywords": [
      "positive mass theorem",
      "spin initial data sets",
      "dominant energy shields",
      "arbitrary ends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}