On the smoothness of the potential function in Riemannian optimal transport

On a closed Riemannian manifold, McCann proved the existence of a unique Borel map pushing a given smooth positive probability measure to another one while minimizing a related quadratic cost functional. The optimal map is obtained as the exponential of the gradient of a c-convex function u. The question of the smoothness of u has been intensively investigated. We present a self-contained partial differential equations approach to this problem. The smoothness question is reduced to a couple of a priori estimates, namely: a positive lower bound on the Jacobian of the exponential map (meant at each fixed tangent space) restricted to the graph of grad u; and an upper bound on the c-Hessian of u. By the Ma-Trudinger-Wang device, the former estimate implies the latter on manifolds satisfying the so-called A3 condition. On such manifolds, it only remains to get the Jacobian lower bound. We get it on simply connected positively curved manifolds which are, either locally symmetric, or two-dimensional with Gauss curvature C-2 close to 1.