Unique determination of quadratic differentials by their admissible functions

Abstract

Let f be an analytic and one-to-one function on the unit disk such that f(0)=0. Let Q(w)dw^2 be a quadratic differential. Suppose that f maps into the complex plane or the unit disk minus analytic arcs w(t) satisfying Q(w(t))(dw/dt)^2<0. We are interested in the question: if Q is unknown but of a specified form, does f determine the quadratic differential Q uniquely? Our main result is that for functions mapping into the unit disk and quadratic differentials with a pole of order 4 at the origin, the quadratic differential is uniquely determined up to exceptional cases. This question arises in the study of extremal functions for functionals over classes of analytic one-to-one maps.