HYPERBOLIC METRIC, PUNCTURED RIEMANN SPHERE, AND MODULAR FUNCTIONS

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