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Gromov hyperbolization of unbounded noncomplete spaces and Hamenstädt metric

Qingshan Zhou

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.

Gromov hyperbolization of unbounded noncomplete spaces and Hamenstädt metric

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.

Paper Structure

This paper contains 14 sections, 14 theorems, 102 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and auxiliary results
  3. Notions and notations
  4. Gromov hyperbolic geometry
  5. Quasihyperbolic metric and uniform spaces
  6. Hyperbolization metric
  7. Proofs of Theorem \ref{['thm-1']} and Theorem \ref{['c-1']}
  8. Proof of Theorem \ref{['thm-1']}
  9. Tools for the proof of Theorem \ref{['c-1']}.
  10. Proof of Theorem \ref{['c-1']}.
  11. Proofs of Theorem \ref{['thm-2']} and Theorem \ref{['c-2']}
  12. Hyperbolizing unbounded uniform spaces
  13. Proof of Theorem \ref{['thm-2']}
  14. Proof of Theorem \ref{['c-2']}

Key Result

Theorem 1.1

Let $(X,d)$ be an unbounded locally complete and incomplete metric space. Then there is a natural $\eta$-quasisymmetric map where $\partial_h X\setminus\{\xi\}$ is the punctured Gromov boundary of hyperbolic space $(X,h)$ equipped with a Hamenstädt metric $h_{b,\varepsilon}$ and $b\in \mathcal{B}_h(\xi)$ is a Busemann function with $\varphi(\xi)=\infty$. The function $\eta$ depends only on $\vare

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.9
  • ...and 9 more