Estimates for the complex monge-ampère equation on hermitian and balanced manifolds

Abstract

We generalize Yau's estimates for the complex Monge-Ampère equation on compact manifolds in the case when the background metric is no longer Kähler. We prove C∞ a priori estimates for a solution of the complex Monge-Ampère equation when the background metric is Hermitian (in complex dimension two) or balanced (in higher dimensions), giving an alternative proof of a theorem of Cherrier. We relate this to recent results of Guan-Li. © 2010 International Press.