Monogenic hull for the n-Cauchy-Fueter operator and twistor theory

TL;DR

This paper introduces the monogenic hull for the n-Cauchy-Fueter operator using twistor theory, showing that monogenic functions extend uniquely to holomorphic functions on this hull, advancing understanding of domains of monogenicity.

Contribution

It establishes the existence of the monogenic hull for the n-Cauchy-Fueter operator using twistor theory, linking monogenic functions to holomorphic extensions.

Findings

Monogenic functions extend uniquely to holomorphic functions on the monogenic hull.

The monogenic hull is explicitly constructed for open subsets of ^n.

Pseudoconvex domains in ^n are shown to be domains of monogenicity.

Abstract

This is the first part in a series of three articles in which are studied the domains of monogenicity for the $n$-Cauchy-Fueter operator. Using the twistor theory, we will in this article show that for a given open subset $U$ of $\mathbb{Q}^n$, there is an open subset $\mathcal{H}(U)$, called the monogenic hull of $U$, of $M_{2n\times 2}^{\mathbb{C}}=\mathbb{Q}^n\otimes\mathbb{C}$ such that each monogenic function in $U$ extends to a unique pair of holomorphic functions on $\mathcal{H}(U)$. In the second part of the series we will exploit the twistor theory furthermore to prove that any pseudoconvex domain in $\mathbb{Q}^n$ is a domain of monogenicity. In the third part of the series, we show the other implication and provide a geometric characterization of the domains of monogenicity.