# Regularity of optimal transport maps on multiple products of spheres

Alessio Figalli; Young-Heon Kim; Robert McCann

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 4, page 1131-1166
- ISSN: 1435-9855

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topFigalli, Alessio, Kim, Young-Heon, and McCann, Robert. "Regularity of optimal transport maps on multiple products of spheres." Journal of the European Mathematical Society 015.4 (2013): 1131-1166. <http://eudml.org/doc/277190>.

@article{Figalli2013,

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. 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 $(C^\{1,\alpha \}/C^\infty )$ of optimal maps for smoother $(C^\alpha /C^\infty )$ 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 $\mathbb \{R\}^n$ 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 [FKM1], where we showed injectivity and continuity of optimal maps on domains in $\mathbb \{R\}^n$ 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.},

author = {Figalli, Alessio, Kim, Young-Heon, McCann, Robert},

journal = {Journal of the European Mathematical Society},

keywords = {optimal transport; functional inequalities; Riemannian geometry; optimal transport; functional inequalities; Riemannian geometry},

language = {eng},

number = {4},

pages = {1131-1166},

publisher = {European Mathematical Society Publishing House},

title = {Regularity of optimal transport maps on multiple products of spheres},

url = {http://eudml.org/doc/277190},

volume = {015},

year = {2013},

}

TY - JOUR

AU - Figalli, Alessio

AU - Kim, Young-Heon

AU - McCann, Robert

TI - Regularity of optimal transport maps on multiple products of spheres

JO - Journal of the European Mathematical Society

PY - 2013

PB - European Mathematical Society Publishing House

VL - 015

IS - 4

SP - 1131

EP - 1166

AB - 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 $(C^{1,\alpha }/C^\infty )$ of optimal maps for smoother $(C^\alpha /C^\infty )$ 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 $\mathbb {R}^n$ 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 [FKM1], where we showed injectivity and continuity of optimal maps on domains in $\mathbb {R}^n$ 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.

LA - eng

KW - optimal transport; functional inequalities; Riemannian geometry; optimal transport; functional inequalities; Riemannian geometry

UR - http://eudml.org/doc/277190

ER -

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