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

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 Brendle's (arXiv:2009.13717) when the intermediate Ricci curvatures are nonnegative.