Holomorphicity of slice-regular functions

TL;DR

This paper explores a new complex geometric approach to better understand the relationship between slice-regular quaternionic functions and classical holomorphic functions, aiming to deepen the theoretical connection between these two areas.

Contribution

It introduces a geometric method to relate the values of slice-regular functions with holomorphic maps, enhancing understanding of their connection.

Findings

Establishes a geometric link between slice-regular and holomorphic functions.

Provides new insights into the behavior of slice-regular functions.

Bridges the gap between quaternionic analysis and complex holomorphic functions.

Abstract

Slice-regular functions of a quaternionic variable have been studied extensively in the last 12 years, resulting, in many ways, quite close to classical holomorphic functions of a complex variable; indeed, there is a correspondence between slice-regular functions and a certain family of holomorphic maps from the complex plane to $\mathbb{C}^4$, as noted by Ghiloni and Perotti. However, such a construction does not seem to offer any insight on the behaviour of slice-regular functions, due to the lack of a connection between the values of the holomorphic map and the values of the associated sliceregular function. The aim of this work is to show that there is indeed a (complex) geometric way to relate the values of this two functions, thus relating more deeply the world of holomorphic functions with that of slice-regular functions.