Analysis of orbit accumulation points and the Greene-Krantz conjecture

TL;DR

This paper classifies certain complex domains with noncompact automorphism groups in c2b2, analyzes their boundary defining functions, and extends previous results on automorphism groups containing a9 under specific conditions.

Contribution

It introduces a method to analyze infinite type boundary defining functions and extends existing results on the structure of automorphism groups in complex domains.

Findings

Classified domains with noncompact automorphism groups in c2b2.

Analyzed defining functions of infinite type boundary.

Proved conditions under which a9 contains a9, extending previous work.

Abstract

In $\mathbb{C}^2$, we classify the domains for which $\rm Aut(\Omega)$ 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, $\rm Aut(\Omega)$ contains $\mathbb{R}$, which is an extension of \cite{Fr86}.