A Study of Some Conformal Metrics and Invariants on Planar Domains

Abstract

The main aim of this thesis is to explain the behaviour of some conformal metrics and invariants near a smooth boundary point of a domain in the complex plane. We will be interested in the invariants associated to the Carathéodory metric such as its higher-order curvatures that were introduced by Burbea and the Aumann-Carathéodory rigidity constant, the Sugawa metric and the Hurwitz metric. The basic technical step in all these is the method of scaling the domain near a smooth boundary point. To estimate the higher-order curvatures using scaling, we generalize an old theorem of Suita on the real analyticity of the Carathéodory metric on planar domains and in the process, we show convergence of the Szeg˝o and Garabedian kernels as well. By using similar ideas we also show that the Aumann-Carathéodory rigidity constant converges to 1 near smooth boundary points. Next on the line is a conformal metric defined using holomorphic quadratic differentials. Thiswas done by T. Sugawa andwe will refer to this as the Sugawa metric. It is shown that this metric is uniformly comparable to the quasi-hyperbolic metric on a smoothly bounded domain. We also study the Hurwitz metric that was introduced by D. Minda. Its construction is similar to the Kobayashi metric but the essential difference lies in the class of holomorphic maps that are considered in its definition. We show that this metric is continuous and also strengthen Minda’s theorem about its comparability with the quasi-hyperbolic metric by estimating the constants in a more natural manner. Finally, we get some weak estimates on the generalized upper and lower curvatures of the Sugawa and Hurwitz metrics.