Journal of the European Mathematical Society

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Volume 15, Issue 4, 2013, pp. 1131–1166
DOI: 10.4171/JEMS/388

Published online: 2013-05-09

Regularity of optimal transport maps on multiple products of spheres

Alessio Figalli[1], Young-Heon Kim[2] and Robert J. McCann[3]

(1) ETH Z├╝rich, Switzerland
(2) The University of British Columbia, Vancouver, Canada
(3) University of Toronto, Canada

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 ($C^{1, \a}/C^\infty$) of optimal maps for smoother ($C^\a/C^\infty$) 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 $\R^n$} 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 \cite{FKM}, where we showed injectivity and continuity of optimal maps on domains in $\R^n$ 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.

Keywords: Optimal transport, functional inequalities and Riemannian geometry

Figalli Alessio, Kim Young-Heon, McCann Robert: Regularity of optimal transport maps on multiple products of spheres. J. Eur. Math. Soc. 15 (2013), 1131-1166. doi: 10.4171/JEMS/388