Metrics with conical singularities on the sphere and sharp extensions of the theorems of Landau and Schottky

TL;DR

This paper derives explicit formulas and bounds for generalized hyperbolic metrics with conical singularities on the thrice-punctured sphere, extending classical theorems of Landau and Schottky with new sharp results.

Contribution

It provides explicit formulas and bounds for the hyperbolic metric with conical singularities, extending classical theorems to new sharp versions.

Findings

Explicit formulas for the hyperbolic metric with singularities on the thrice-punctured sphere.

Sharp lower bounds for the generalized hyperbolic metric.

Applications to sharp Landau- and Schottky-type theorems for meromorphic functions.

Abstract

An explicit formula for the generalized hyperbolic metric on the thrice--punctured sphere $\P \backslash \{z_1, z_2, z_3\}$ with singularities of order $\alpha_j \le 1$ at $z_j$ is obtained in all possible cases $\alpha_1+\alpha_2+\alpha_3 >2$. The existence and uniqueness of such a metric was proved long time ago by Picard \cite{Pic1905} and Heins \cite{Hei62}, while explicit formulas for the cases $\alpha_1=\alpha_2=1$ were given earlier by Agard \cite{AG} and recently by Anderson, Sugawa, Vamanamurthy and Vuorinen \cite{A}. We also establish precise and explicit lower bounds for the generalized hyperbolic metric. This extends work of Hempel \cite{Hem79} and Minda \cite{Min87b}. As applications, sharp versions of Landau-- and Schottky--type theorems for meromorphic functions are obtained.