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

摘要

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 (L-p-logarithmic Sobolev inequality or Faber-Krahn-type inequality), then a global non-collapsing n-dimensional volume growth holds, i.e., there exists a universal constant C-0 > 0 such that m(B-x (rho)) >= C-0 rho(n) for all x is an element of M and rho >= 0, where B-x(rho) = {y is an element of M : d(x, y) < rho}. 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.