{
  "id": "2311.16924",
  "version": "v1",
  "published": "2023-11-28T16:28:44.000Z",
  "updated": "2023-11-28T16:28:44.000Z",
  "title": "Fill-ins with scalar curvature lower bounds and applications to positive mass theorems",
  "authors": [
    "Stephen McCormick"
  ],
  "comment": "19 pages, no figures",
  "categories": [
    "math.DG",
    "gr-qc"
  ],
  "abstract": "Given a constant C and a smooth closed $(n-1)$-dimensional Riemannian manifold $(\\Sigma, g)$ equipped with a positive function $H$, a natural question to ask is whether this manifold can be realised as the boundary of a smooth $n$-dimensional Riemannian manifold with scalar curvature bounded below by C. That is, does there exist a fill-in of $(\\Sigma,g,H)$ with scalar curvature bounded below by C? We use variations of an argument due to Miao and the author [arXiv:1701.04805] to explicitly construct fill-ins with different scalar curvature lower bounds, where we permit the fill-in to contain another boundary component provided it is a minimal surface. Our main focus is to illustrate the applications of such fill-ins to geometric inequalities in the context of general relativity. By filling in a manifold beyond a boundary, one is able to obtain lower bounds on the mass in terms of the boundary geometry through positive mass theorems and Penrose inequalities. We consider fill-ins with both positive and negative scalar curvature lower bounds, which from the perspective of general relativity corresponds to the sign of the cosmological constant, as well as a fill-in suitable for the inclusion of electric charge.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-28T16:28:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C99"
    ],
    "keywords": [
      "positive mass theorems",
      "dimensional riemannian manifold",
      "applications",
      "negative scalar curvature lower bounds",
      "general relativity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}