The twistor lifts of surfaces in 4-spaces

TL;DR

The article surveys the twistor lifts of surfaces in four-dimensional spaces, focusing on $E^4$ and formulating the Gauss maps and twistor lifts via orthogonal complex structures. It develops the Gauss-Weingarten framework for $E^4$, defines the mean curvature vector, and introduces isothermal coordinates to simplify the fundamental equations, showing how minimal surfaces yield harmonic coordinate functions and holomorphic relations among auxiliary quantities. The core twistorial construction then represents oriented 2-planes and twistor lifts through $\Sigma_+$ and $\Sigma_-$, with explicit expressions for the lifts $F_{\pm}$ in terms of holomorphic data $\psi^i$ and bivectors $I_{\varepsilon,k}$, and proves that minimality enforces holomorphicity of associated data like $g_+$. The work further connects these local twistorial objects to global 4D geometry, isotropy criteria, and, in the sphere case $S^4$, to Penrose-style twistor maps, thereby linking classical surface theory in $E^4$ with twistorial methods in 4-manifolds. Overall, it provides a detailed, calculable bridge between minimal surface theory in four dimensions and twistorial geometry, highlighting when twistor lifts are holomorphic or constant and how isotropy emerges from the underlying linear algebra of $SO(4)$ and its coverings.

Abstract

This is a survey of the twistor lifts of surfaces in 4-spaces. In most part of this survey, the space is the Euclidean $4$-space $E^4$. The definitions of the Gauss maps and the twistor lifts of surfaces in $E^4$ are given by complex structures of $E^4$. Based on these definitions, we can understand that the Gauss maps of minimal surfaces are holomorphic and characterize isotropicity of minimal surfaces.

Theorems & Definitions (20)

• Proposition 1.1
• Proposition 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 2.5
• Proposition 2.6
• Proposition 3.1