Automorphism groups of domains that depend on fewer than the maximal number of parameters

TL;DR

This paper investigates complex domains with automorphism groups that depend on fewer parameters than the ambient space, providing geometric conditions for such 'thin' automorphism groups and illustrating with examples.

Contribution

It introduces a geometric criterion for identifying domains with automorphism groups of lower dimension than the ambient space, expanding understanding of symmetry in complex analysis.

Findings

Established a sufficient geometric condition for 'thin' automorphism groups

Provided examples illustrating the geometric ideas

Enhanced understanding of automorphism group dependence in complex domains

Abstract

We study domains in complex $n$-space with automorphism group that does not depend on the full $n$ dimensions of the ambient space. A sufficient geometric condition is obtained to guarantee that a domain has such a "thin" automorphism group. Examples are provided to illustrate the ideas.