The Carathéodory and Kobayashi Metrics and Applications in Complex Analysis

The Carathéodory and Kobayashi metrics have proved to be important tools in the function theory of several complex variables. But they are less familiar in the context of one complex variable. Our purpose here is to gather in one place the basic ideas about these important invariant metrics for domains in the plane and to provide some illuminating examples and applications. 0 Prefatory Thoughts In the late nineteenth century, Henri Poincaré (1854–1912) introduced the profoundly original idea of equipping the unit disc D in the complex plane with a metric that is invariant under conformal self-maps of D. One may recall (see [GRK]) that the conformal maps of the disc are generated by the rotations ρθ : ζ 7−→ e ζ for 0 ≤ θ < 2π and the Möbius transformations φa : ζ 7−→ ζ − a 1− aζ for a ∈ C, |a| < 1. While rotations certainly preserve Euclidean distance, the Möbius transformations do not—see Figure 1. It is most convenient to describe the Poincaré metric in infinitesimal form. In fact we set ρ(ζ) = 1 1− |ζ |2 .