Monge’s transport problem on a Riemannian manifold

Abstract

Monge’s problem refers to the classical problem of optimally transporting mass: given Borel probability measures μ + ≠ μ − \mu ^+ \ne \mu ^- , find the measure-preserving map s : M ⟶ M s:M \longrightarrow M between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold M M — and assuming absolute continuity of μ + \mu ^+ — an optimal map will be shown to exist. Aspects of its uniqueness are also established.