$L^p$-Sobolev inequalities on minimal submanifolds

TL;DR

The paper develops $L^p$-Sobolev inequalities on complete minimal submanifolds of Euclidean space using Optimal Mass Transport on submanifolds to obtain integral Monge–Ampère relations with explicit constants. It yields a codimension-free constant $S(n,p)$ for $p\ge 2$ and an asymptotically sharp regime as $n\to\infty$, along with a codimension-dependent constant $\tilde{S}(n,m,p)$ for $1<p<2$, improving upon traditional bounds in certain ranges. Additionally, it provides a unified OMT-based proof of Brendle's isoperimetric inequalities that removes the compactness assumption and sharpens constants in codimensions $m=1$ and $m=2$, while clarifying the obstacles for higher codimension. Overall, the work bridges optimal transport, Monge–Ampère techniques, and geometric analysis to produce sharp, explicit Sobolev constants on minimal submanifolds and to refine isoperimetric inequalities in this setting.

Abstract

The paper is devoted to proving Allard-Michael-Simon-type $L^p$-Sobolev inequalities $(p>1)$ with explicit constants in the setting of Euclidean minimal submanifolds of arbitrary codimension. Our results require separate discussions for the cases $p\geq 2$ and $1<p<2$, respectively. In particular, for $p\geq 2$, we obtain an asymptotically sharp and codimension-free Sobolev constant. Our argument is based on optimal mass transport theory on Euclidean submanifolds and also provides an alternative, unified proof of the recent isoperimetric inequalities of Brendle (J. Amer. Math. Soc., 2021) and Brendle and Eichmair (Notices Amer. Math. Soc., 2024).

Theorems & Definitions (12)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Corollary 1.1
• Theorem 2.1
• Remark 2.1
• Theorem 3.1
• Lemma 3.1
• proof