#### Date of Completion

5-7-2020

#### Embargo Period

5-7-2020

#### Keywords

Kahler-Einstein metric, hyperbolic geometry, modular form, modular functions, Schwarzian derivative, punctured sphere, quasi-projective manifold, asymptotic metric

#### Major Advisor

Damin Wu

#### Associate Advisor

Liang Xiao

#### Associate Advisor

Guozhen Lu

#### Associate Advisor

Maria Gordina

#### Field of Study

Mathematics

#### Degree

Doctor of Philosophy

#### Open Access

Open Access

#### Abstract

In the first chapter, 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. This new technique is at the intersection of analysis and algebra. 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 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.

The second chapter gives an explicit formula of the asymptotic expansion of the Kobayashi-Royden metric on the punctured sphere $\mathbb{CP}^1\backslash\{0,1,\infty\}$ in terms of the exponential Bell polynomials. We prove a local quantitative version of the Little Picard's theorem as an application of the asymptotic expansion. Furthermore, the explicit formula of the metric and the conclusion regarding the coefficients apply to more general cases of $\mathbb{CP}^1\backslash\{a_1,\ldots,a_n\}$, $n\ge 3$ as well, and the metric on $\mathbb{CP}^1\backslash\{0,\frac{1}{3},-\frac{1}{6}\pm\frac{\sqrt{3}}{6}i\}$ will be given as a concrete example of our results.

#### Recommended Citation

Qian, Junqing, "Modular Functions and Asymptotic Geometry on Punctured Riemann Spheres" (2020). *Doctoral Dissertations*. 2516.

https://digitalcommons.lib.uconn.edu/dissertations/2516