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

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

2009-11-04

arxiv.org/abs/0911.0758v1

Mathematics

Complex Variables

Scientific paper

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.

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