A Gromov Hyperbolic metric and M\"obius transformations

TL;DR

This paper compares a Gromov hyperbolic metric with classical hyperbolic metrics in the unit ball and upper half space, establishing sharp inequalities and analyzing distortions under Möbius transformations.

Contribution

It introduces sharp comparison and distortion inequalities between Gromov hyperbolic and classical hyperbolic metrics, enhancing understanding of their geometric relationships.

Findings

Established sharp comparison inequalities between Gromov and classical hyperbolic metrics.

Derived sharp distortion inequalities for Gromov hyperbolic metric under Möbius transformations.

Provided insights into the geometric behavior of Gromov hyperbolic metrics.

Abstract

We compare a Gromov hyperbolic metric with the hyperbolic metric in the unit ball or in the upper half space, and prove sharp comparison inequalities between the Gromov hyperbolic metric and some hyperbolic type metrics. We also obtain several sharp distortion inequalities for the Gromov hyperbolic metric under some families of M\"{o}bius transformations.