Positivity and Completeness of Invariant Metrics

We present a method for constructing global holomorphic peak functions from local holomorphic support functions for broad classes of unbounded domains in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^n$$\end{document}. 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.