On the linearity of origin-preserving automorphisms of quasi-circular domains in $\mathbb C^n$

TL;DR

This paper extends Cartan's theorem, showing that under certain conditions, origin-preserving automorphisms of quasi-circular domains in complex n-space are linear, with applications to specific cases and existing theorems.

Contribution

It generalizes Cartan's linearity result to quasi-circular domains using Bergman's theory and verifies the Braun-Kaup-Upmeier theorem in this context.

Findings

Automorphisms are linear under certain conditions.

Criteria established for the case n=3.

Braun-Kaup-Upmeier theorem holds for quasi-circular domains.

Abstract

A theorem due to Cartan asserts that every origin-preserving automorphism of bounded circular domains with respect to the origin is linear. In the present paper, by employing the theory of Bergman's representative domain, we prove that under certain circumstances Cartan's assertion remains true for quasi-circular domains in $\mathbb C^n$. Our main result is applied to obtain some simple criterions for the case $n=3$ and to prove that Braun-Kaup-Upmeier's theorem remains true for our class of quasi-circular domains.