$L^p$-Sobolev inequalities on minimal submanifolds

TL;DR

This paper establishes explicit $L^p$-Sobolev inequalities on Euclidean minimal submanifolds, providing sharp constants and unifying proofs for related isoperimetric inequalities using optimal mass transport theory.

Contribution

It introduces new $L^p$-Sobolev inequalities with explicit constants for minimal submanifolds, covering different $p$ ranges and offering a unified proof approach.

Findings

Derived sharp, codimension-free Sobolev constants for $p extgreater 2$

Unified proof of recent isoperimetric inequalities

Explicit constants in $L^p$-Sobolev inequalities

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).