About regularity of optimal transport maps on Riemannian manifolds

The Wasserstein space over a Riemannian manifold is the space of probability measures endowed with the so-called Wasserstein distance. The study of this space and its geodesic is directly linked to the optimal transport problem introduced by Monge and generalized by Kantorovich. The fact that there exists a unique solution to the Monge problem is due to Brenier in the Euclidean case and to McCann in the case of closed Riemannian manifold. However, even if we consider two smooth probability measures, the optimal transport map may fail to be smooth. Caffarelli showed that the right geometric condition in order to obtain smoothness of the optimal transport map is to impose is the convexity of the supports of the smooth probability measures considered. The case of Riemannian manifolds is way harder. By using PDE methods, Ma, Trudinger and Wang identified a necessary condition on the geometry of the base manifold to get regularity of the optimal transport map: the sectional curvature has to be non-negative. Finally, Villani and Loeper highlighted some sufficient geometric conditions to get regularity. This talk is about all what I said before. If time permits I will present some interesting subspaces of the Wasserstein space such as the space of Gaussian measures on the Euclidean space.