Regularity of optimal transport maps on locally nearly spherical manifolds

Yuxin Ge; Jian Ye

doi:10.5802/afst.1678

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

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 2, pp. 353-409.

Abstract
Abstract

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 $exp (grad u)$ , 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 $exp (grad u)$ , 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: 2021-07-09

DOI: 10.5802/afst.1678

Classification: 35R01, 53C21, 49N60
Keywords: regularity, optimal transport maps
Mot clés : régularité, application du transport optimal

Author's affiliations:

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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     author = {Yuxin Ge and Jian Ye},
     title = {Regularity of optimal transport maps on locally nearly spherical manifolds},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {353--409},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 30},
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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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