On the Calabi-Yau equation in the Kodaira-Thurston manifold

TL;DR

This paper reviews and extends results on solving the Calabi-Yau equation on the Kodaira-Thurston manifold, reducing it to a Monge-Ampère equation under certain conditions and exploring non-invariant structures and higher dimensions.

Contribution

It provides a reduction of the Calabi-Yau problem to a Monge-Ampère equation using an invariant ansatz and extends analysis to non-invariant structures and higher-dimensional cases.

Findings

Reduction to Monge-Ampère equation under invariant conditions

Extension to non-invariant almost-complex structures

Generalization to higher dimensions

Abstract

We review some previous results about the Calabi-Yau equation on the Kodaira-Thurston manifold equipped with an invariant almost-Kaehler structure and assuming the volume form invariant by the action of a torus. In particular, we observe that under some restrictions the problem is reduced to a Monge-Amp\`ere equation by using the ansatz $\tilde \omega=\Omega-dJdu+da$, where $u$ is a $T^2$-invariant function and $a$ is a $1$-form depending on $u$. Furthermore, we extend our analysis to non-invariant almost-complex structures by considering some basic cases and we finally take into account a generalization to higher dimensions.