University of Connecticut OpenCommons@UConn Doctoral Dissertations University of Connecticut Graduate School 5-7-2020 Modular Functions and Asymptotic Geometry on Punctured Riemann Spheres Junqing Qian University of Connecticut - Storrs,
[email protected] Follow this and additional works at: https://opencommons.uconn.edu/dissertations Recommended Citation Qian, Junqing, "Modular Functions and Asymptotic Geometry on Punctured Riemann Spheres" (2020). Doctoral Dissertations. 2516. https://opencommons.uconn.edu/dissertations/2516 Modular Functions and Asymptotic Geometry on Punctured Riemann Spheres Junqing Qian, Ph.D. University of Connecticut, 2020 ABSTRACT In the first chapter, we derive a precise asymptotic expansion of the complete K¨ahler-Einsteinmetric on the punctured Riemann sphere with three or more omit- ting points. This new technique is at the intersection of analysis and algebra. By using Schwarzian derivative, we prove that the coefficients of the expansion are poly- nomials on the two parameters which are uniquely determined by the omitting points. Futhermore, we use the modular form and Schwarzian derivative to explicitly deter- mine the coefficients in the expansion of the complete K¨ahler-Einstein metric for punctured Riemann sphere with 3; 4; 6 or 12 omitting points. The second chapter gives an explicit formula of the asymptotic expansion of the 1 Kobayashi-Royden metric on the punctured sphere CP nf0; 1; 1g in terms of the ex- ponential Bell polynomials. We prove a local quantitative version of the Little Picard's theorem as an application of the asymptotic expansion. Furthermore, the explicit for- mula of the metric and the conclusion regarding the coefficients apply to more general p 1 1 1 1 3 cases of CP nfa1; : : : ; ang, n ≥ 3 as well, and the metric on CP nf0; 3 ; − 6 ± 6 ig will be given as a concrete example of our results.