Revisiting Kobayashi hyperbolicity on planar domains

TL;DR

The paper provides new elementary proofs for the Kobayashi hyperbolicity of certain planar domains, simplifying classical results and offering a new characterization of hyperbolicity in planar domains.

Contribution

It introduces elementary proofs that do not rely on universal covers or negatively curved metrics, and offers a new characterization of Kobayashi hyperbolicity for planar domains.

Findings

Elementary proofs of hyperbolicity for the twice-punctured plane

Proof that bounded domains are complete hyperbolic

A new characterization of hyperbolicity for planar domains

Abstract

We give two new elementary proofs of the complete Kobayashi hyperbolicity of the twice-punctured complex plane. We also present an extremely short proof that bounded domains are complete Kobayashi hyperbolic. Our proofs rely neither on the fact that the universal cover of the twice-punctured plane is the disk nor on the existence of negatively curved metrics. As applications, we present concise proofs of the classical theorems of Landau, Schottky, and Picard. Finally, we provide a characterization of Kobayashi hyperbolicity for planar domains inspired by a similar result of Hahn.