{
  "id": "2108.02135",
  "version": "v1",
  "published": "2021-08-04T16:04:48.000Z",
  "updated": "2021-08-04T16:04:48.000Z",
  "title": "Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds",
  "authors": [
    "Francesco Nobili",
    "Ivan Yuri Violo"
  ],
  "categories": [
    "math.DG",
    "math.AP",
    "math.MG"
  ],
  "abstract": "We prove that if $M$ is a closed $n$-dimensional Riemannian manifold, $n \\ge 3$, with ${\\rm Ric}\\ge n-1$ and for which the optimal constant in the critical Sobolev inequality equals the one of the $n$-dimensional sphere $\\mathbb{S}^n$, then $M$ is isometric to $\\mathbb{S}^n$. An almost-rigidity result is also established, saying that if equality is almost achieved, then $M$ is close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in the ${\\rm RCD}$-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds. An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compact ${\\rm CD}$ space, extending to the non-smooth setting a classical result by Aubin. Our arguments are based on a new concentration compactness result for mGH-converging sequences of ${\\rm RCD}$ spaces and on a Polya-Szego inequality of Euclidean-type in ${\\rm CD}$ spaces. As an application of the technical tools developed we prove both an existence result for the Yamabe equation and the continuity of the generalized Yamabe constant under measure Gromov-Hausdorff convergence, in the ${\\rm RCD}$-setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-04T16:04:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower ricci curvature bounds",
      "sobolev inequality",
      "synthetic lower ricci curvature",
      "compact spaces",
      "satisfying synthetic lower ricci"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}