caa a22 4500 445836121 CHVBK 20180317145330.0 cr unu---uuuuu 170323e20110401xx s 000 0 eng 10.1007/s00209-009-0649-x doi (NATIONALLICENCE)springer-10.1007/s00209-009-0649-x Metrics with conical singularities on the sphere and sharp extensions of the theorems of Landau and Schottky [Elektronische Daten] [Daniela Kraus, Oliver Roth, Toshiyuki Sugawa] 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 etal. (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. Springer-Verlag, 2009 Kraus Daniela Institut für Mathematik, Universität Würzburg, Am Hubland, 97074, Würzburg, Germany aut Roth Oliver Institut für Mathematik, Universität Würzburg, Am Hubland, 97074, Würzburg, Germany aut Sugawa Toshiyuki Division of Mathematics, Graduate School of Information Sciences, Tohoku University, 980-8579, Sendai, Japan aut Mathematische Zeitschrift Springer-Verlag 267/3-4(2011-04-01), 851-868 0025-5874 267:3-4<851 2011 267 209 https://doi.org/10.1007/s00209-009-0649-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00209-009-0649-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Kraus Daniela Institut für Mathematik, Universität Würzburg, Am Hubland, 97074, Würzburg, Germany aut NATIONALLICENCE 700 1- Roth Oliver Institut für Mathematik, Universität Würzburg, Am Hubland, 97074, Würzburg, Germany aut NATIONALLICENCE 700 1- Sugawa Toshiyuki Division of Mathematics, Graduate School of Information Sciences, Tohoku University, 980-8579, Sendai, Japan aut NATIONALLICENCE 773 0- Mathematische Zeitschrift Springer-Verlag 267/3-4(2011-04-01), 851-868 0025-5874 267:3-4<851 2011 267 209 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer