The Calabi-Yau equation on the Kodaira-Thurston manifold

TL;DR

This paper demonstrates that the Calabi-Yau equation can be solved on the Kodaira-Thurston manifold for all T^2-invariant volume forms, supporting the extension of Yau's theorem to certain symplectic four-manifolds.

Contribution

It proves the solvability of the Calabi-Yau equation on a specific non-Kähler manifold, extending the understanding of Yau's theorem beyond complex manifolds.

Findings

Calabi-Yau equation solvable on Kodaira-Thurston manifold

Supports conjecture on extension of Yau's theorem to symplectic four-manifolds

Provides explicit solutions for T^2-invariant volume forms

Abstract

We prove that the Calabi-Yau equation can be solved on the Kodaira-Thurston manifold for all given $T^2$-invariant volume forms. This provides support for Donaldson's conjecture that Yau's theorem has an extension to symplectic four-manifolds with compatible but non-integrable almost complex structures.