Twistor interpretation of slice regular functions

Abstract

Given a slice regular function $f:\Omega \subset \mathbb{H}\to \mathbb{H}$, with $\Omega \cap \mathbb{R}\ne \varnothing$, it is possible to lift it to a surface in the twistor space ${\mathbb{CP}}^3$ of ${\mathbb{S}}^4\simeq \mathbb{H}\cup \{\infty \}$ (see~\cite{gensalsto}). In this paper we show that the same result is true if one removes the hypothesis $\Omega \cap \mathbb{R}\ne \varnothing$ on the domain of the function $f$. Moreover we find that if a surface $\mathcal{S}\subset {\mathbb{CP}}^3$ contains the image of the twistor lift of a slice regular function, then $\mathcal{S}$ has to be ruled by lines. Starting from these results we find all the projective classes of algebraic surfaces up to degree 3 in ${\mathbb{CP}}^3$ that contain the lift of a slice regular function. In addition we extend and further explore the so-called twistor transform, that is a curve in $\mathbb{G}{r}_2({\mathbb{C}}^4)$ which, given a slice regular function, returns the arrangement of lines whose lift carries on. With the explicit expression of the twistor lift and of the twistor transform of a slice regular function we exhibit the set of slice regular functions whose twistor transform describes a rational line inside $\mathbb{G}{r}_2({\mathbb{C}}^4)$, showing the role of slice regular functions not defined on $\mathbb{R}$. At the end we study the twistor lift of a particular slice regular function not defined over the reals. This example shows the effectiveness of our approach and opens some questions.

29 pages

References in corpus (4)

• A Bloch-Landau Theorem for slice regular functions
• Luigi Cremona and cubic surfaces
• Landau-Toeplitz theorems for slice regular functions over quaternions
• The Bohr Theorem for slice regular functions

Cited by in corpus (9)

• Slice-Polynomial Functions and Twistor Geometry of Ruled Surfaces in ${\mathbb{CP}}^3$
• On a class of orientation-preserving maps of ${\mathbb{R}}^4$
• S-regular functions which preserve a complex slice
• S--polyregular Bargmann spaces
• A representation formula for slice regular functions over slice-cones in several variables
• Twistor lines on algebraic surfaces
• Eigenvalue problems for slice functions
• Algebraic surfaces with infinitely many twistor lines
• Three Topological Results on the Twistor Discriminant Locus in the 4-Sphere