Sharp Sobolev inequalities on noncompact Riemannian manifolds with Ric 0 via optimal transport theory

Abstract In their seminal work, Cordero-Erausquin, Nazaret and Villani (Adv Math 182 (2): 307-332, 2004) proved sharp Sobolev inequalities in Euclidean spaces via Optimal Transport, raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. By using L¹ L 1 -optimal transport approach, the compact case has been successfully treated by Cavalletti and Mondino (Geom Topol 21: 603-645, 2017), even on metric measure spaces verifying the synthetic lower Ricci curvature bound. In the present paper we affirmatively answer the above question for noncompact Riemannian manifolds with non-negative Ricci curvature; namely, by using Optimal Transport theory with quadratic distance cost, sharp Lᵖ L p -Sobolev and Lᵖ L p -logarithmic Sobolev inequalities (both for p>1 p > 1 and p=1 p = 1) are established, where the sharp constants contain the asymptotic volume ratio arising from precise asymptotic properties of the Talentian and Gaussian bubbles, respectively. As a byproduct, we give an alternative, elementary proof to the main result of do Carmo and Xia (Math 140: 818-826, 2004) and subsequent results, concerning the quantitative volume non-collapsing estimates on Riemannian manifolds with non-negative Ricci curvature that support Sobolev inequalities.