$C^{1,1}$ regularity of degenerate complex Monge-Amp\`{e}re equations and some applications

In this paper, we prove a $C^{1,1}$ estimate for solutions of complex Monge-Amp`{e}re equations on compact almost Hermitian manifolds. Using this $C^{1,1}$ estimate, we show existence of $C^{1,1}$ solutions to the degenerate Monge-Amp`{e}re equations, the corresponding Dirichlet problems and the singular Monge-Amp`{e}re 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-Amp`{e}re type equations. As a geometric application, we prove the $C^{1,1}$ regularity of geodesics in the space of Sasakian metrics.