Donaldson's a priori estimates conjecture for the Calabi–Yau equation

Let $(M,\omega)$ be a compact symplectic manifold of real dimension $4$, equipped with an almost complex structure $J$ tamed by $\omega$. Let $\tilde{\omega}$ be another symplectic form compatible with $J$, cohomologous to $\omega$, and satisfying

$$\tilde{\omega}^2=e^F\omega^2$$

for some smooth function $F$. Donaldson's conjecture. There are $C^{\infty}$ a priori estimates for $\tilde{\omega}$ depending only on $M$, $J$, $\omega$, and $F$.

This conjecture concerns a priori control for solutions of the Calabi–Yau equation on symplectic four-manifolds. It remains open in general; the paper proves two consequences of the Aleksandrov–Bakelman–Pucci estimate.