Conformal Mappings Through the Lens of Invariant Metrics

TL;DR

This paper explores the properties of invariant metrics in hyperbolic planar domains, providing new proofs of classical conformal mapping results and characterizing the structure of isotropy groups.

Contribution

It offers novel proofs for key theorems in conformal mapping theory using invariant metrics, including the finiteness of metric balls and the nature of isotropy groups.

Findings

Balls under invariant metrics are finitely-connected.

Conformal self-maps fixing three points are identities.

Isotropy groups are finite or the domain is simply-connected.

Abstract

The main objective of this paper is to show that balls under invariant metrics on hyperbolic planar domains are finitely-connected. As applications, we give new and transparent proofs of classical results on conformal mappings of planar domains. In particular, we show that any conformal self-map of a hyperbolic planar domain with three fixed points is the identity. We also give a new and very simple proof of the theorem by Aumann and Carath\'eodory that states that the isotropy groups of a hyperbolic planar domain are either finite or the domain is simply-connected.