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

Abstract

Let $(M,\text{d},\text{m})$ be a metric measure space which satisfies the Lott-Sturm-Villani curvature-dimension condition $\text{CD}(K,n)$ for some $K\ge 0$ and $n\ge 2$, and a lower $n-$density assumption at some point of $M$. We prove that if $(M,\text{d},\text{m})$ 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 $\text{m}(B_x(\rho ))\ge C_0{\rho }^n$ for all $x\in M$ and $\rho \ge 0,$ where $B_x(\rho )=\{y\in M:\mathsf{d}(x,y)<\rho \}$. 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., 2010). Further rigidity results are also presented on some reversible Finsler manifolds.

arXiv comment21 pages [The whole class of Gagliardo-Nirenberg inequalities is considered on metric measure spaces w.r.t. the first version, where only one of the limiting cases was discussed.] To appear in Calculus of Variations and Partial Differential Equations