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

TL;DR

This paper extends optimal transport theory to submanifolds within manifolds having intermediate Ricci curvature bounds, establishing new Michael-Simon-Sobolev inequalities that generalize existing results.

Contribution

It generalizes McCann's optimal transport theorem to submanifolds and proves new Michael-Simon-Sobolev inequalities under intermediate Ricci curvature bounds.

Findings

Established a generalized optimal transport theorem for submanifolds.

Proved Michael-Simon-Sobolev inequalities in manifolds with intermediate Ricci curvature bounds.

Included a variant of the sharp inequality for nonnegative intermediate Ricci curvatures.

Abstract

We generalize McCann's theorem of optimal transport to a submanifold setting and 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 work arXiv:2009.13717 when the intermediate Ricci curvatures are nonnegative.