Sharp Sobolev inequalities on noncompact Riemannian manifolds with ${\sf Ric}\geq 0$ via Optimal Transport theory

TL;DR

This paper extends sharp Sobolev inequalities to noncompact Riemannian manifolds with non-negative Ricci curvature using Optimal Transport theory, providing new inequalities and volume estimates.

Contribution

It introduces a novel application of Optimal Transport to establish sharp Sobolev inequalities on manifolds with non-negative Ricci curvature, including elementary proofs of volume non-collapsing.

Findings

Sharp $L^p$-Sobolev inequalities on manifolds with ${ m Ric}\, ext{ extgreater}=0$

Elementary proof of volume non-collapsing estimates

Optimal constants linked to volume growth and bubble asymptotics

Abstract

In their seminal work, Cordero-Erausquin, Nazaret and Villani [Adv. Math., 2004] proved sharp Sobolev inequalities in Euclidean spaces via Optimal Mass Transportation, raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. In this paper we affirmatively answer their question for Riemannian manifolds with non-negative Ricci curvature; namely, by using Optimal Mass Transportation with quadratic distance cost, sharp $L^p$-Sobolev and $L^p$-logarithmic Sobolev inequalities (both for $p>1$ and $p=1$) are established, where the optimal constants contain the asymptotic volume growth arising from precise asymptotic properties of the Talentian and Gaussian bubbles. As a byproduct, we give an alternative, elementary proof to the main result of do Carmo and Xia [Compos. Math., 2004] (and subsequent results) concerning the quantitative volume non-collapsing estimates on Riemannian manifolds with non-negative Ricci curvature that support certain Sobolev inequalities.