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 mM, 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.

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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.

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