Riemann Surfaces and the L ² $\bar{\partial}$-Method for Scalar-Valued Forms

Abstract

In this chapter, we consider some elementary properties of Riemann surfaces, as well as a fundamental technique called the L ² $\bar{\partial}$-method, Radó’s theorem on second countability of Riemann surfaces, and analogues of the Mittag-Leffler theorem and the Runge approximation theorem for open Riemann surfaces. Viewing holomorphic functions as solutions of the homogeneous Cauchy–Riemann equation $\partial f/\partial\bar{z}=0$ in ℂ allows one to very efficiently obtain their basic properties (see Chap. 1). The intrinsic form of the homogeneous Cauchy–Riemann equation on a Riemann surface is given by $\bar{\partial}f=0$ (see Sect. 2.5). In order to obtain holomorphic functions (and holomorphic 1-forms) on a Riemann surface (even on an open subset of ℂ), it is useful to consider the inhomogeneous Cauchy–Riemann equation $\bar{\partial}\alpha=\beta$. One well-known approach to solving this differential equation (as well as differential equations in many other contexts) is to consider weak solutions in L ². This is the approach taken in this book. In order to do so, we must develop suitable versions of an L ² space of differential forms (see Sect. 2.6) and an (intrinsic) distributional $\bar{\partial}$ operator (see Sect. 2.7). The relatively simple approaches to the above appearing in this book are, in part, special to Riemann surfaces; but they do contain important elements of the higher-dimensional versions (see, for example, L. Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland, 1990 or J.-P. Demailly, Complex Analytic and Differential Geometry, online book, for the higher-dimensional versions).