Continuity, curvature, and the general covariance of optimal transportation

Young-Heon Kim ; Robert J. McCann

Journal of the European Mathematical Society, Tome 12 (2010) no. 4, pp. 1009-1040

Résumé

Let M and Mˉ be n-dimensional manifolds equipped with suitable Borel probability measures ρ and ρˉ​. For subdomains M and Mˉ of Rn, Ma, Trudinger Wang gave sufficient conditions on a transportation cost c∈C4(M×M) to guarantee smoothness of the optimal map pushing ρ forward to ρˉ​; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via non-negativity of the sectional curvature of certain null-planes in a novel but natural pseudo-Riemannian geometry which the cost c induces on the product space M×Mˉ. We also explore some connections between optimal transportation and spacelike Lagrangian submanifolds in symplectic geometry.

Détail

DOI : 10.4171/jems/221

Classification : 35-XX, 49-XX, 58-XX, 90-XX
Keywords: Optimal transportation, regularity of optimal maps, Hölder continuity, curvature, covariance, pseudo-Riemannian, semi-Riemannian, para-Kähler, spacelike Lagrangian, lightlike submanifold, signature (n, n), Monge–Kantorovich, measure-preserving homeomorphism