On the Bergman metric of an annulus.

Abstract

In this thesis we investigate the distortion caused by an analytic map f : B → W, between the Bergman metric dsB of a bounded domain B in the complex z-plane and the Euclidean metric of the complex w-plane W. We deduce sharp inequalities in the cases of the unit disc, the punctured hyperbolic plane and the annuli, and the minimizing maps are determined.