On a generalized Monge-Amp\`ere equation on closed almost K\"ahler surfaces

TL;DR

This paper proves existence and uniqueness of solutions to a generalized Monge-Ampère equation on closed almost Kähler surfaces and confirms Donaldson's conjecture for tamed almost complex 4-manifolds.

Contribution

It introduces a new existence and uniqueness result for a generalized Monge-Ampère equation on almost Kähler surfaces and verifies a significant conjecture in the field.

Findings

Existence and uniqueness of solutions established

Donaldson's conjecture confirmed for tamed almost complex 4-manifolds

Applicable to closed almost Kähler surfaces

Abstract

We show the existence and uniqueness of solutions to a generalized Monge-Amp\`{e}re equation on closed almost K\"ahler surfaces, where the equation depends only on the underlying almost K\"ahler structure. As an application, we prove Donaldson's conjecture for tamed almost complex 4-manifolds.