Optimal transportation on non-compact manifolds

TL;DR

This paper extends existing optimal transportation results from compact to non-compact manifolds, demonstrating that certain cost functions satisfy the Monge Transport Problem without curvature restrictions.

Contribution

It generalizes known optimal transport results to non-compact manifolds for costs derived from Tonelli Lagrangians, removing curvature restrictions.

Findings

Results hold for costs like d^r, r>1, on non-compact manifolds

No curvature restrictions are needed for these transportation results

Extends Monge Transport Problem solutions to broader geometric settings

Abstract

In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type $d^r,r>1$, where $d$ is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction.