- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:09
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
15
2013
4
Regularity of optimal transport maps on multiple products of spheres
Alessio
Figalli
University of Texas at Austin, AUSTIN, UNITED STATES
Young-Heon
Kim
The University of British Columbia, VANCOUVER, CANADA
Robert
McCann
University of Toronto, TORONTO, ONTARIO, CANADA
Optimal transport, functional inequalities and Riemannian geometry
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.
Calculus of variations and optimal control; optimization
Partial differential equations
Differential geometry
General
1131
1166
10.4171/JEMS/388
http://www.ems-ph.org/doi/10.4171/JEMS/388