# Mass Transportation on sub-Riemannian structures of rank two in dimension four

## TL;DR

This paper investigates the Monge optimal transport problem on rank-two sub-Riemannian manifolds of dimension four, focusing on existence and uniqueness of optimal maps despite the presence of many singular geodesics.

## Contribution

It extends previous results by establishing existence and uniqueness of optimal transport maps in complex sub-Riemannian structures with numerous singular geodesics.

## Key findings

- Existence of optimal transport maps in the specified structures
- Uniqueness of these maps under certain conditions
- Handling of singular minimizing geodesics in the analysis

## Abstract

This paper is concerned with the study of the Monge optimal transport problem in sub-Riemannian manifolds where the cost is given by the square of the sub-Riemannian distance. Our aim is to extend previous results on existence and uniqueness of optimal transport maps to cases of sub-Riemannian structures which admit many singular minimizing geodesics. We treat here the case of sub-Riemannian structures of rank two in dimension four.