Holomorphic retracts in the Lie ball and the tetrablock

TL;DR

This paper investigates the properties and classifications of holomorphic retracts within Lempert domains, focusing on the Lie ball and tetrablock, providing new examples and a comprehensive description of these structures.

Contribution

It establishes connections between holomorphic and linear retracts, offers new examples, and fully characterizes holomorphic retracts in the Lie ball and tetrablock domains.

Findings

Holomorphic retracts are linked to linear ones in Lempert domains.

Complete description of retracts in the Lie ball and tetrablock.

New non-trivial examples of holomorphic retracts.

Abstract

In this article, we study various properties of holomorphic retracts in Lempert domains. We associate the existence and the related form of holomorphic retracts with the linear ones, provide non-trivial examples and discuss their properties in a quite general setting. Later we specialize on two Lempert domains which are the Lie ball of dimension three and its $2$-proper holomorphic image, that is, the tetrablock and give a complete description of holomorphic retracts in these domains.