Logarithmic-Sobolev inequalities on non-compact Euclidean submanifolds: sharpness and rigidity
Zoltán M. Balogh, Alexandru Kristály
TL;DR
The paper proves sharp Michael–Simon-type $L^p$-logarithmic–Sobolev inequalities on complete, non-compact Euclidean submanifolds via optimal transport on submanifolds, yielding a sharp $p=2$ inequality with a mean-curvature term and a rigidity statement: equality implies $\Sigma$ is isometric to $\mathbb{R}^n$ and the extremizer is Gaussian. It extends to general $p\ge 2$ with codimension-dependent constants, and to minimal submanifolds in codimension-free form; an ABP/optimal-transport framework is developed to handle non-compact supports and equality cases. Applications include hypercontractivity estimates for Hopf–Lax semigroups on submanifolds, both of Euclidean-type (bounded mean curvature) and Gaussian-type (self-similar shrinkers with Gaussian measure), with sharp constants and complete equality characterizations. The results bridge optimal transport, geometric analysis on submanifolds, and nonlinear semigroup theory, providing tools for sharp entropy and hypercontractivity analyses in non-compact geometric settings.
Abstract
The paper is devoted to provide Michael-Simon-type $L^p$-logarithmic-Sobolev inequalities on complete, not necessarily compact $n$-dimensional submanifolds $Σ$ 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 $Σ$. 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 $Σ$ 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.