Donaldson’s conjecture on a priori bounds for the Calabi–Yau equation on tamed almost complex 4-manifolds

Establish that for any compact 4-manifold M equipped with an almost complex structure J tamed by a symplectic form Ω, and for any smooth volume form σ on M satisfying ∫_M σ = ∫_M Ω^2, if an almost Kähler form \tilde{ω} cohomologous to Ω solves the Calabi–Yau equation \tilde{ω}^2 = σ, then there exist C^∞ a priori bounds on \tilde{ω} depending only on Ω, J, and M.

Background

The conjecture, attributed to Donaldson, seeks uniform C^∞ a priori estimates for almost Kähler solutions \tilde{ω} to the Calabi–Yau equation \tilde{ω}^2 = σ in the cohomology class [Ω] on compact almost complex 4-manifolds where Ω tames J. This is an analogue of Yau’s a priori estimates in the integrable Kähler setting, adapted to the tamed almost complex case.

In this paper, the authors introduce a generalized Monge–Ampère equation on closed almost Kähler surfaces using the operator \mathcal{D}_J^+. They prove existence and uniqueness together with C^∞ a priori bounds, and derive as a corollary bounds that address Donaldson’s conjecture in the tamed almost complex 4-manifold setting.