The Calabi-Yau equation for $T^2$-bundles over $\mathbb{T}^2$: the non-Lagrangian case

TL;DR

This paper solves the Calabi-Yau equation on certain 4-manifolds with a non-Lagrangian almost-K"ahler structure by reducing it to a Monge-Amp e \

Contribution

It demonstrates existence of solutions for the Calabi-Yau problem on $T^2$-bundles over $ ^2$ with invariant non-Lagrangian structures, extending previous results.

Findings

Unique solutions to the Monge-Amp e equation for invariant data

Existence of Calabi-Yau metrics on all such $T^2$-bundles with invariant structures

Reduction of the problem to a solvable Monge-Amp e equation

Abstract

In the spirit of [10,2], we study the Calabi-Yau equation on $T^2$-bundles over $\mathbb{T}^2$ endowed with an invariant non-Lagrangian almost-K\"ahler structure showing that for $T^2$-invariant initial data it reduces to a Monge-Amp\`ere equation having a unique solution. In this way we prove that for every total space $M^4$ of an orientable $T^2$-bundle over $\mathbb{T}^2$ endowed with an invariant almost-K\"ahler structure the Calabi-Yau problem has a solution for every normalized $T^2$-invariant volume form.