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

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

1 Department of Mathematics and Statistics [Texas Tech]
2 JAD - Laboratoire Jean Alexandre Dieudonné
3 ICJ - Institut Camille Jordan

In this paper we investigate the regularity of optimal transport maps for the squared distance cost on Riemannian manifolds. First of all, we provide some general necessary and sufficient conditions for a Riemannian manifold to satisfy the so-called Transport Continuity Property. Then, we show that on surfaces these conditions coincide. Finally, we give some regularity results on transport maps in some specific cases, extending in particular the results on the flat torus and the real pro jective space to a more general class of manifolds.

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. Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds. Tohoku mathematical journal, 2011, 63 (4), pp.855-876. ⟨10.2748/tmj/1325886291⟩. ⟨hal-00923320⟩