Homogeneous optimal transport maps between oblique cones

TL;DR

This paper constructs homogeneous optimal transport maps between convex cones with homogeneous densities under obliqueness, impacting boundary regularity in optimal transport and the existence of Calabi-Yau metrics.

Contribution

It introduces a method to construct optimal transport maps between convex cones satisfying an obliqueness condition, extending the theory to degenerate densities.

Findings

Existence of homogeneous optimal transport maps under obliqueness

Applications to boundary regularity in convex domain transport

Relevance to Calabi-Yau metric existence on quasi-projective varieties

Abstract

We construct homogeneous optimal transport maps for the quadratic cost between convex cones with homogeneous, possibly degenerate, densities when the cones satisfy an obliqueness condition. The existence of such maps plays a central role in the boundary regularity theory for optimal transport maps between convex domains. Our results are also relevant for the existence of complete Calabi-Yau metrics on certain quasi-projective varieties.