Optimal transport approach to Michael–Simon–Sobolev inequalities in manifolds with intermediate Ricci curvature lower bounds

Abstract

We generalize McCann's theorem of optimal transport to a submanifold setting and use it to prove Michael-Simon-Sobolev inequalities for submanifolds in manifolds with lower bounds on intermediate Ricci curvatures. The results include a variant of the sharp Michael-Simon-Sobolev inequality in S. Brendle's [arXiv:2009.13717] when the intermediate Ricci curvatures are nonnegative.