Continuity and bi-Lipschitz properties of the Hurwitz and its invariant metrics

TL;DR

This paper investigates the continuity of the Hurwitz metric in complex domains and introduces a new invariant metric that is bi-Lipschitz equivalent to it, along with its fundamental properties.

Contribution

It presents a new invariant metric bi-Lipschitz equivalent to the Hurwitz metric and analyzes its continuity and basic properties in hyperbolic domains.

Findings

The Hurwitz metric is continuous in arbitrary proper subdomains.

A new invariant metric is introduced and shown to be bi-Lipschitz equivalent to the Hurwitz metric.

The invariant metric exhibits lower semi-continuity and other fundamental properties.

Abstract

This paper attempts to study the continuity of the Hurwitz metric in arbitrary proper subdomains of the complex plane and to introduce a new invariant metric bi-Lipschitz equivalent to the Hurwitz metric in hyperbolic domains. The lower semi-continuity and other basic properties of this invariant metric are also presented.