Gromov hyperbolicity I: the dimension-free Gehring-Hayman inequality for quasigeodesics
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2025
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| author | Guo, Chang-Yu Huang, Manzi Wang, Xiantao |
| author_facet | Guo, Chang-Yu Huang, Manzi Wang, Xiantao |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_02930 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gromov hyperbolicity I: the dimension-free Gehring-Hayman inequality for quasigeodesics Guo, Chang-Yu Huang, Manzi Wang, Xiantao Complex Variables Metric Geometry 30C65 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. |
| title | Gromov hyperbolicity I: the dimension-free Gehring-Hayman inequality for quasigeodesics |
| topic | Complex Variables Metric Geometry 30C65 |
| url | https://arxiv.org/abs/2502.02930 |