Rigidity of proper holomorphic self-mappings of the hexablock

TL;DR

This paper proves that all proper holomorphic self-maps of the hexablock are automorphisms, confirming a conjecture about its automorphism group structure and contributing to complex analysis in several variables.

Contribution

It establishes the rigidity of proper holomorphic self-maps of the hexablock, showing they are necessarily automorphisms, and resolves a conjecture about its automorphism group.

Findings

Every proper holomorphic self-map of the hexablock is an automorphism.

Confirmed the conjecture that the automorphism group equals the group of all proper self-maps.

Resolved the structure of the automorphism group of the hexablock.

Abstract

The hexablock \(\mathbb{H}\), introduced by Biswas-Pal-Tomar \cite{Hexablock}, is a Hartogs domain in \(\mathbb{C}^4\) fibered over the tetrablock \(\mathbb{E}\) in \(\mathbb{C}^3\), arising in the context of \(\mu\)-synthesis problems. In this paper, we prove that every proper holomorphic self-map of \(\mathbb{H}\) is necessarily an automorphism. Consequently, we resolve the conjecture \(G(\mathbb{H}) = \mathrm{Aut}(\mathbb{H})\) on the automorphism group structure, originally posed by Biswas-Pal-Tomar in \cite{Hexablock}.