Regularity of optimal transport on compact, locally nearly spherical, manifolds

Abstract

Given a couple of smooth positive measures of same total mass on a compact connected Riemannian manifold M, we look for a smooth optimal transportation map G, pushing one measure to the other at a least total squared distance cost, directly by using the continuity method to produce a classical solution of the elliptic equation of Monge–Ampère type satisfied by the potential function u, such that G = exp(grad u). This approach boils down to proving an a priori upper bound on the Hessian of u, which was done on the flat torus by the first author. The recent local C² estimate of Ma–Trudinger–Wang enabled Loeper to treat the standard sphere case by overcoming two difficulties, namely: in collaboration with the first author, he kept the image G(m) of a generic point m ∈ M, uniformly away from the cut-locus of m; he checked a fourth-order inequality satisfied by the squared distance cost function, proving its so-called (strict) regularity. In the present paper, we treat along the same lines the case of manifolds with curvature sufficiently close to 1 in C² norm—specifying and proving a conjecture stated by Trudinger.