Regularity of optimal transport maps on locally nearly spherical manifolds. (Régularité de l’application du transport optimal sur les variétés riemanniennes localement proches de La sphère.) (English. French summary) Zbl 1471.49034
Summary: 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 \(\exp (\operatorname{grad}\,u)\), where the potential function \(u\) satisfies a Monge-Ampère type equation. P. Delanoë [Commun. Anal. Geom. 23, No. 1, 11–89 (2015; Zbl 1310.53036)] 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 \({\mathcal{C}} \)-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.
MSC:
| 49Q22 | Optimal transportation |
| 35R01 | PDEs on manifolds |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 49N60 | Regularity of solutions in optimal control |
Citations:
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