Non-smooth paths having unit speed with respect to the Kobayashi metric, and their applications

TL;DR

This paper explores whether non-constant absolutely continuous paths can be reparametrized to have unit speed in the Kobayashi metric, providing an answer and discussing various applications of this property.

Contribution

It establishes conditions under which such reparametrizations are possible and explores their implications in complex analysis.

Findings

Confirmed the existence of reparametrizations to unit speed in the Kobayashi metric under certain conditions

Identified subtleties involved in reparametrizing paths to unit speed

Discussed applications in complex geometry and analysis

Abstract

We investigate the question of whether a non-constant absolutely continuous path can be reparametrized to be of unit speed with respect to the Kobayashi metric and be absolutely continuous. Even when the answer is "Yes," which isn't always the case, its proof involves some subtleties. We answer the above question and discuss several applications.