Estimates for the Complex Monge-Ampère Equation on Hermitian and Balanced Manifolds

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.

Keywords

Complex Monge-Ampère equation, Hermitian manifold, balanced manifold

2010 Mathematics Subject Classification

Primary 32W20. Secondary 32Q25, 53C55.

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Published 1 January 2010