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

TL;DR

This paper establishes a dimension-free Gehring-Hayman inequality for quasigeodesics, extending previous results to infinite-dimensional spaces and more general classes of curves, thus advancing the understanding of Gromov hyperbolicity.

Contribution

It introduces a new approach to achieve a dimension-free inequality for quasigeodesics and extends the results to Banach spaces, answering longstanding open problems.

Findings

Dimension-free multiplicative constant achieved

Inequality extended to quasigeodesics and coarsely quasihyperbolic equivalence

Valid in Banach spaces, affirming previous open questions

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. V\"{a}is\"{a}l\"{a} in 2005.