The complex Monge–Ampère equation on some compact Hermitian manifolds

Jianchun Chu

Vol. 276 (2015), No. 2, 369–386

DOI: 10.2140/pjm.2015.276.369

Abstract

We consider the complex Monge–Ampère equation on compact manifolds when the background metric is a Hermitian metric (in complex dimension 2) or a Hermitian metric satisfying an additional condition (in higher dimensions). We prove that the Laplacian estimate holds when $F$ is in $W^{\mathfrak{1},q_{\mathfrak{0}}}$ for any $q_{\mathfrak{0}}>\mathfrak{2}n$. As an application, we show that, up to scaling, there exists a unique classical solution in $W^{\mathfrak{3},q_{\mathfrak{0}}}$ for the complex Monge–Ampère equation when $F$ is in $W^{\mathfrak{1},q_{\mathfrak{0}}}$.

Keywords

complex Monge–Ampère equation, compact Hermitian manifold

Mathematical Subject Classification 2010

Primary: 35J96, 53C55

Milestones

Received: 15 May 2014

Revised: 22 October 2014

Accepted: 23 December 2014

Published: 15 July 2015