Monge's transport problem on a Riemannian manifold

Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures mu(+) not equal mu(-) find the measure-preserving map s : M --> M between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold M - and assuming absolute continuity of mu(+) - an optimal map will be shown to exist. Aspects of its uniqueness are also established.