A study of the Carathéodory and Szegő metrics on planar domains

Abstract

We study several intrinsic properties of the Carathéodory and Szeg˝o metrics on finitely connected planar domains. Among them are the existence of closed geodesics and geodesic spirals, boundary behaviour of Gaussian curvatures, and L^2-cohomology. A formula for the Szeg˝o 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 obtain optimal universal upper bounds for their Gaussian curvatures while no universal lower bounds exist for their Gaussian curvatures. Moreover, it follows that there are domains where the Gaussian curvatures of the Szeg˝o metric assume both negative and positive values. Furthermore, we have established the existence of domains where the Gaussian curvatures of the Bergman and Szeg˝o metrics have opposite signs. Lastly, it is also observed that there is no universal upper bound for the ratio of the Szeg˝o and Carathéodory metrics.