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 mea-sure to another one while minimizing a related quadratic cost func-tional. 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 PDE 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 gradu; 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 con-nected positively curved manifolds which are, either locally symmetric, or 2-dimensional with Gauss curvature C2 close to 1