Geometry of quasi-circular domains and applications to tetrablock

TL;DR

This paper investigates the geometry of quasi-circular domains, their boundary invariance under holomorphic mappings, and applies these findings to analyze the tetrablock, including its boundary structure and non-convexity.

Contribution

It establishes boundary invariance for proper holomorphic maps in quasi-circular domains and characterizes the Shilov boundary for several complex domains, including the tetrablock.

Findings

Shilov boundary is invariant under proper holomorphic mappings

No non-trivial proper holomorphic self-mappings of the tetrablock exist

Tetrablock is not $ ext{C}$-convex

Abstract

We prove that the Shilov boundary is invariant under proper holomorphic mappings between some classes of domains (containing among others quasi-balanced domains with the continuous Minkowski functionals). Moreover, we obtain an extension theorem for proper holomorphic mappings between quasi-circular domains. Using these results we show that there are no non-trivial proper holomorphic self-mappings in the tetrablock. Another important result of our work is a description of Shilov boundaries of a large class of domains (containing among other the symmetrized polydisc and the tetrablock). It is also shown that the tetrablock is not $\mathbb C$-convex.