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 $\mathrm{A}3$ 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 $1$.

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Published 24 November 2014

This paper was revised on February 9, 2015.