Calculus of Variations and Geometric Measure Theory

A. Figalli - Y. H. Kim - R. J. McCann

Regularity of optimal transport maps on multiple products of spheres

created by figalli on 24 May 2011
modified on 17 Oct 2011


Accepted Paper

Inserted: 24 may 2011
Last Updated: 17 oct 2011

Journal: J. Eur. Math. Soc. (JEMS)
Year: 2011


This article addresses regularity of optimal transport maps for cost$=$``squared distance'' on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved. Under boundedness and non-vanishing assumptions on the transfered source and target densities we show that optimal maps stay away from the cut-locus (where the cost exhibits singularity), and obtain injectivity and continuity of optimal maps. This together with the result of Liu, Trudinger and Wang also implies higher regularity ($C^{1,\alpha}/C^\infty$) of optimal maps for smoother ($C^\alpha/C^\infty$) densities. These are the first global regularity results which we are aware of concerning optimal maps on Riemannian manifolds which possess some vanishing sectional curvatures, beside the totally flat case of $\mathbb R^n$ and its quotients. Moreover, such product manifolds have potential relevance in statistics and in statistical mechanics (where the state of a system consisting of many spins is classically modeled by a point in the phase space obtained by taking many products of spheres). For the proof we apply and extend the method developed in a previous paper, where we showed injectivity and continuity of optimal maps on domains in $\mathbb R^n$ for smooth non-negatively cross-curved cost. The major obstacle in the present paper is to deal with the non-trivial cut-locus and the presence of flat directions.