Notes on the Level Curves of a Meromorphic Function

TL;DR

This paper investigates the properties and arrangements of bounded level curves of meromorphic functions, establishing their continuity, conformal equivalences, and providing a new proof of the Gauss–Lucas theorem using level curves.

Contribution

It introduces a detailed analysis of level curves of meromorphic functions, including their continuity, conformal decompositions, and a novel proof of the Gauss–Lucas theorem.

Findings

Bounded level curves are studied in isolation and in relation to each other.

Level curves vary continuously with respect to their modulus parameter.

A natural decomposition of the domain into regions where the function is conformally equivalent to a power map.

Abstract

The subject of this paper is the bounded level curves of a meromorphic function $f$ with domain $G$ such that each component of $\partial{G}$ consists of a level curve of $f$. (A primary example of such a function being a ratio of finite Blaschke products of different degrees, with domain $\mathbb{D}$.) We will first prove several facts about a single bounded level curve of a $f$ in isolation from the other level curves of $f$. We will then study how the level curves of $f$ lie with respect to each other. It is natural to expect that the sets $\{z:|f(z)|=\epsilon\}$ vary continuously as $\epsilon$ varies. We will make this notion explicit, and use this continuity to prove several results about the bounded level curves of $f$. It is well known that if $z_0$ is a zero or a pole of $f$, then $f$ is conformally equivalent to the function $z\mapsto{z^k}$ (for some $k\in\mathbb{Z}$) in a neighborhood of $z_0$. We generalize this fact by finding a natural decomposition of $G$ into finitely many sub-regions (also bounded by level curves of $f$), on each of which $f$ is conformally equivalent to $z\mapsto{z^k}$ (for some $k\in\mathbb{Z}$). Also included is a new proof, using level curves, of the Gauss--Lucas theorem that the critical points of a polynomial are contained in the convex hull of the polynomial's zeros.