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Gromov hyperbolicity of intrinsic metrics from isoperimetric inequalities

Tianqi Wang, Andrew Zimmer

Abstract

In this paper we investigate the Gromov hyperbolicity of the classical Kobayashi and Hilbert metrics, and the recently introduced minimal metric. Using the linear isoperimetric inequality characterization of Gromov hyperbolicity, we show if these metrics have an "expanding property" near the boundary, then they are Gromov hyperbolic. This provides a new characterization of the convex domains whose Hilbert metric is Gromov hyperbolic, a new proof of Balogh-Bonk's result that the Kobayashi metric is Gromov hyperbolic on a strongly pseudoconvex domain, a new proof of the second author's result that the Kobayashi metric is Gromov hyperbolic on a convex domain with finite type, and a new proof of Fiacchi's result that the minimal metric is Gromov hyperbolic on a strongly minimally convex domain. We also characterize the smoothly bounded convex domains where the minimal metric is Gromov hyperbolic.

Gromov hyperbolicity of intrinsic metrics from isoperimetric inequalities

Abstract

In this paper we investigate the Gromov hyperbolicity of the classical Kobayashi and Hilbert metrics, and the recently introduced minimal metric. Using the linear isoperimetric inequality characterization of Gromov hyperbolicity, we show if these metrics have an "expanding property" near the boundary, then they are Gromov hyperbolic. This provides a new characterization of the convex domains whose Hilbert metric is Gromov hyperbolic, a new proof of Balogh-Bonk's result that the Kobayashi metric is Gromov hyperbolic on a strongly pseudoconvex domain, a new proof of the second author's result that the Kobayashi metric is Gromov hyperbolic on a convex domain with finite type, and a new proof of Fiacchi's result that the minimal metric is Gromov hyperbolic on a strongly minimally convex domain. We also characterize the smoothly bounded convex domains where the minimal metric is Gromov hyperbolic.

Paper Structure

This paper contains 26 sections, 54 theorems, 342 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Expanding metrics
  3. Generalized quasi-hyperbolic metrics
  4. Outline of paper
  5. Preliminaries
  6. Possibly ambiguous notations
  7. Geodesic, length, and proper metric spaces
  8. Gromov hyperbolic spaces
  9. Pseudometrics
  10. Expanding metrics
  11. Quasi-normal vectors and distance to the boundary
  12. Uniformly transverse splittings
  13. The upper bound
  14. The lower bound
  15. The Moreover Part
  16. ...and 11 more sections

Key Result

Theorem 1.1

Suppose $M$ is a compact manifold and $\left\|\cdot\right\|$ is a strongly integrable pseudo-metric on $M \times (0,1]$ with the following properties: Then the distance on $M \times (0,1]$ induced by $\left\|\cdot\right\|$ is Gromov hyperbolic.

Figures (1)

  • Figure 1: The many points in the proof of Proposition \ref{['prop:definite contraction']}

Theorems & Definitions (98)

  • Theorem 1.1: see Theorem \ref{['thm:expanding metrics']}
  • Corollary 1.2: Balogh--Bonk BaloghBonk2000
  • proof
  • Corollary 1.3: Fiacchi Fiacchi2022
  • Example 1.4
  • Theorem 1.5: see Theorems \ref{['thm:2 implies 1 in main theorem']} and \ref{['thm:1 implies 2 in main theorem']}
  • Theorem 1.6: see Theorem \ref{['thm:smooth case']}
  • Corollary 1.7: Z. ZimmerMathAnn2016
  • Corollary 1.8: see Corollary \ref{['cor:characterization in smooth case for minimal metric in paper']}
  • Proposition 2.1
  • ...and 88 more