On the boundedness of invariant hyperbolic domains

Abstract

In this paper, we generalize a theorem of A. Kodama about boundedness of hyperbolic circular domains. We will prove that if $K$ is a compact Lie group which acts linearly on $\mathbb{C}^n$ with $\mathcal{O}(\mathbb{C}^n)^K=\mathbb{C}$, and $\Omega $ is a $K$-invariant orbit convex domain in $\mathbb{C}^n$ which contains $0$, then $\Omega $ is bounded if and only if $\Omega $ is Kobayashi hyperbolic.