Rational maps as Schwarzian primitives

Guizhen Cui (1) , Yan Gao (2) , Lei Tan (3) , Rugh Hans Henrik (4)

1 Institute of Mathematics
2 SCU - Sichuan University [Chengdu]
3 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
4 Faculté des Sciences d’Orsay, Université Paris-Sud, 91405 Orsay Cedex, France.

We study necessary and sufficient conditions for a meromorphic quadratic differential with prescribed poles to be the Schwarzian derivative of a rational map. We give geometric interpretations of these conditions. We also study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case, the analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball.

Mots clés: Schwarzian derivaties; rational maps; meromorphic differential

Domaines: Variables complexes [math.CV]

Guizhen Cui, Yan Gao, Lei Tan, Rugh Hans Henrik. Rational maps as Schwarzian primitives. 2016. ⟨hal-01228525v2⟩