The five gradients inequality on differentiable manifolds

TL;DR

This paper extends the five gradients inequality in optimal transport to general cost functions on locally compact Lie groups and compact Riemannian manifolds, broadening its applicability in geometric analysis.

Contribution

It derives the five gradients inequality for optimal transport on two classes of differentiable manifolds, generalizing previous results to new geometric settings.

Findings

Established the inequality for locally compact Lie groups

Extended the inequality to compact Riemannian manifolds

Provided new tools for geometric optimal transport analysis

Abstract

The goal of this paper is to derive the so-called five gradients inequality for optimal transport theory for general cost functions on two class of differentiable manifolds: locally compact Lie groups and compact Riemannian manifolds.