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The most important result on open Riemann surfaces is the analog of Runge’ theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge’ theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.

The paper is a continuation of the preceding papers [4], [5] where description of spin structures and harmonic spinors on the hyperelliptic Riemann surfaces was presented. Here the case of nonhyperelliptic surfaces of genera 3 ≤ g ≤ 6 is studied in details by means of algebraic geometry.

As we have said already the theorems of Weierstrass and Mittag-Leffler proved originally for domains in ℂ were proved in 1948 by Behnke-Stein for arbitrary open Riemann surfaces. Caratheodory conjectured that on every open Riemann surface there exists a non constant global holomorphic function f ∈ O(X). The conjecture that on a surface X every meromorphic function h ∈ M (X) is a quotient of two meromorphic functions was proved in the Behnke-Stein paper as well. In this chapter we will reproduce proofs of these theorems, and the fact that every open Riemann surface is a Stein manifold using the method due to B. Malgrange based on sheaf theory and the theory of d″ operator.

The present part is central for the whole of this book and its title is borrowed from the celebrated monograph of the young Hermann Weyl Idee der Riemannschen Fasche. (Of course, our presentation does not have much in common with the Weyl’s classic, however in this way I want to express admiration and gratitude to the greatest, most influential master of mathematics of our century.)

As we know from the Riemann-Roch theorem the topological classification of complex vector bundles E → M over compact Riemann surface is ‘very simple’: they are classified by one integer c 1(E), the first Chern class of E → M. The situation is quite different in the case of classification of holomorphic category. This could have been expected from our experience with the Teichmuller theory, the main chapter of the Riemann moduli problem.

Let Г be a lattice in ℂ, then the elliptic Г periodic functions form a field E Г; indeed, meromorphic functions (on ℂ/Г) form a field. Riemann and Weierstrass knew it already that every two functions f 0, f l ∈ E Г are algebraically dependent, moreover, Weierstrass showed that E Г = ℂ(℘, ℘′), that is, every elliptic Г periodic function is a rational function of the Weierstrass function ℘ and its derivative. Similar theorem was later proved for automorphic function of one variable. As we know the problem of uniformization of compact Riemann surface S led F. Klein and H. Poincare to creation of the theory of automorphic functions.

This motto is the first sentence of the beautiful opening address at the International Congress of Mathematicians held in Stockholm in August, 1962 Teichmuller Spaces given by the great expert on the Riemann moduli problem. Lars V. Ahlfors was the first winner of the Fields medal (in 1936, together with J. Douglas).

This chapter contains a general introduction to minimal surfaces, without boundary, in Euclidean space. In Section 1 we review the Riemannian and conformal geometries of surfaces: The overriding observation in intrinsic surface theory is that isothermal coordinates — which always exist by the Korn-Lichtenstein theorem — provide an intimate link between the two geometries. This observation together with a bit of Hodge theory is explained in Section 1. The next section represents an excursion into the method of moving frames and a review of the geometry of submanifolds. The exposition in Section 2 is on a fairly abstract footing — this was to indicate the generality of our method. Several results of this section are used later when we discuss the Gauss map of a minimal surface. In Section 3 we prove that a conformal immersion from a Riemann surface into R n is minimal if and only if it is harmonic.

In this chapter we will deal exclusively with complete minimal surfaces in R 3 with finite total curvature; for the sake of convenience we will call such a surface an algebraic minimal surface. In earlier papers [Y4, Y5, Y6, Y7] the author began a study of the Puncture Number Problem for algebraic minimal surfaces. Given a compact Riemann surface M g of genus g, a positive integer r is called a puncture number of M g if M g can be conformally immersed in R 3 as an algebraic minimal surface with exactly r punctures. The set of all puncture numbers for M g is denoted by P (M g ).

The aim of the paper is a description of some relations between spin-structures, the Dirac operators and harmonic spinors on the one side and the complex geometry of Riemann surfaces on the other side. The case of hyperelliptic surfaces is studied in more details.

Let M be a compact oriented smooth manifold with boundary ∂M (possibly ∂M = O). Also let f: M → (N, ds2) be an immersion into a Riemannian manifold N. By a smooth variation of f we mean a smooth map F: I × M → N, I =(−1, 1) such that i) Ft = F(t, ·): M → N is an immersion; ii) F0 = f; iii) Ft | ∂M = f| ∂M for every t ∈ I.

Much attention has been paid to the investigation of minimal surfaces in ℂPn (and in more general spaces) in recent years. Inspired by the work of physicists Din and Zakrzewski [D—Z] Eells and Wood [E—W] first gave a rigorous mathematical treatment of the classification of minimal two—spheres in ℂPn. [R] later gave a classification of minimal two—spheres in the complex Grassmannian ℂG4,2. [U, C—W2] gave a description of minimal two—spheres in ℂGn,k in general. [G1, Y4] constructed a class of minimal surfaces in ℍPn.

A minimal surface in Sn “is” a minimal cone (see §5 for a precise statement of this) in ℝn+l Thus the study of minimal surfaces in Sn is in some sense more restricted in scope than that of minimal surfaces in Euclidean space.

In [2] for the first time there have been given a complete proof of the following statement of A. Hurwitz: For any integer g > 2 there exists a compact Riemann surface of genus g, whose group of all conformai automorphisms is trivial. Then Greenberg [3] has shown that for g > 2 almost all points in the Teichmuller space Tg, except perhaps for an analytic subset, correspond to Riemann surfaces with the trivial group of conformal automorphisms. Nevertheless, only a few constructive examples of such Riemann surfaces are known. One of them is given by Accola [1]. However, the method of Accola does not let us describe analytically the fundamental set of a Fuchsian group which uniformizes that surface. In the present paper, announced in [5], we construct in an explicit way the fundamental set of a Fuchsian group which uniformizes a compact Riemann surface with the trivial group of conformai automorphisms.