C1,1 regularity of degenerate complex Monge–Ampère equations and some applications

Abstract

We prove a $C^{1,1}$ estimate for solutions of complex Monge–Ampère equations on compact almost Hermitian manifolds. Using this $C^{1,1}$ estimate, we show the existence of $C^{1,1}$ solutions to the degenerate Monge–Ampère equations, the corresponding Dirichlet problems and the singular Monge–Ampère equations. We also study the singularities of the pluricomplex Green’s function. In addition, the proof of the above $C^{1,1}$ estimate is valid for a kind of complex Monge–Ampère-type equation. As a geometric application, we prove the $C^{1,1}$ regularity of geodesics in the space of Sasakian metrics.