Monge's transport problem on a Riemannian manifold (Q2781378)

The paper deals with the problem of optimal transportation of mass, raised by Monge in 1781. This problem admits many variants, but the one considered here is very close to the classical problem, as the cost function involved is the distance between the initial and final point: the only difference is that the ambient space is a \(C^3\), complete and connected Riemannian manifold. In this situation new difficulties arise, due to the lack of differentiability in the large of the cost function. This difficulty has already been overcome in a previous paper by the second author, in the case when the cost function is the \(p\)-th power of the distance, with \(p>1\). The main result of the paper is the existence of optimal transport maps when the initial and final distribution of mass are absolutely continuous with respect to the volume measure of the manifold. It is also shown that the only degree of freedom, as in the Euclidean case, occurs along transport rays, and that forcing monotonicity of the map along transport rays leads to the uniqueness of the transport map. This also leads to the fact that the problem has a nice invariant, the so-called transport density. On the technical side, the methods are close to the ones used in previous papers by several authors (Sudakov, Evans-Gangbo, Caffarelli-Feldman-McCann, Trudinger-Wang, Ambrosio), i.e., the strategy is to solve the 1-dim. problem in each transport ray and then glue all partial transport maps to obtain a global one. An essential point of this construction is the countable Lipschitz property of transport rays, whose proof in the Riemannian context is more demanding.