We present a method for constructing global holomorphic peak functions from local holomorphic support functions for broad classes of unbounded domains. As an application, we establish a method for showing the positivity and completeness of invariant
metrics including the Bergman metric mainly for the unbounded domains.
The Kohn-Nireberg domains are unbounded domains in the complex Euclidean space of dimension 2 upon which many outstanding questions are yet to be explored. The primary aim of this article is to demonstrate that the Bergman and Caratheodory metrics of
any Kohn-Nirenberg domains are positive and complete.
