Multi-marginal optimal transport on sub-Riemannian manifolds

Abstract

In this paper, we study multi-marginal optimal transport on ideal sub-Riemannian manifolds. We establish a sub-Riemannian barycentric balance condition and prove the existence of a solution to the Monge problem under an additional assumption. We also investigate the absolute continuity of Wasserstein barycenters. Our results extend and improve upon those of Pass, Pinamonti, and Vedovato (Methods Appl. Anal. 28 (2021), 61-76) on the Heisenberg group, answering two open questions raised in their work.