Asymptotic properties of the hyperbolic metric on the sphere with three conical singularities
Bull. Korean Math. Soc. 2014 Vol. 51, No. 5, 1485-1502
https://doi.org/10.4134/BKMS.2014.51.5.1485
Printed September 30, 2014
Tanran Zhang
Tohoku University
Abstract : The explicit formula for the hyperbolic metric $\lambda_{\alpha,\,\beta,\,\gamma}(z)|dz|$ on the thrice-punctured sphere $\mathbb{P} \backslash \{0,\,1,\,\infty\}$ with singularities of order $0<\alpha,\,\beta<1$, $\gamma \leq 1$, $\alpha+\beta+\gamma>2$ at $0,\,1,\,\infty$ was given by Kraus, Roth and Sugawa in \cite{Rothhyper}. In this article we investigate the asymptotic properties of the higher order derivatives of $\lambda_{\alpha,\,\beta,\,\gamma}(z)$ near the origin and give more precise descriptions for the asymptotic behavior of $\lambda_{\alpha,\,\beta,\,\gamma}(z)$.
Keywords : conical singularities, hyperbolic metrics, special functions
MSC numbers : 30L10, 33D15
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