C1,1 REGULARITY OF DEGENERATE COMPLEX MONGE-AMPERE EQUATIONS AND SOME APPLICATIONS

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