Some remarks on the Carath\'eodory and Szeg\"o metrics on planar domains

TL;DR

This paper investigates intrinsic properties of Carathéodory and Szeg"o metrics on finitely connected planar domains, including geodesics, boundary behavior, curvature bounds, and metric ratios, providing new formulas and bounds.

Contribution

It introduces a formula for the Szeg"o metric using the Weierstrass -function and establishes optimal curvature bounds and the variability of the metrics on planar annuli.

Findings

Existence of closed geodesics and spirals in these metrics

Optimal universal upper bounds for Gaussian curvatures

No universal lower bounds for Gaussian curvatures and unbounded metric ratios

Abstract

We study several intrinsic properties of the Carath\'eodory 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 $\wp$-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 and show that no universal lower bounds exist for their Gaussian curvatures. Moreover, it follows that there are domains where the Gaussian curvature of the Szeg\"o metric assumes both negative and positive values. Lastly, it is also observed that there is no universal upper bound for the ratio of the Szeg\"o and Carath\'eodory metrics.