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- W2509796004 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 $Kgeq 0$ and $ngeq 2$, 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 $xin {M}$ and $rhogeq 0,$ where $B_x(rho)={yin M:{sf d}(x,y)<rho}$. 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." @default.
- W2509796004 created "2016-09-16" @default.
- W2509796004 creator A5069656791 @default.
- W2509796004 date "2013-12-23" @default.
- W2509796004 modified "2023-09-27" @default.
- W2509796004 title "Metric measure spaces supporting Gagliardo-Nirenberg inequalities: volume non-collapsing and rigidities" @default.
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