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

TL;DR

The paper develops ABP-based proofs of sharp logarithmic Sobolev inequalities in curved spaces. It proves two main results for manifolds with nonnegative Ricci curvature: sharp $L^1$ and $L^p$ logarithmic Sobolev inequalities on domains, with constants governed by the asymptotic volume ratio $\theta$, and sharp extensions to submanifolds in manifolds with nonnegative sectional curvature, incorporating a mean curvature term $|H|^2$. The proofs hinge on the ABP method of Brendle, employing transport maps $\Phi_r$ and Jacobian estimates, Neumann boundary problems, and scaling arguments to optimize constants, with sharp constants involving $\theta$ and the Euclidean volume constants $\omega_n$. These results connect ABP techniques with curvature-driven geometric analysis, complementing optimal transport approaches and enriching the toolbox for sharp functional inequalities on manifolds and submanifolds.

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.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['thm 1.1']}