Automorphisms of the Worm Domain

TL;DR

This paper investigates the automorphism group of the worm domain, a significant example in complex analysis, revealing boundary properties and exceptions that influence its symmetry structure.

Contribution

It provides a detailed analysis of the automorphism group of the worm domain, highlighting boundary behavior and local sphericity conditions.

Findings

Boundary is locally spherical everywhere except at the exceptional locus and caps.

Automorphism group structure is characterized in relation to boundary geometry.

The worm domain lacks a Stein neighborhood basis, impacting its automorphisms.

Abstract

The Diederich-Forn{\ae}ss worm domain, an important example of a smoothly bounded pseudoconvex domain without a Stein neighborhood basis, provides key counterexamples in the theory of Several Complex Variables. In this paper, we examine its automorphism group and observe that its boundary is locally spherical everywhere except at the exceptional locus and the caps.