Given a slice regular function f:Ω⊂H→H, with Ω∩R≠∅, it is possible to lift it to surfaces in the twistor space CP3 of S4≃H∪{∞} (see Gentili et al., 2014). In this paper we show that the same result is true if one removes the hypothesis Ω∩R≠∅ on the domain of the function f. Moreover we find that if a surface S⊂CP3 contains the image of the twistor lift of a slice regular function, then 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 CP3 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 Gr2(C4) 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 Gr2(C4), showing the role of slice regular functions not defined on 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.

Twistor interpretation of slice regular functions

Altavilla A.
2018

Abstract

Given a slice regular function f:Ω⊂H→H, with Ω∩R≠∅, it is possible to lift it to surfaces in the twistor space CP3 of S4≃H∪{∞} (see Gentili et al., 2014). In this paper we show that the same result is true if one removes the hypothesis Ω∩R≠∅ on the domain of the function f. Moreover we find that if a surface S⊂CP3 contains the image of the twistor lift of a slice regular function, then 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 CP3 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 Gr2(C4) 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 Gr2(C4), showing the role of slice regular functions not defined on 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11586/265128
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