Regularity of optimal transport maps on multiple products of spheres

  • Young-Heon Kim

    The University of British Columbia, Vancouver, Canada
  • Robert J. McCann

    University of Toronto, Canada
  • Alessio Figalli

    ETH Zürich, Switzerland


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 () of optimal maps for smoother () 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 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 [FKM1], where we showed injectivity and continuity of optimal maps on domains in 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.

Cite this article

Young-Heon Kim, Robert J. McCann, Alessio Figalli, Regularity of optimal transport maps on multiple products of spheres. J. Eur. Math. Soc. 15 (2013), no. 4, pp. 1131–1166

DOI 10.4171/JEMS/388