On the Ma-Trudinger-Wang condition on surfaces

Alessio Figalli (1) , Ludovic Rifford (2) , Cédric Villani (3)

1 Department of Mathematics and Statistics [Texas Tech]
2 JAD - Laboratoire Jean Alexandre Dieudonné
3 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We investigate the properties of the Ma-Trudinger-Wang nonlocal curvature tensor in the case of surfaces. In particular, we prove that a strict form of the Ma-Trudinger-Wang condition is stable under C 4 perturbation if the nonfocal domains are uniformly convex; and we present new examples of positively curved surfaces which do not satisfy the Ma-Trudinger-Wang condition. As a corollary of our results, optimal transport maps on a "sufficiently flat" ellipsoid are in general nonsmooth.

Domaines: Equations aux dérivées partielles [math.AP]; Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]; Optimisation et contrôle [math.OC]

Alessio Figalli, Ludovic Rifford, Cédric Villani. On the Ma-Trudinger-Wang condition on surfaces. Calculus of Variations and Partial Differential Equations, 2010, 39 (3-4), pp.307. ⟨hal-00923304⟩