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Regularity of optimal transport maps on locally nearly spherical manifolds
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 2, pp. 353-409.

Given a compact connected n-dimensional Riemannian manifold, we investigate the smoothness of the optimal transport map between the smooth densities with respect to the squared Riemannian distance cost. The optimal map is characterized by exp(gradu), where the potential function u satisfies a Monge–Ampère type equation. Delanoë [7] showed the smoothness of u on the Riemannian surfaces when the scalar curvature is close to 1 in C 2 norm. In this work, we study the regularity issue on Riemannian manifolds with curvature sufficiently close to curvature of round sphere in C 2 norm in all dimensions and prove that the 𝒞-curvature on such Riemannian manifolds satisfies an improved Ma-Trudinger-Wang condition and the Jacobian of the exponential map is positive. As a consequence, we imply the smoothness of the optimal transport map by the continuity method.

Etant donné une variété riemannienne compacte connexe de dimension n, nous étudions la régularité de l’application du transport optimal entre les densités lisses par rapport au coût de la distance riemannienne au carré. L’application du transport optimal est caractérisée par exp(gradu), où la fonction potentielle u satisfait une équation de type Monge–Ampère. Delanoë [7] a montré la régularité de u sur les surfaces riemanniennes lorsque la courbure scalaire est proche de 1 dans la norme C 2 . Dans ce travail, nous étudions le problème de régularité sur les variétés riemanniennes avec courbure suffisamment proche de la courbure de la sphère usuelle dans la norme C 2 en toutes les dimensions et prouvons que la 𝒞-courbure sur de telles variétés riemanniennes satisfait une condition Ma-Trudinger-Wang améliorée et le jacobien de l’application exponentielle est strictement positive. Par conséquent, nous impliquons la régularité de l’application du transport optimal par la méthode de continuité.

Published online:
DOI: 10.5802/afst.1678
Classification: 35R01,  53C21,  49N60
Keywords: regularity, optimal transport maps
Yuxin Ge 1; Jian Ye 2

1 Institut de Mathématiques de Toulouse, Université Toulouse 3, 118, route de Narbonne, 31062 Toulouse Cedex, France
2 School of Mathematical Sciences, University of Science and Technology of China, 96 Jinzhai Road, Baohe District, Hefei, Anhui, China
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Yuxin Ge; Jian Ye. Regularity of optimal transport maps on locally nearly spherical manifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 2, pp. 353-409. doi : 10.5802/afst.1678. https://afst.centre-mersenne.org/articles/10.5802/afst.1678/

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