Geometric and analytic problems of the theory of ramified coverings of the sphere

TL;DR

This paper explores geometric and analytic aspects of ramified coverings of the Riemann sphere, focusing on elliptic functions, conformal mappings, and applications to approximation theory, providing new methods for analyzing complex tori and related structures.

Contribution

It introduces a system of ODEs for critical points of rational functions and elliptic functions, aiding in conformal mapping and uniformization of complex tori.

Findings

Derived ODE systems for critical points and modules of tori

Applied complex tori to Pade-Hermit approximations

Analyzed structure of quadratic differentials and Riemann surface partitions

Abstract

This is a course of lectures given for students of the Regional Mathematical Center of the Novosibirsk State University from October 20 to November 3, 2017. The course is devoted to some geometric problems of ramified coverings of the Riemann sphere. A special attention is payed to compact surfaces of genus one (complex tori). In the first section we give a short introduction to the theory of elliptic functions. Section 2 is devoted to one-parametric families of holomorphic and meromorphic functions. We recall the role of such families on Loewner's equation in solving some problems of the theory of univalent functions. Further we deduce a system of ODEs expressing dependence of critical points of a family of rational functions from their critical values. This gives an approximate method to find a conformal mapping of the Riemann sphere onto a given simply-connected compact Riemann surface over the sphere. Thereafter a similar problem is solved for elliptic functions uniformizing complex tori over the Riemann sphere. The corresponding system of ODEs also contains a differential equation for the modules of complex tori. In Section 3 we apply complex tori in some problems connected to Pade-Hermit approximations. We study the partition of a 3-sheeted Riemann surface into sheets induced by some abelian integrals (so-called Nattall's partition). In the symmetric case we describe the structure of trajectories of quadratic differentials connected with the abelian integrals.