Visible quasihyperbolic geodesics

Abstract

In this paper, motivated by the work of Bonk, Heinonen, and Koskela (Asterisque, 2001), we consider the problem of the equivalence of the Gromov boundary and Euclidean boundary. Our strategy to study this problem comes from the recent work of Bharali and Zimmer (Adv. Math., 2017) and Bracci, Nikolov, and Thomas (Math. Z., 2021). We present the concept of a quaihyperbolic visibility domain (QH-visibility domain) for domains that meet the visibility property in relation to the quasihyperbolic metric. By utilizing this visibility property, we offer a comprehensive solution to this problem. Indeed, we prove that such domains are precisely the QH-visibility domains that have no geodesic loops in the Euclidean closure. Furthermore, we establish a general criterion for a domain to be the QH-visibility domain. Using this criterion, one can determine that uniform domains, John domains, and domains that satisfy quasihyperbolic boundary conditions are QH-visibility domains. We also compare the visibility of hyperbolic and quasihyperbolic metrics for planar hyperbolic domains. As an application of the visibility property, we study the homeomorphic extension of quasiconformal maps. Moreover, we also study the QH-visibility of unbounded domains in $\mathbb{R}^n$. Finally, we present a few examples of QH-visibility domains that are not John domains or QHBC domains.