Logarithmic-Sobolev inequalities on non-compact Euclidean submanifolds: sharpness and rigidity

TL;DR

This paper establishes sharp logarithmic-Sobolev inequalities on complete Euclidean submanifolds, characterizes equality cases, and applies optimal mass transport theory to derive hypercontractivity estimates, advancing understanding of geometric analysis on submanifolds.

Contribution

It provides new sharp $L^p$-logarithmic-Sobolev inequalities involving mean curvature, characterizes extremizers, and develops optimal transport tools for non-compact submanifolds.

Findings

Sharp $p=2$ inequality involving mean curvature

Equality only for Euclidean space with Gaussian extremizer

Hypercontractivity estimates for submanifolds with bounded mean curvature

Abstract

The paper is devoted to provide Michael-Simon-type $L^p$-logarithmic-Sobolev inequalities on complete, not necessarily compact $n$-dimensional submanifolds $\Sigma$ of the Euclidean space $\mathbb R^{n+m}$. Our first result, stated for $p=2$, is sharp, it is valid on general submanifolds, and it involves the mean curvature of $\Sigma$. It implies in particular the main result of S. Brendle [Comm. Pure Appl. Math.}, 2022]. In addition, it turns out that equality can only occur if and only if $\Sigma$ is isometric to the Euclidean space $\mathbb R^{n}$ and the extremizer is a Gaussian. The second result is a general $L^p$-logarithmic-Sobolev inequality for $p\geq 2$ on Euclidean submanifolds with constants that are codimension-free in case of minimal submanifolds. In order to prove the above results - especially, to deal with the equality cases - we elaborate the theory of optimal mass transport on submanifolds between measures that are not necessarily compactly supported. Applications are provided to sharp hypercontractivity estimates of Hopf-Lax semigroups on submanifolds. The first hypercontractivity estimate is for general submanifolds with bounded mean curvature vector, the second one is for self-similar shrinkers endowed with the natural Gaussian measure. The equality cases are characterized here as well.