Proper holomorphic mappings between symmetrized ellipsoids

TL;DR

This paper characterizes proper holomorphic mappings between symmetrized ellipsoids, showing their non-existence for non-trivial self-mappings and describing their automorphism groups, advancing understanding of these special complex domains.

Contribution

It provides a complete characterization of proper holomorphic mappings between symmetrized ellipsoids and describes their automorphism groups, a novel result in complex analysis.

Findings

No non-trivial proper holomorphic self-mappings exist for symmetrized ellipsoids.

Proper holomorphic mappings between these domains are characterized explicitly.

Automorphism groups of symmetrized ellipsoids are described.

Abstract

We characterize the existence of proper holomorphic mappings in the special class of bounded $(1,2,...,n)$-balanced domains in $\mathbb{C}^n$, called the symmetrized ellipsoids. Using this result we conclude that there are no non-trivial proper holomorphic self-mappings in the class of symmetrized ellipsoids. We also describe the automorphism groupof these domains.