Shu-Jing Gao | Searcharxiv

arXiv2026

Uniformizing non-proper Gromov Hyperbolic Spaces

In this paper, we extend a large part of the uniformization theory of Bonk-Heinonen-Koskela [Asterisque 2001] to length spaces that are not necessarily proper or geodesic. Among other things, we show that there is a one-to-one correspondence between the quasiisometry classes of complete roughly starlike Gromov hyperbolic spaces and the quasisimilarity classes of bounded uniform spaces, which provides an affirmative solution to an open question of Bonk-Heinonen-Koskela. Our approach relies crucially on the work of Väisälä [Expo. Math. 2005], who investigated in depth Gromov hyperbolic spaces that are not necessarily proper or geodesic. One key new ingredient is to use the so-called (quasihyperbolic) $(c,μ)$-quasigeodesic as a suitable substitute for quasihyperbolic geodesic.

math.CV↗

arXiv2025

Characterizations of quasihyperbolic John domains and uniform domains in metric spaces

In a recent work of Zhou and Ponnusamy [Ann. Sc. Norm. Super. Pisa Ci. Sci. 2025], the authors studied the following natural question: find sufficient and necessary conditions for a domain $Ω$ in a metric space $X$ to be quasihyperbolic John. It was proved that Gromov hyperbolic John domains are quasihyperbolic John, quantitatively. As an application, they obtained a characterization of uniform domains in Ahlfors regular spaces. In a recent work, using a deep improved characterization of Gromov hyperbolicity, Guo, Huang and Wang [arXiv 2025] proved the quantitative equivalence bteween inner uniformity and the quasihyperbolic John condition in metric doubling spaces. However, the proof does not yield a similar characterization for uniform domains. In this article, we find a new elementary approach to successfully extend the above characterization to uniform domains: a domain $Ω$ in a doubling length space $X$ is uniform if and only if it is linearly locally connected (LLC) and satisfies the ball separation condition, if and only if it is LLC-1 and quasihyperbolic John, quantitatively. This substantially improved the corresponding results of Zhou and Ponnusamy. Our new approach also allows us to give an alternative proof of the inner uniformity result of Guo-Huang-Wang without using the improved characterization on Gromov hyperbolicity.

math.CV↗