TY - JOUR
T1 - Metrics with conical singularities on the sphere and sharp extensions of the theorems of Landau and Schottky
AU - Kraus, Daniela
AU - Roth, Oliver
AU - Sugawa, Toshiyuki
N1 - Funding Information:
D. Kraus and O. Roth were supported by a DFG grant (RO 3462/3-1). T. Sugawa was supported in part by JSPS Grant-in-Aid for Scientific Research (B), 17340039 and for Exploratory Research, 19654027.
PY - 2011/4
Y1 - 2011/4
N2 - An explicit formula for the generalized hyperbolic metric on the thrice-punctured sphere ℙ\{z1, z2, z3} with singularities of order αj ≤ 1 at zj is obtained in all possible cases α1 + α2 + α3 > 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 α1 = α2 = 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.
AB - An explicit formula for the generalized hyperbolic metric on the thrice-punctured sphere ℙ\{z1, z2, z3} with singularities of order αj ≤ 1 at zj is obtained in all possible cases α1 + α2 + α3 > 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 α1 = α2 = 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.
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U2 - 10.1007/s00209-009-0649-x
DO - 10.1007/s00209-009-0649-x
M3 - Article
AN - SCOPUS:79952574397
VL - 267
SP - 851
EP - 868
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
SN - 0025-5874
IS - 3
ER -