$L^p$-Sobolev inequalities on minimal submanifolds

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).