Regularity of optimal transport maps on multiple products of spheres

Figalli, 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 -