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

Abstract

The paper is devoted to providing 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. 75 (2022), 449–454]. In addition, it turns out that equality can occur if and only if $\Sigma$ is isometric to the Euclidean space ${\mathbb{R}}^n$ and the extremizer is a Gaussian. Our second result is a general $L^p$-logarithmic-Sobolev inequality for $p\ge 2$ on Euclidean submanifolds with constants that are codimension-free in the 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. Two 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, while the second is for self-similar shrinkers endowed with the natural Gaussian measure. The equality cases are characterized here as well.