Which Fueter-regular functions are holomorphic?

TL;DR

This paper classifies Fueter-regular quaternionic functions based on the complex linearity of their differentials, revealing that most are not holomorphic with respect to any complex structure, with special cases being uniquely holomorphic or affine transformations.

Contribution

It provides a classification of Fueter-regular functions by analyzing the complex linearity of their differentials, linking holomorphy to specific geometric structures.

Findings

Most Fueter-regular functions are not holomorphic with respect to any complex structure.

When holomorphic, functions are typically uniquely so with respect to a specific complex structure.

Holomorphicity with respect to multiple structures is limited to affine transformations or constants.

Abstract

We provide a classification of Fueter-regular quaternionic functions $f$ in terms of the degree of complex linearity of their real differentials $df$. Quaternionic imaginary units define orthogonal almost-complex structures on the tangent bundle of the quaternionic space by left or right multiplication. Every map of two complex variables that is holomorphic with respect to one of these structures defines a Fueter-regular function. We classify the differential $df$ of a Fueter-regular function $f$, roughly speaking, in terms of how many choices of complex structures make $df$ complex linear. It turns out that, generically, $f$ is not holomorphic with respect to any choice of almost-complex structures. In the special case when it is indeed holomorphic, generically there is a unique choice of almost-complex structures making it holomorphic. The case of holomorphy with respect to several choices of almost-complex structures is limited to conformal real affine transformations or constants.