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

Date
Authors
Kraus, Daniela
Roth, Oliver
Sugawa, Toshiyuki
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
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.
Keywords
Mathematics - Complex Variables, Mathematics - Differential Geometry, 30F45
Citation
Collections