Given a slice regular function $f:\Omega\subset\mathbb{ H}\to \mathbb{ H}$, with $\Omega\cap\mathbb{ R}\neq \emptyset$, 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 $\Omega\cap\mathbb{ R}\neq \emptyset$ 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.
Categories: math.CV math.DG