Rational maps as Schwarzian primitives

TL;DR

This paper investigates the conditions under which a meromorphic quadratic differential can be realized as the Schwarzian derivative of a rational map, providing geometric insights and analyzing pole dependencies, especially in the cubic case.

Contribution

It establishes necessary and sufficient conditions for such quadratic differentials to be Schwarzian derivatives of rational maps and explores geometric and pole-dependency aspects.

Findings

Conditions characterized for quadratic differentials as Schwarzian derivatives

Geometric interpretation of these conditions provided

Pole dependency failure linked to regular ideal tetrahedron configuration

Abstract

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.