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

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.

12 pages

References in corpus (3)

• Taming symplectic forms and the Calabi-Yau equation
• $C^{2,\alpha }$ estimates for nonlinear elliptic equations in complex and almost complex geometry
• (1,1) forms with specified Lagrangian phase: A priori estimates and algebraic obstructions