Rigidity of Proper Holomorphic Self-mappings of the Pentablock

TL;DR

This paper proves that all proper holomorphic self-mappings of the pentablock are automorphisms, establishing a rigidity property for these mappings in a complex geometric setting.

Contribution

It demonstrates that any proper holomorphic self-map of the pentablock is necessarily an automorphism, confirming a rigidity phenomenon for this domain.

Findings

Proper holomorphic self-maps of the pentablock are automorphisms.

The result extends understanding of automorphism groups of complex domains.

The paper characterizes the structure of self-mappings in a non-smooth, pseudoconvex domain.

Abstract

The pentablock is a Hartogs domain over the symmetrized bidisc. The domain is a bounded inhomogeneous pseudoconvex domain, and does not have a $\mathcal{C}^{1}$ boundary. Recently, Agler-Lykova-Young constructed a special subgroup of the group of holomorphic automorphisms of the pentablock, and Kosi\'nski completely described the group of holomorphic automorphisms of the pentablock. The purpose of this paper is to prove that any proper holomorphic self-mapping of the pentablock must be an automorphism.