Asymptotics and Convergence for the Complex Monge-Ampère Equation

We study the asymptotics of complete Kähler-Einstein metrics on strictly pseudoconvex domains in \(\mathbb {C}^n\) ${\mathbb{C}}^n$ and derive a convergence theorem for solutions to the corresponding Monge-Ampère equation. If only a portion of the boundary is analytic, the solutions satisfy Gevrey type estimates for tangential derivatives. A counterexample for the model linearized equation suggests that there is no local convergence theorem for the complex Monge-Ampère equation.