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Gromov hyperbolicity I: the dimension-free Gehring-Hayman inequality for quasigeodesics

Chang-Yu Guo, Manzi Huang, Xiantao Wang

Abstract

This is the first article of a series of our recent works, addressing an open question of Bonk-Heinonen-Koskela [5], to study the relationship between (inner) uniformality and Gromov hyperbolicity in infinite dimensional spaces. Our main focus of this paper is to establish a dimension-free Gehring-Hayman inequality for quasigeodesics. A well-known theorem of J. Heinonen and S. Rohde in 1993 states that if $D\subset \mathbb{R}^n$ is quasiconformally equivalently to an uniform domain, then the Gehring-Hayman inequality holds in $D$: quasihyperbolic geodesics in $D$ minimizes the Euclidean length among all curves in $D$ with the same end points, up to a universal dimension-dependent multiplicative constant. In this paper, we develop a new approach to strengthen the above result in the following three aspects: 1) obtain a dimension-free multiplicative constant in the Gehring-Hayman inequality; 2) relax the class of quasihyperbolic geodesics to more general quasigeodesics; 3) relax the quasiconformal equivalence to more general coarsely quasihyperbolic equivalence. As a byproduct of our general approach, we are able to prove that the above improved Gehring-Hayman inequality indeed holds in Banach spaces. This answers affirmatively an open problem raised by J. Heinonen and S. Rohde in 1993 and reformulated by J. Väisälä in 2005.

Gromov hyperbolicity I: the dimension-free Gehring-Hayman inequality for quasigeodesics

Abstract

This is the first article of a series of our recent works, addressing an open question of Bonk-Heinonen-Koskela [5], to study the relationship between (inner) uniformality and Gromov hyperbolicity in infinite dimensional spaces. Our main focus of this paper is to establish a dimension-free Gehring-Hayman inequality for quasigeodesics. A well-known theorem of J. Heinonen and S. Rohde in 1993 states that if is quasiconformally equivalently to an uniform domain, then the Gehring-Hayman inequality holds in : quasihyperbolic geodesics in minimizes the Euclidean length among all curves in with the same end points, up to a universal dimension-dependent multiplicative constant. In this paper, we develop a new approach to strengthen the above result in the following three aspects: 1) obtain a dimension-free multiplicative constant in the Gehring-Hayman inequality; 2) relax the class of quasihyperbolic geodesics to more general quasigeodesics; 3) relax the quasiconformal equivalence to more general coarsely quasihyperbolic equivalence. As a byproduct of our general approach, we are able to prove that the above improved Gehring-Hayman inequality indeed holds in Banach spaces. This answers affirmatively an open problem raised by J. Heinonen and S. Rohde in 1993 and reformulated by J. Väisälä in 2005.

Paper Structure

This paper contains 24 sections, 53 theorems, 412 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Outline of the proof
  5. Three applications of the techniques and results in following-up works
  6. Preliminaries
  7. List of reserved constants
  8. Coarse quasihyperbolicity and quasiconformality
  9. Quasigeodesics, $\lambda$-curves and uniform domains
  10. Gromov hyperbolicity and Rips spaces
  11. Compactness of $\lambda$-pairs with $\mathfrak{C}$-Condition
  12. Main result
  13. Auxiliary lemmas
  14. Proof of Theorem \ref{['9-15-1']}
  15. Proofs of the main results
  16. ...and 9 more sections

Key Result

Theorem 1

Let $D\subset \mathbb{R}^n$ be a domain that is $K$-quasiconformally equivalent to a $c$-uniform domain. If $\vartheta$ is a quasihyperbolic geodesic in $D$ and if $\gamma$ is any other arc in $D$ with the same end points as $\vartheta$, then where $b>0$ depends only on the dimension $n$, the uniformility constant $c$ and the quasiconformality coefficient $K$.

Figures (12)

  • Figure 1: Illustration of Lemma \ref{['lem-8-12']}
  • Figure 2: Illustration for the proof of second case in Lemma \ref{['lem-2.4']}
  • Figure 3: Illustration for the proof of Theorem \ref{['9-15-1']}
  • Figure 4: Illustration for the construction of $\zeta$-sequence
  • Figure 5: Illustration for the proofs of Lemmas \ref{['lem-3.4-0']} and \ref{['lem-22-09-20']}
  • ...and 7 more figures

Theorems & Definitions (81)

  • Definition 1.1
  • Theorem 1: HR)
  • Theorem 2: HR)
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • ...and 71 more