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

TL;DR

This paper studies the detailed asymptotic behavior of higher derivatives of the hyperbolic metric on a thrice-punctured sphere with conical singularities, extending previous explicit formulas.

Contribution

It provides a refined analysis of the asymptotic properties of higher derivatives of the hyperbolic metric near singularities, building on prior explicit formulas.

Findings

Derived asymptotic formulas for higher derivatives near singularities

Enhanced understanding of the hyperbolic metric's local behavior

Extended previous explicit metric formulas with detailed asymptotics

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 $\alpha,\,\beta,\,\gamma \leq 1$ 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.