REGULARITY OF OPTIMAL TRANSPORT MAPS (after Ma{Trudinger{Wang and Loeper)

In the special case “cost=squared distance” on R, the problem was solved by Caffarelli [Caf1, Caf2, Caf3, Caf4], who proved the smoothness of the map under suitable assumptions on the regularity of the densities and on the geometry of their support. However, a major open problem in the theory was the question of regularity for more general cost functions, or for the case “cost=squared distance” on a Riemannian manifold. A breakthrough in this problem has been achieved by Ma, Trudinger and Wang [MTW] and Loeper [Loe1], who found a necessary and sufficient condition on the cost function in order to ensure regularity. This condition, now called MTW condition, involves a combination of derivatives of the cost, up to the fourth order. In the special case “cost=squared distance” on a Riemannian manifold, the MTW condition corresponds to the non-negativity of a new curvature tensor on the manifold (the so-called MTW tensor), which implies strong geometric consequences on the geometry of the manifold and on the structure of its cut-locus.