# Level Curve Configurations and Conformal Equivalence of Meromorphic Functions

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## Abstract

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 configuration of 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 its 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 use this to 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 Bôcher (which is an extension of the Gauss–Lucas theorem to rational functions) using level curves.

## Keywords

Complex analysis Meromorphic functions Level curves Critical points Critical values## Mathematics Subject Classification

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