Modular functions and Asymptotic geometry on punctured Riemann sphere

Abstract. The asymptotic expansion of the complete K-E metric on the punctured sphere is the dimension one case of the asymptotic expansion on a quasi-projective manifold M\D, which was proposed by S. T. Yau. Several people have worked on this problem by using techniques from partial differential equations. In this talk, I will use the analytic properties of the covering map, the Schwarzian derivatives, and the modular form to derive a precise asymptotic expansion on the punctured sphere. More precisely, the coefficients in the expansion will be uniquely determined up to two parameters. As a consequence, the Kobayashi-Royden metric is given in joint work with G. Cho.