Hyperbolic Metric, Punctured Riemann Sphere and Modular Functions

TL;DR

This paper derives detailed asymptotic expansions of the Kähler-Einstein metric on punctured Riemann spheres with multiple omitted points, using Schwarzian derivatives and modular forms to explicitly compute coefficients.

Contribution

It provides explicit formulas for the metric's expansion coefficients, linking them to modular functions and Schwarzian derivatives, advancing understanding of complex hyperbolic geometry.

Findings

Explicit asymptotic expansion of the metric for multiple punctures

Polynomial coefficients determined by the positions of omitted points

Explicit formulas for special cases with 3, 4, 6, or 12 punctures

Abstract

We derive a precise asymptotic expansion of the complete K\"{a}hler-Einstein metric on the punctured Riemann sphere with three or more omitting points. By using Schwarzian derivative, we prove that the coefficients of the expansion are polynomials on the two parameters which are uniquely determined by the omitting points. Futhermore, we use the modular form and Schwarzian derivative to explicitly determine the coefficients in the expansion of the complete K\"{a}hler-Einstein metric for punctured Riemann sphere with $3, 4, 6$ or $12$ omitting points.