Complex scaling and domains with non-compact automorphism group

Abstract

In this paper we study the boundary orbit accumulation points of smoothly bounded domains in $\mathbf{C}^2$ with non-compact automorphism group. We prove that a boundary orbit accumulation point cannot be exponentially flat. This confirms a version of a conjecture of Greene and Krantz. The proof uses a new result on the a priori convergence of convex scaling methods, which in particular implies the equivalence of two different scaling methods on convex domains.