Asymptotics and convergence for the complex Monge-Ampere equation

TL;DR

This paper investigates the asymptotic behavior and convergence properties of solutions to the complex Monge-Ampère equation in the context of complete Kähler-Einstein metrics on pseudoconvex domains, revealing conditions for convergence and regularity.

Contribution

It provides a convergence theorem for solutions of the Monge-Ampère equation and analyzes their regularity, especially when boundary analyticity is partial, including Gevrey estimates.

Findings

Derived a convergence theorem for Monge-Ampère solutions.

Established Gevrey type estimates for tangential derivatives.

Identified limitations of local convergence through a counterexample.

Abstract

We study the asymptotics of complete Kaehler-Einstein metrics on strictly pseudoconvex domains in C^n and derive a convergence theorem for solutions to the corresponding Monge-Ampere 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-Ampere equation