Semicontinuity of the Automorphism Groups of Domains with Rough Boundary

TL;DR

This paper investigates the semicontinuity of automorphism groups of complex domains with Lipschitz boundaries, revealing it fails in higher dimensions but holds in one dimension, and explores related geometric concepts.

Contribution

It demonstrates the failure of semicontinuity in higher dimensions and establishes its validity in one dimension, also developing related geometric and mapping concepts.

Findings

Semicontinuity fails for Lipschitz boundary domains in $ ext{C}^n$, n > 1.

Semicontinuity holds for Lipschitz boundary domains in $ ext{C}^1$.

Develops concepts like Bergman curvature and stability properties for Lipschitz domains.

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.