The automorphism group of domains with boundary points of infinite type

Abstract

Let $\Omega\subset\mathbb C^2$ be a smoothly bounded domain. We prove that if $\partial \Omega$ contains a (small) smooth curve of points of infinity type, then the automorphism group $\Aut(\Omega)$ is compact. This result implies the Greene-Krantz conjecture for a special class of domains. The proof makes no use of scaling techniques.