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An elementary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2,C)

Masatoshi Kokubu·Masaaki Umehara·Kotaro Yamada
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Abstract

For null curves in PSL(2,C), there exists a representation formula in terms of two meromorphic functions and their derivatives (Small's formula). In this paper, we give an elementary proof of Small's formula. Moreover, a similar formula for Legendrian curves in PSL(2,C) is given. As null curves in PSL(2,C) are related to mean curvature one surfaces in hyperbolic 3-space H^3, Legendrian curves are related to flat surfaces in H^3. So, as an application of Small-type formula for Lengendrian curves, we give new examples of flat surfaces in H^3.

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Masatoshi Kokubu, Masaaki Umehara, Kotaro Yamada. 2003-02-21. An elementary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2,C). https://arxiv.org/abs/math/0209258

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