Gromov hyperbolicity, John spaces and quasihyperbolic geodesics

TL;DR

This paper establishes that in certain geometric spaces, quasihyperbolic geodesics are cone arcs, linking hyperbolic geometry properties with space uniformity through a new approach in metric geometry.

Contribution

It introduces a novel method connecting Gromov hyperbolicity and quasihyperbolic geodesics in John spaces, addressing a longstanding elementary metric geometry question.

Findings

Quasihyperbolic geodesics are cone arcs in John spaces with hyperbolic quasihyperbolization.

A geometric condition is identified that relates space uniformity to Gromov hyperbolic quasihyperbolization.

Provides a new perspective on elementary metric geometry problems.

Abstract

We show that every quasihyperbolic geodesic in a John space admitting a roughly starlike Gromov hyperbolic quasihyperbolization is a cone arc. This result provides a new approach to the elementary metric geometry question, formulated in \cite[Question 2]{Hei89}, which has been studied by Gehring, Hag, Martio and Heinonen. As an application, we obtain a simple geometric condition connecting uniformity of the space with the existence of Gromov hyperbolic quasihyperbolization.