A domain in $\mathbb C^4$ and its connection with $\mu$-synthesis problem

TL;DR

This paper investigates a specific domain in complex four-dimensional space, compares it with known domains related to $$-synthesis, and establishes its unique properties and connections within complex analysis.

Contribution

It introduces a new domain $$ in $^4$, characterizes it differently, and explores its relationship with domains arising from $$-synthesis problems, clarifying their biholomorphic distinctions.

Findings

The domain $$ is not biholomorphic to the hexablock.

Alternative characterizations of $$ are provided.

$$ is connected to domains associated with $$-synthesis, such as the symmetrized bidisc and tetrablock.

Abstract

We explore a domain $\mathbb F$ in $\mathbb C^4$ that has structural similarities with the hexablock $\mathbb{H} \subset \mathbb C^4$. It leads to the question if these two domains are biholomorphic. In this paper, we answer this question in negative. We provide alternative characterizations of the domain $\mathbb{F}$ and find its connection with the domains associated with $\mu$-synthesis problem such as the symmetrized bidisc, the tetrablock, the pentablock and the hexablock. We also address the following question: does $\mathbb{F}$ (which is biholomorphic with $\mathbb L_4$) arise from a $\mu$-synthesis problem in the same manner as $\mathbb{G}_2$ and $\mathbb{E}$ ?