Gromov hyperbolization of unbounded noncomplete spaces and Hamenst\"adt metric

TL;DR

This paper explores Gromov hyperbolization techniques applied to unbounded, locally complete, and incomplete metric spaces using various hyperbolic metrics, providing new characterizations of uniform domains in Banach spaces.

Contribution

It introduces a Gromov hyperbolic framework for unbounded noncomplete metric spaces with different hyperbolic metrics, linking them to uniform domains in Banach spaces.

Findings

Gromov hyperbolization characterizes unbounded uniform domains in Banach spaces.

Establishes relationships between hyperbolic metrics and space completeness.

Provides new tools for analyzing metric space hyperbolicity.

Abstract

In this paper, we investigate Gromov hyperbolizations of unbounded locally complete and incomplete metric spaces associated with three hyperbolic type metrics: the hyperbolization metric introduced by Ibragimov, the distance ratio metric, and the quasihyperbolic metric. As an application, we obtain a Gromov hyperbolic characterization of unbounded uniform domains in Banach spaces.