A McLean Theorem for the moduli space of Lie solutions to mass transport equations

TL;DR

This paper proves the existence of volume-preserving 'fake' solutions to optimal transport equations on non-simply connected manifolds, revealing a geometric structure related to the manifold's topology and establishing a Hodge-Helmholtz decomposition.

Contribution

It introduces a new class of solutions to optimal transport problems on complex manifolds and connects geometric analysis with topological invariants.

Findings

Existence of 'fake' solutions on non-simply connected manifolds.

The solution set forms a manifold with dimension equal to the first Betti number.

Established a Hodge-Helmholtz decomposition for vector fields.

Abstract

On compact manifolds which are not simply connected, we prove the existence of "fake" solutions to the optimal transportion problem. These maps preserve volume and arise as the exponential of a closed 1 form, hence appear geometrically like optimal transport maps. The set of such solutions forms a manifold with dimension given by the first Betti number of the manifold. In the process, we prove a Hodge-Helmholtz decomposition for vector fields. The ideas are motivated by the analogies between special Lagrangian submanifolds and solutions to optimal transport problems.