On Small-type formulae for curves in PSL$(2,\mathbb{C}$)

Abstract

Taking the point of view of Legendrian curves in complex projective 3-space and using the classical Lie–Klein duality of algebraic geometry, we revisit various meromorphic representation formulae for Legendrian and null curves in PSL$(2,\mathbb{C}$. These include Small’s formula for null curves, the formulae for Legendrian curves given by Kokubu, Umehara and Yamada, by Gálvez, Martínez and Milán, and more recently by Ejiri and Takahashi. We discuss their relationships and provide a dictionary for their meromorphic data. Some examples related to the $W$-curves of complex projective 3-space are discussed.