A Sudakov Decomposition in Riemannian Manifolds with Positive Curvature

TL;DR

This paper extends the theory of optimal transport to Riemannian manifolds with positive curvature, establishing existence and structure of optimal transport plans using a generalized variational approach.

Contribution

It generalizes a variational method from Euclidean spaces to positively curved Riemannian manifolds, providing explicit measure disintegration along optimal rays.

Findings

Existence of a transport density on curved manifolds.

Explicit disintegration of measures along optimal rays.

Extension of Euclidean optimal transport methods to Riemannian geometry.

Abstract

In this paper, we study Monge's problem on Riemannian manifolds $(M, g)$ with positive sectional curvature. Assuming that the source and target measures are absolutely continuous with respect to the Riemannian volume measure, we generalize a variational method from the Euclidean setting to establish the existence of a transport density and an explicit disintegration of measures along optimal rays. These results extend the approach of Bianchini-Caravenna to the Riemannian context.