Asymptotic properties of the hyperbolic metric on the sphere with three conical singularities

Abstract

The explicit formula for the hyperbolic metric ${\lambda }_{\alpha ,\beta ,\gamma }(z)|dz|$ on the thrice-punctured sphere $\mathbb{P}\backslash \{z_1,z_2,z_3\}$ with singularities of order $\alpha ,\beta ,\gamma \le 1$ with $\alpha +\beta +\gamma >2$ at $z_1,z_2,z_3$ was given by Kraus, Roth and Sugawa in \cite{Rothhyper}. In this paper we investigate the asymptotic properties of the higher order derivatives of ${\lambda }_{\alpha ,\beta ,\gamma }(z)$ near the singularity and give some more precise description for the asymptotic behavior.