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 CP^3 .