Metric measure spaces supporting Gagliardo–Nirenberg inequalities: volume non-collapsing and rigidities

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Abstract

Let (M, d, m) be a metric measure space which satisfies the Lott–Sturm–Villani curvature-dimension condition CD(K, n) for some K≥ 0 and n≥ 2 , and a lower n-density assumption at some point of M. We prove that if (M, d, m) supports the Gagliardo–Nirenberg inequality or any of its limit cases (Lp-logarithmic Sobolev inequality or Faber–Krahn-type inequality), then a global non-collapsingn-dimensional volume growth holds, i.e., there exists a universal constant C0> 0 such that m(Bx(ρ)) ≥ C0ρn for all x∈ M and ρ≥ 0 , where Bx(ρ) = { y∈ M: d(x, y) < ρ}. Due to the quantitative character of the volume growth estimate, we establish several rigidity results on Riemannian manifolds with non-negative Ricci curvature supporting Gagliardo–Nirenberg inequalities by exploring a quantitative Perelman-type homotopy construction developed by Munn (J Geom Anal 20(3):723–750, 2010). Further rigidity results are also presented on some reversible Finsler manifolds.

Original languageEnglish
Article number112
JournalCalculus of Variations and Partial Differential Equations
Volume55
Issue number5
DOIs
Publication statusPublished - Oct 1 2016

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Keywords

  • 53C60
  • Primary 53C23
  • Secondary 35R06

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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