Rational maps as Schwarzian primitives

We examine when a meromorphic quadratic differential ϕ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of ϕ around each pole c, the most singular term should take the form (1 − d2)/(2(z − c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles (i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko (2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by ϕ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative. Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball.