{
  "id": "2012.11862",
  "version": "v1",
  "published": "2020-12-22T07:13:07.000Z",
  "updated": "2020-12-22T07:13:07.000Z",
  "title": "Explicit sharp constants in Sobolev inequalities on Riemannian manifolds with nonnegative Ricci curvature",
  "authors": [
    "Alexandru Kristály"
  ],
  "comment": "21 pages",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "Let $(M,g)$ be a noncompact, complete $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and Euclidean volume growth, i.e., $0<$AVR$(g)\\leq 1$, where AVR$(g)$ stands for the asymptotic volume ratio of $(M,g)$. The main purpose of the paper is to prove that the sharp Sobolev constants in various $L^p$-Sobolev inequalities on $(M,g)$ (both for $p<n$ and $p>n$) can be explicitly given by means of the geometric invariant AVR$(g)$. The equality cases are also characterized by stating that nonzero extremal functions occur if and only if AVR$(g)=1$, i.e., $(M,g)$ is isometric to the standard Euclidean space $(\\mathbb R^n,g_0).$ The proofs are based on an isoperimetric inequality established by S. Brendle (2020), combined with appropriate symmetrization techniques and optimal volume non-collapsing properties. We also provide examples of Riemannian manifolds with their explicit asymptotic volume ratios, which represent typical geometric settings we are working in.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-22T07:13:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonnegative ricci curvature",
      "riemannian manifold",
      "explicit sharp constants",
      "sobolev inequalities",
      "inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}