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

Let $({M},\textsf{d},\textsf{m})$ be a metric measure space which satisfies the Lott-Sturm-Villani curvature-dimension condition $\textsf{CD}(K,n)$ for some $K\geq 0$ and $n\geq 2$, and a lower $n-$density assumption at some point of $M$. We prove that if $({M},\textsf{d},\textsf{m})$ 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 $\textsf{m}( B_x(\rho))\geq C_0 \rho^n$ for all $x\in {M}$ and $\rho\geq 0,$ where $B_x(\rho)=\{y\in M:{\sf d}(x,y)<\rho\}$. 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.