度量测度空间中具有黎曼曲率维数条件的Monge问题
Monge problem in metric measure spaces with Riemannian curvature-dimension condition
摘要：
我们证明了Monge极小化问题的解的存在性，解决了度量测度空间$(X,d,m)$享有黎曼曲率维条件$\RCD(K,N)$，与$N < \infty$。对于第一个边际测度，我们假设$μ_{0} \ll m$。作为推论，我们得到Monge问题和它的松弛形式Monge-Kantorovich问题，达到相同的极小值。证明了$d$ -循环单调集的一个结构定理：当忽略一个零$m$ -测度集时，它们不包含任何分支结构，即它们可以表示为不相交测地线族的象的不相交并.
英文摘要：
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 $μ_{0} \ll m$. 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.
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