Level Curve Configurations and Conformal Equivalence of Meromorphic Functions

Let $f=B_1/B_2$ be a ratio of finite Blaschke products having no critical points on $\partial\mathbb{D}$. Then $f$ has finitely many critical level curves (level curves containing critical points of $f$) in the disk, and the non-critical level curves interpolate smoothly between the critical level curves. Thus, to understand the geometry of all the level curves of $f$, one needs only understand the finitely many critical level curves of $f$. In this paper, we show that in fact such a function $f$ is determined not just geometrically but conformally by the configuration of critical level curves. That is, if $f_1$ and $f_2$ have the same configuration of critical level curves, then there is a conformal map $\phi$ such that $f_1\equiv f_2\circ\phi$. We then show that every configuration of critical level curves which could come from an analytic function is instantiated by a polynomial. We also include a new proof of a theorem of B\^{o}cher (which is an extension of the Gauss--Lucas theorem to rational functions) using level curves.