Carathéodory Convergence of Domains and Hyperbolic Geodesics

In this chapter, we relate the convergence of a sequence of conformal maps to the convergence of the sequence of simply connected image domains. The latter form of convergence was introduced by Carathéodory and is known as kernel convergence. It is shown that if a sequence of proper simply connected domains converges in the sense of Carathéodory then so do their hyperbolic metrics. In the second part of this chapter, we turn to a more in-depth study of hyperbolic geodesics in simply connected domains and, in particular, how the shape of a domain affects the shape of its geodesics. Since geodesics are length-minimising curves with respect to the hyperbolic metric and since, by Koebe’s \(1/4\) -Theorem, the hyperbolic metric is comparable to the reciprocal of the distance to the boundary in a simply connected domain, one would expect hyperbolic geodesics to stay as far away from the boundary as possible. Some results of Jorgensen are described that give this heuristic idea concrete form. The chapter ends with a theorem of Pommerenke to the effect that a hyperbolic geodesic never enters a disk to which it is tangent.