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We generalize McCann's theorem of optimal transport to a submanifold settingand prove Michael-Simon-Sobolev inequalities for submanifolds in manifolds withlower bounds on intermediate Ricci curvatures. The results include a variant ofthe sharp Michael-Simon-Sobolev inequality in S. Brendle's workarXiv:2009.13717 when the intermediate Ricci curvatures are nonnegative.

Optimal Transport Approach to Michael-Simon-Sobolev Inequalities in Manifolds with Intermediate Ricci Curvature Lower Bounds