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 $${\mathbb {P} \backslash \{z_1, z_2, z_3\}}$$ with singularities of order α j ≤ 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 (J Reine Angew Math 130:243–258, 1905) and Heins (Nagoya Math J 21:1–60, 1962), while explicit formulas for the cases α 1 = α 2 = 1 were given earlier by Agard (Ann Acad Sci Fenn Ser A I 413, 1968) and recently by Anderson et al. (Math Z, to appear, doi: 10.1007/s00209-009-0560-5 ). We also establish precise and explicit lower bounds for the generalized hyperbolic metric. This extends work of Hempel (J Lond Math Soc II Ser 20:435–445, 1979) and Minda (Complex Var 8:129–144, 1987). As applications, sharp versions of Landau- and Schottky-type theorems for meromorphic functions are obtained.