Calculus of Variations and Geometric Measure Theory

A. Figalli - L. Rifford - C. Villani

Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds

created by figalli on 24 May 2011
modified on 13 Jun 2011


Accepted Paper

Inserted: 24 may 2011
Last Updated: 13 jun 2011

Journal: Tohoku Math. J.
Year: 2011


In this paper we continue the investigation of the regularity of optimal transport maps on Riemannian manifolds, in relation with the geometric conditions of Ma--Trudinger--Wang and the geometry of the cut locus. We derive some sufficient and some necessary conditions to ensure that the optimal transport map is always continuous. In dimension two, we can sharpen our result into a necessary and sufficient condition. We also provide some sufficient conditions for regularity, and review existing results.