Pseudo-Riemannian geometry calibrates optimal transportation

TL;DR

This paper introduces a pseudo-Riemannian geometric framework with calibration forms to characterize optimal transportation maps as maximal submanifolds, providing a new geometric perspective on optimal transport problems.

Contribution

It develops a pseudo-metric and calibration form on the product space to identify optimal maps as calibrated maximal submanifolds, bridging optimal transport and pseudo-Riemannian geometry.

Findings

Optimal maps correspond to calibrated maximal submanifolds.

A new pseudo-metric and calibration form are constructed.

The approach extends to defining mass of space-like currents in indefinite metrics.

Abstract

Given a transportation cost $c: M \times\bar M \to\mathbf{R}$, optimal maps minimize the total cost of moving masses from $M$ to $\bar M$. We find a pseudo-metric and a calibration form on $M\times\bar M$ such that the graph of an optimal map is a calibrated maximal submanifold. We define the mass of space-like currents in spaces with indefinite metrics.