Regularity of optimal transport maps on multiple products of spheres

By Alessio Figalli, Young-Heon Kim, Robert J. McCann. 2010

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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 allows to be used to with the results of Liu, Trudinger and Wang [LTW] to conclude higher regularity (C 1; =C 1 ) of optimal maps for smoother (C =C 1 ) 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 [Ca3] and its quotients [Co]. 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 product structure of the cut-locus and the presence of flat directions. Optimal transport, functional inequalities and Riemannian geometry

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4fdf330ffed39c95e3569f032598fd305946c732

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Citations in the corpus

On the Ma–Trudinger–Wang curvature on surfaces (2010, background): “…17 Appendix: On the Riemannian cut locus of surfaces 27 References 31 Introduction The Ma–Trudinger–Wang (MTW) tensor is a nonlocal generalization of sectional curvature, involving fourth-order derivatives of the squared distance function [4, 7, 9, 10, 11, 18, 22, 25, 26, 27, 28, 29].”
The Monge–Ampère equation and its link to optimal transportation (2013, background): “) While this property has been proven to hold true in some special cases [41, 42, 53], it is still unknown in general and this creates several difficulties in the proof of regularity of optimal maps.”
Partial regularity for optimal transport maps (2012, background): “Let us also mention that, up to now, the MTW condition is known to be satisfied only for very special classes of Riemannian manifolds, such as spheres, their products, their quotients and submersions, and their perturbations [10, 11, 20, 22, 25, 29, 32], and for instance it is known to fail on suffi”
Hölder Continuity and Injectivity of Optimal Maps (2011, background): “…‡Department of Mathematics, University of British Columbia, Vancouver BC Canada yhkim@math.ubc.ca and School of Mathematics, Institute for Advanced Study at Princeton NJ USA §Department of Mathematics, University of Toronto, Toronto Ontario Canada M5S 2E4 mccann@math.toronto.edu Contents”
Unidimensional and Evolution Methods for Optimal Transportation (2013, unknown): “McCann [29].”
NECESSARY AND SUFFICIENT CONDITIONS FOR CONTINUITY OF OPTIMAL TRANSPORT MAPS ON RIEMANNIAN MANIFOLDS (2011, unknown): “As already observed in [11], the continuity and injectivity of T combined with the result”