Semicontinuity of the Automorphism Groups of Domains with Rough Boundary

Abstract

Based on some ideas of Greene and Krantz, we study the semicontinuity of automorphism groups of domains in one and several complex variables. We show that semicontinuity fails for domains in $\CC^n$, $n > 1$, with Lipschitz boundary, but it holds for domains in $\CC^1$ with Lipschitz boundary. Using the same ideas, we develop some other concepts related to mappings of Lipschitz domains. These include Bergman curvature, stability properties for the Bergman kernel, and also some ideas about equivariant embeddings.