Smoothness of the Optimal Transport Map on Riemannian Products of Spheres

Abstract

This paper approaches the smoothness of the optimal transport map on Riemannian products manifolds. By classical continuity method, we give an alternative proof of the smoothness of the optimal transport map on Riemannian products of the round spheres. In addition, we also prove that the smoothness of the optimal transport map is not stable in Riemannian products of round spheres.

Keywords

Monge–Ampère type equation / Optimal transport map / Continuity method / C-convex function / Maximum principle

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School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature