Gears, Pregears and Related Domains

TL;DR

This paper investigates conformal mappings from the unit disk to gear-shaped domains using the Schwarzian derivative, introducing pregear domains and analyzing their properties and univalence regions.

Contribution

It introduces the concept of pregear domains, relates accessory parameters to conformal modulus, and precisely characterizes the univalence region for these mappings.

Findings

Schwarzian derivative of gear mappings is a rational function.

Accessory parameters relate to conformal modulus.

Univalence region is explicitly determined.

Abstract

We study conformal mappings from the unit disk to one-toothed gear-shaped planar domains from the point of view of the Schwarzian derivative. Gear-shaped (or "gearlike") domains fit into a more general category of domains we call "pregears" (images of gears under Mobius transformations), which aid in the study of the conformal mappings for gears and which we also describe in detail. Such domains being bounded by arcs of circles, the Schwarzian derivative of the Riemann mapping is known to be a rational function of a specific form. One accessory parameter of these mappings is naturally related to the conformal modulus of the gear (or pregear) and we prove several qualitative results relating it to the principal remaining accessory parameter. The corresponding region of univalence (parameters for which the rational function is the Schwarzian derivative of a conformal mapping) is determined precisely.