Analysis of orbit accumulation points and the Greene-Krantz conjecture

Abstract

In ${\mathbb{C}}^2$, we classify the domains for which $\mathrm{Aut}(\mathrm{\Omega })$ is noncompact and describe these domains by their defining functions. This note is based on the technique of the scaling method introduced by Frankel \cite{Fr86} and Kim \cite{Ki90}. One feature of this article is that we are able to analyze the defining functions of infinite type boundary. As a corollary, we also prove a result that under some conditions, $\mathrm{Aut}(\mathrm{\Omega })$ contains $\mathbb{R}$, which is an extension of \cite{Fr86}.

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