Regularity of optimal transport maps on locally nearly spherical manifolds

Ge, Yuxin; Ye, Jian

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

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 $exp\left(\mathrm{grad}u\right)$, 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.

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 $exp\left(\mathrm{grad}u\right)$, 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.

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

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