Monge problem in metric measure spaces with Riemannian curvature-dimension condition

Abstract

We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space $(X,d,m)$ enjoying the Riemannian curvature-dimension condition $\RCD(K,N)$, with $N<\infty$. For the first marginal measure, we assume that ${\mu }_0\ll m$. As a corollary, we obtain that the Monge problem and its relaxed version, the Monge-Kantorovich problem, attain the same minimal value. Moreover we prove a structure theorem for $d$-cyclically monotone sets: neglecting a set of zero $m$-measure they do not contain any branching structures, that is, they can be written as the disjoint union of the image of a disjoint family of geodesics.