{
  "id": "2112.05105",
  "version": "v2",
  "published": "2021-12-09T18:44:04.000Z",
  "updated": "2023-06-05T11:40:12.000Z",
  "title": "Sobolev Inequalities and Convergence For Riemannian Metrics and Distance Functions",
  "authors": [
    "Brian Allen",
    "Edward Bryden"
  ],
  "comment": "26 pages, 1 figure. v2: Final published version in AGAG",
  "journal": "Annals of Global Analysis and Geometry, June 2023",
  "doi": "10.1007/s10455-023-09906-z",
  "categories": [
    "math.DG",
    "math.MG"
  ],
  "abstract": "If one thinks of a Riemannian metric, $g_1$, analogously as the gradient of the corresponding distance function, $d_1$, with respect to a background Riemannian metric, $g_0$, then a natural question arises as to whether a corresponding theory of Sobolev inequalities exists between the Riemannian metric and its distance function. In this paper we study the sub-critical case $p < \\frac{m}{2}$ and show a Sobolev inequality exists where an $L^{\\frac{p}{2}}$ bound on a Riemannian metric implies an $L^q$ bound on its corresponding distance function. We then use this result to state a convergence theorem and show how this theorem can be useful to prove geometric stability results by proving a version of Gromov's conjecture for tori with almost non-negative scalar curvature in the conformal case. Examples are given to show that the hypotheses of the main theorems are necessary.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-06-05T11:40:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sobolev inequality",
      "corresponding distance function",
      "convergence",
      "background riemannian metric",
      "natural question arises"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}