Regularity of optimal transport maps on multiple products of spheres

TL;DR

This paper proves the regularity, injectivity, and continuity of optimal transport maps on products of spheres with arbitrary sizes and dimensions, extending previous results to non-flat Riemannian manifolds with vanishing sectional curvatures.

Contribution

It establishes the first global regularity results for optimal transport maps on non-flat Riemannian manifolds with some flat directions, specifically on products of spheres.

Findings

Optimal maps stay away from the cut-locus under certain conditions.

Optimal maps are injective and continuous.

Higher regularity of optimal maps is achieved for smooth densities.

Abstract

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 [KM2]. 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 [LTW] also implies higher regularity (C^{1,\alpha}/C^\infty) of optimal maps for more smooth (C^\alpha /C^\infty)) densities. These are the first global regularity results which we are aware of concerning optimal maps on non-flat Riemannian manifolds which possess some vanishing sectional curvatures. Moreover, such product manifolds have potential relevance in statistics (see [S]) 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 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.