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 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 2-dimensional with Gauss curvature C2 close to 1.
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Philippe Delanoë. On the smoothness of the potential function in Riemannian optimal transport. Communications in Analysis and Geometry, 2015, 23 (1), pp.11-89. ⟨10.4310/CAG.2015.v23.n1.a2⟩. ⟨hal-00957161v2⟩

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