$L^p$-Sobolev inequalities on Euclidean submanifolds

Abstract: The paper is devoted to prove Allard-Michael-Simon-type $L^p$-Sobolev $(p>1)$ inequalities with explicit constants on Euclidean submanifolds of any codimension. Such inequalities contain, beside the Dirichlet $p$-energy, a term involving the mean curvature of the submanifold. Our results require separate discussions for the cases $p\geq 2$ and $1<p<2$, respectively; in particular, for $p\geq 2$, the coefficient in front of the Dirichlet $p$-energy is asymptotically sharp and codimension-free. Our argument is based on optimal mass transport theory on Euclidean submanifolds and it 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).