This thesis consists of three contributions to the theory of complex approximation on Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E, then it is “usually” not possible to approximate f uniformly by functions holomorphic on all of R. In Chapter 2, we show, however, that for every open Riemann surface...

The purpose of these notes is to give an overview of Riemann surfaces, and their moduli spaces. In general, theorems will be presented without proof, although references will be provided. These notes were originally prepared for the summer 2004 session of the UIUC Graduate String Seminar1. Proofs of nearly all the theorems may be found, unless otherwise noted, in [FK80]. CONTENTS 1. Mathematica...

Solutions to the Riemann-Hilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The solutions are given in terms of meromorphic bidifferentials defined on the underlying Riemann surface. The relationship between different classes of Frobenius ma...

Introduction. The rst major goal of this paper is to prove the existence of complete minimal surfaces of each genus p > 1 which minimize the total curvature (equivalently, the degree of the Gau map) for their genus. The genus zero version of these surfaces is known as Enneper's surface (see [Oss2]) and the genus one version is due to Chen-Gackstatter ([CG]). Recently, experimental evidence for ...

The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-)periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation are given. First, a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision. This formula ...

The goal of this paper is to make mathematically precise, then illustrate, how to “email” a Riemann surface. We assume each of our Riemann surfaces are orientable and connected. We will follow the notation from [2] and will cite theorems from this source whenever possible. To email a Riemann surface we want a unique Riemann surface to correspond to a set L of finitely many matrices whose entrie...