Abstract
We describe, in terms of generalized elliptic integrals, the hyperbolic metric of the twice-punctured sphere with one conical singularity of prescribed order. We also give several monotonicity properties of the metric and a couple of applications.
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Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions. Dover, New York (1972)
Agard S.: Distortion theorems for quasiconformal map**s. Ann. Acad. Sci. Fenn. A I Math. 413, 1–12 (1968)
Ahlfors L.V.: Conformal Invariants. McGraw-Hill, New York (1973)
Anderson G.D., Qiu S.-L., Vamanamurthy M.K., Vuorinen M.: Generalized elliptic integrals and modular equations. Pac. J. Math. 192, 1–37 (2000)
Andrews G., Askey R., Roy R.: Encyclopedia of Mathematics and its Applications, Special Functions, vol. 71. Cambridge University Press, Cambridge (1999)
Beardon A.F., Pommerenke Ch.: The Poincaré metric of plane domains. J. Lond. Math. Soc. (2) 18, 475–483 (1978)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G. (eds): Bateman Manuscript Project Higher. Transcendental Functions, vol. I. McGraw-Hill, New York (1953)
Heins M.: On a class of conformal metrics. Nagoya J. Math. 21, 1–60 (1962)
Kraus D., Roth O.: The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point. Math. Proc. Cambridge Philos. Soc. 145, 643–667 (2008)
Lehto O., Virtanen K.I., Väisälä J.: Contributions to the distortion theory of quasiconformal map**s. Ann. Acad. Sci. Fenn. A I 273, 1–14 (1959)
Nehari Z.: Conformal Map**s. McGraw-Hill, New York (1952)
Nitsche J.: Über die isolierten Singularitäten der Lösungen von Δu = e u. Math Z. 68, 316–324 (1957)
Picard E.: De l’intégration de l’équation Δu = e u sur une surface de Riemann fermée. J. Reine Angew. Math. 130, 243–258 (1905)
Schumacher G., Trapani S.: Variation of cone metrics on Riemann surfaces. J. Math. Anal. Appl. 311, 218–230 (2005)
Yu, A.: Solynin. Functional inequalities via polarization. Algebra i Analiz 8, 148–185 (1996) (in Russian); English transl. in St. Petersburg Math. J. 8, 1015–1038 (1997)
Sugawa T., Vuorinen M.: Some inequalities for the Poincaré metric of plane domains. Math. Z. 250, 885–906 (2005)
Takhtajan L., Zograf P.: Hyperbolic 2-spheres with conical singularities, accessory parameters and K ähler metrics on \({\mathcal {M}_{0,n}}\) . Trans. Am. Math. Soc. 355, 1857–1867 (2003)
Weitsman A.: A symmetric property of the Poincaré metric. Bull Lond. Math. Soc. 11, 295–299 (1979)
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After submission of this paper the authors learned of the death of the third author, M. K. Vamanamurthy, who was born on 5 September 1934 in Mysore, India and died on 6 April 2009 in Auckland, New Zealand. The remaining authors wish to express their appreciation for his co-authorship on this paper, as well as on a series of earlier joint publications, where the foundations for this work were laid.
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Anderson, G.D., Sugawa, T., Vamanamurthy, M.K. et al. Twice-punctured hyperbolic sphere with a conical singularity and generalized elliptic integral. Math. Z. 266, 181–191 (2010). https://doi.org/10.1007/s00209-009-0560-5
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DOI: https://doi.org/10.1007/s00209-009-0560-5