Hyperbolic Geometry in Simply Connected Domains

Abstract

In this chapter we introduce hyperbolic metric and hyperbolic distance in simply connected domains and we introduce the basic notion of geodesics as distance minimizing length curves. It turns out that there is a one-to-one correspondence between geodesic rays and prime ends and the cluster set of each geodesic is the principal part of the corresponding prime end. Then we consider useful estimates of hyperbolic metric and distance in simply connected domains depending on the distance from the boundary. The chapter ends with some detailed estimates on the hyperbolic distance in the half-plane.