The Aleksandrov-Bakelman-Pucci estimate and the Calabi-Yau equation

TL;DR

This paper applies the Aleksandrov-Bakelman-Pucci estimate to the Calabi-Yau equation on symplectic four-manifolds, demonstrating solvability in specific cases and reducing a broader conjecture to a measure bound problem.

Contribution

It introduces new applications of the Aleksandrov-Bakelman-Pucci estimate to solve the Calabi-Yau equation in particular settings and links Donaldson's conjecture to measure estimates.

Findings

Solvability of the Calabi-Yau equation on the Kodaira-Thurston manifold under certain conditions.

Reduction of Donaldson's conjecture to a measure bound on superlevel sets.

Extension of previous results to almost-Kahler structures with $S^1$-invariance.

Abstract

We give two applications of the Aleksandrov-Bakelman-Pucci estimate to the Calabi-Yau equation on symplectic four-manifolds. The first is solvability of the equation on the Kodaira-Thurston manifold for certain almost-Kahler structures assuming $S^1$-invariance, extending a result of Buzano-Fino-Vezzoni. The second is to reduce the general case of Donaldson's conjecture to a bound on the measure of a superlevel set of a scalar function.