Geometric theory of meromorphic functions

TL;DR

This paper surveys the relationship between the geometric properties of simply connected Riemann surfaces spread over the sphere and their uniformizing functions, highlighting the interplay between complex analysis and geometric structures.

Contribution

It provides a comprehensive overview of how geometric features of Riemann surfaces influence their uniformizing functions, based on lecture material from 2006.

Findings

Connections between surface geometry and uniformizing functions elucidated

Key results linking geometric and analytic properties summarized

Survey consolidates known results in geometric function theory

Abstract

This is a survey of results on the following problem. Consider a simply connected Riemann surface spread over the Riemann sphere. How are the properties of the uniformizing function of this surface related to the geometric properties of the surface? Based on the lecture in U. Michigan in May 2006.