Regularity for the optimal transportation problem with Euclidean distance squared cost on the embedded sphere

TL;DR

This paper establishes conditions under which the optimal transportation problem on the sphere with squared Euclidean distance cost results in a smooth, invertible map, advancing understanding of geometric optimal transport on curved spaces.

Contribution

It provides new sufficient conditions for the regularity of optimal transport maps on the sphere with quadratic cost, extending previous Euclidean results to curved manifolds.

Findings

Conditions ensuring the solution is a diffeomorphism on the sphere.

Extension of regularity results to spherical geometry.

Enhanced understanding of optimal transport on curved spaces.

Abstract

We give sufficient conditions on initial and target measures supported on the sphere $\S^n$ to ensure the solution to the optimal transport problem with the cost $|x-y|^2/2$ is a diffeomorphism.