Logarithmic Sobolev inequality in manifolds with nonnegative curvature via the ABP method

TL;DR

This paper proves optimal logarithmic Sobolev inequalities on manifolds with nonnegative curvature using the ABP method, providing sharp constants related to the manifold's volume growth.

Contribution

It introduces a novel application of the ABP method to establish sharp logarithmic Sobolev inequalities in nonnegative curvature settings.

Findings

Established optimal $L^p$ logarithmic Sobolev inequality for manifolds with nonnegative Ricci curvature.

Derived sharp $L^2$ logarithmic Sobolev inequality for submanifolds in nonnegatively curved ambient spaces.

Constants depend on the asymptotic volume ratio of the manifold.

Abstract

In this paper, we employ the ABP method developed by Brendle to establish the optimal $L^p$ logarithmic Sobolev inequality on manifolds with nonnegative Ricci curvature, as well as a sharp $L^2$ logarithmic Sobolev inequality for submanifolds in manifolds with nonnegative sectional curvature. The sharp constants in both inequalities depend on the asymptotic volume ratio of the ambient manifold.