Twistor transforms of quaternionic functions and orthogonal complex structures

Abstract

The theory of slice-regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains $\Omega$ of ${\mathbb{R}}^4$. 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 $\mathbb{C}{P}^3$.