Complex Monge-Ampère Equation

In this thesis,we mainly study the regularity of solutions to complex Monge-Ampere equations and related Liouville type theorems.In Chapter 2,we introduce some basic notations and results on complex Monge-Ampere equations.In Chapter 3,we consider the Dirichlet problem on a ball in Cn.According to the assumptions on regularity of boundary data and the right hand member(being C0,?continuous or C1,? continuous),we obtain Holder estimates for the solution and its first-order derivatives respectively.This generalizes Bedford and Taylor's interior C1,1 esti-mate.In Chapter 4.we obtain a local C2,? estimate for a solution,under the assumption that the solution is C1,?(? sufficiently close to l)continuous and the right hand member is C? continuous.Compared with existing results,the assumption on the regularity of the solution is weakened.In Chapter 5,we consider the product manifold of Cn and a compact Kahler man-ifold with nonnative Ricci curvature.We prove that a Kahler form satisfy the complex Monge-Ampere equation and some other conditions must be paralleled with respect to some product metric.In Chapter 6,we generalize our work in Chapter 3 and 4 to general cases on Her-mitian manifolds.