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

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.