Monge's transport problem in the Heisenberg group

TL;DR

This paper proves the existence of solutions to Monge's optimal transport problem within the Heisenberg group, under specific measure conditions, advancing understanding in geometric measure theory and optimal transport in non-Euclidean spaces.

Contribution

It establishes the existence of solutions to Monge's problem in the Heisenberg group with certain measure assumptions, a novel extension of classical optimal transport theory.

Findings

Existence of solutions proven for Monge's problem in the Heisenberg group.

Results apply when the initial measure is absolutely continuous.

Advances the theory of optimal transport in sub-Riemannian geometries.

Abstract

We prove the existence of solutions to Monge transport problem between two compactly supported Borel probability measures in the Heisenberg group equipped with its Carnot-Caratheodory distance assuming that the initial measure is absolutely continuous with respect to the Haar measure of the group.