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

Abstract

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 $$K\ge 0$$ and $$n\ge 2$$ , 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 $$\textsf {m}( B_x(\rho ))\ge C_0 \rho ^n$$ 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 20(3):723–750, 2010). Further rigidity results are also presented on some reversible Finsler manifolds.