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arXiv · 0911.0758

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

Daniela Kraus·Oliver Roth·Toshiyuki Sugawa
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Abstract

An explicit formula for the generalized hyperbolic metric on the thrice--punctured sphere $¶\backslash \{z_1, z_2, z_3\}$ with singularities of order $α_j \le 1$ at $z_j$ is obtained in all possible cases $α_1+α_2+α_3 >2$. 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 $α_1=α_2=1$ 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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Daniela Kraus, Oliver Roth, Toshiyuki Sugawa. 2009-11-04. Metrics with conical singularities on the sphere and sharp extensions of the theorems of Landau and Schottky. https://arxiv.org/abs/0911.0758

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