Asymptotic Behavior of Homogeneous Complex Monge-Ampere Equations on ALE Kahler manifolds

Abstract

This paper is a sequel to the author's earlier work and investigates the homogeneous complex Monge--Ampere equation (HCMA) on the product space $X \times D$, where $X$ is an asymptotically locally Euclidean (ALE) Kahler manifold and $D subset C$ is the unit disc. We establish precise asymptotic behavior of the solution to the HCMA equation, showing that the decay rate of the solution matches that of the prescribed boundary data and that uniform control in weighted Holder norms can be achieved. The analysis combines two main ingredients: a redevelopment of pluripotential theory on the noncompact space $X \times D$ and a PDE-based construction of holomorphic disc foliations on the end of $X$, inspired by the works of Semmes and Donaldson. As an application in general Kahler manifolds, the techniques developed in this paper also imply a local regularity result for the HCMA equation.