Calculus of Variations and Geometric Measure Theory

A. Figalli - L. Rifford

Continuity of optimal transport maps and convexity of injectivity domains on small deformations of S^2

created by figalli on 19 May 2009
modified on 15 Jun 2009


Accepted Paper

Inserted: 19 may 2009
Last Updated: 15 jun 2009

Journal: Comm. Pure Appl. Math.
Year: 2009


Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so-called Ma-Trudinger-Wang condition and to show that this condition, together with the strict convexity on the nonfocal domains, implies the continuity of the optimal transport map. Moreover our new condition, again combined with the strict convexity of the nonfocal domains, allows to prove that all injectivity domains are strictly convex too. These results apply for instance on any small $C^4$-deformation of the two-sphere.