Examples of domains with non-compact automorphism groups

TL;DR

This paper constructs specific examples of bounded, pseudoconvex domains in complex spaces with non-compact automorphism groups that are not equivalent to Reinhardt domains, highlighting new phenomena in complex analysis.

Contribution

It provides explicit examples of complex domains with non-compact automorphism groups that are not biholomorphically equivalent to Reinhardt domains, including smooth and non-smooth cases.

Findings

Existence of non-Reinhardt domains with non-compact automorphism groups in dimensions ≥2

Smooth, pseudoconvex, circular domains with these properties in dimension ≥3

A non-smooth example in dimension two at a boundary point

Abstract

We give, in dimensions three or greater, an example of a bounded, pseudoconvex, circular domain in complex space with smooth real analytic boundary and non-compact automorphism group which is not biholomorphically equivalent to any Reinhardt domain. We give an analogous example in dimension two, but the domain fails to be smooth at one boundary point---indeed it is not in any Lipschitz class at the exceptional point.