The quasihyperbolic metric and associated estimates on the hyperbolic metric

Abstract

In this paper we investigate the geometry of the quasihyperbolic metric of domains in R". This metric arises from the conformally flat generalized Riemannian metric d(x, aD)-'ldx I. Due to the fact that the density cl(x, aD) -t is not necessarily differentiable, the classical theories of Riemannian geometry do not apply to this metric. The quasihyperbolic metric has been found to have many interesting and varied applications in geometric function theory. In particular, quasiconformal mappings are quasi-isometries of this metric for sufficiently far-lying points, also bounds on the quasihyperbolic metric in terms of other metrics imply that a domain is uniform which then implies certain injectivity criteria for locally-Lipschitz mappings, amongst others. In fact there is quite a strong relationship between uniform domains and the quasihyperbolic metric. Most of these basic results on the quasihyperbolic metric can be found in [3], [2] and [5]. We note here that the quasihyperbolic metric is complete and generates the usual topology on a proper subdomain of R". Further, geodesics (length minimizing curves) always exist for this metric and these geodesics have Lipschitz continuous first derivatives, which is in fact best possible. We begin by calculating the quasihyperbolic metric, its curvature and geodesics for some nontrivial examples in R'. We also calculate the possible isometries and show that they are conformal mappings, and so when n > 2 are M6bius transformations. We then turn to the planar case for a more detailed investigation. Here it becomes possible to compute the Gaussian curvature of the quasihyperbolic metric in some basic examples and use these for comparison in more general cases. In particular, we show that the quasihyperbolic metric of a planar domain is an S-K metric, in the sense of Heins, if and only if the domain is convex. We denote euclidean n-space by R" and its one point compactification by R'.