The Caratheodory and Kobayashi metrics and applications in complex analysis

TL;DR

This paper reviews the Caratheodory and Kobayashi metrics, highlighting their significance in complex analysis of one and several variables, with examples and applications to deepen understanding.

Contribution

It consolidates fundamental concepts of these invariant metrics for planar domains and illustrates their applications, which are less familiar in one-variable complex analysis.

Findings

Provides basic ideas and definitions of the metrics.

Includes illustrative examples and applications.

Highlights importance in complex analysis.

Abstract

The Caratheodory and Kobayashi metrics have proved to be important tools in the function theory of several complex variables. But they are less familiar in the context of one complex variable. Our purpose here is to gather in one place the basic ideas about these important invariant metrics for domains in the plane and to provide some illuminating examples and applications.