Some Remarks on the Carathéodory and Szegő Metrics on Planar Domains

Abstract

We study several intrinsic properties of the Carathéodory and Szegő metrics on finitely connected planar domains. Among them are the existence of closed geodesics and geodesic spirals, boundary behaviour of Gaussian curvatures, and -cohomology. A formula for the Szegő metric in terms of the Weierstrass -function is obtained. Variations of these metrics and their Gaussian curvatures on planar annuli are also studied. Consequently, we observe that the optimal universal upper bound for the Gaussian curvature of the Szegő metric is 4 and that no universal lower bounds exist for the Gaussian curvatures of the Carathéodory and Szegő metrics. Moreover, it follows that there are domains where the Gaussian curvature of the Szegő metric assumes both negative and positive values. Lastly, it is also observed that there is no universal upper bound for the ratio of the Szegő and Carathéodory metrics.