Twistor transforms of quaternionic functions and orthogonal complex structures

TL;DR

This paper explores the application of slice regular quaternionic functions to orthogonal complex structures on R^4, revealing their twistor transforms as holomorphic curves and analyzing specific cases like the parabola complement.

Contribution

It introduces a novel connection between quaternionic slice regular functions and twistor geometry, providing explicit descriptions of their transforms in special cases.

Findings

Twistor transform of quaternionic functions as holomorphic curves in the Klein quadric

Explicit description of the case where the domain is the complement of a parabola

Representation of the transform by a rational quartic surface in CP^3

Abstract

The theory of slice regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains \Omega\ of R^4. When \Omega\ is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which \Omega\ is the complement of a parabola is studied in detail and described by a rational quartic surface in the twistor space CP^3.