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 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 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 con- nected positively curved manifolds which are, either locally symmetric, or 2-dimensional with Gauss curvature C2 close to 1.
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https://hal.univ-cotedazur.fr/hal-00957161
Contributor : Philippe Delanoë <>
Submitted on : Sunday, March 9, 2014 - 11:29:34 AM
Last modification on : Thursday, January 11, 2018 - 6:12:16 AM
Long-term archiving on: Monday, June 9, 2014 - 10:36:49 AM

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Philippe Delanoë. On the smoothness of the potential function in Riemannian optimal transport. 2014. ⟨hal-00957161v1⟩

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