On the smoothness of the potential function in Riemannian optimal transport

Abstract

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 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 xed tangent space) restricted to