Compact Riemann surfaces. An introduction to contemporary mathematics (Q5921630)

In complex analytic geometry as well as in complex algebraic geometry, Riemann surfaces provide the first typical examples for complex manifolds and (smooth) projective varieties. Historically, the theory of Riemann surfaces can be traced back to the beginning of the 19th century, when the fundamental discoveries of N. H. Abel and C. G. J. Jacobi prepared the ground for B. Riemann's ingenious and pioneering conceptual work on geometric function theory (or nonlinear complex geometry, respectively), anticipating the basic objects of study in contemporary differential geometry, complex analysis, algebraic geometry, algebraic topology, and (to a growing extent) in mathematical physics. Compact Riemann surfaces, i.e., one-dimensional compact complex manifolds, together with their allied theory of Abelian functions, were among the most prominent advocates of those disciplines whose extensive development significantly characterizes the science of pure mathematics in our century. Being a time-honoured, fascinating, tremendously rich and particularly far-developed subject of mathematics, compact Riemann surfaces are still in the center of interest of both analysis and geometry, and over the past twenty years they have even undergone another striking renascence. The latter phenomenon is due to the fact that Riemann surfaces and their moduli spaces, both invented by B. Riemann over more than one century ago, recently turned out to be crucial tools for constructing and analyzing quantum field theories, integrable Hamiltonian systems, and classifying spaces for other geometric objects. The ubiquity of compact Riemann surfaces in contemporary mathematics and mathematical physics, their nearly unparalleled rôle as an example for testing, demonstrating and exploring the interrelation of various mathematical principles and methods, just as the fact that Riemann surfaces seem to be an inexhaustible spring of new inspirations, undoubtedly makes them an ideal topic for providing beginners with an introduction to some of the basic, faceted and mutually intertwined methods in modern analysis, geometry and topology. The present textbook on compact Riemann surfaces, another one among many excellent others, distinctly aims at providing just such a faceted introduction to the subject and, along this way, to several analytic-geometrical principles of classical and/or contemporary mathematics. While most texts on compact Riemann surfaces focus on one or two approaches to them, and develop the according theory with more or less strict purity and in achievable depth, the text under review discusses several viewpoints of the subject, certainly not so with respective comprehensiveness, but correspondingly in a more panoramic fashion. Working through the material presented here, the reader will get acquainted with many fundamental concepts of algebraic topology, two-dimensional Riemannian geometry, regularity theory for elliptic partial differential equations, functional analysis, calculus of variations, and with some ideas and results from complex algebraic geometry, all through motivated and illustrated by the theory of compact Riemann surfaces. Three of the highlights in Riemann surface theory serve as the strategic-methodical corner stones for the arrangement of the text: the uniformization theorem (Riemann-Poincaré-Koebe), Teichmüller's theorem on the variation of conformal structures on a surface, and the Riemann-Roch theorem for curves. These three fundamental theorems represent the complex-analytic, real-analytic and differential-geometric, and the algebro-geometric aspects, respectively. Their proofs, and the different methods for obtaining them, are given in full detail, whereas at least two features of this new text on Riemann surfaces, apart from its already mentioned panoramic structure, make it somehow unique among the others. Namely, Teichmüller theory (which is usually treated in books exclusively dedicated to this subject) is essentially integrated in the discussion of Riemann surfaces and, beyond that, it is presented in the framework of harmonic maps instead of the classical approach via quasiconformal maps. The latter feature must be seen as highly welcome, since the approach via harmonic maps throws another important, intradisciplinary bridge to recent developments in complex geometry and physics (Kähler manifolds, symmetric spaces, Monge-Ampère equations, Yang-Mills theory, Donaldson theory, Seiberg-Witten equations, etc.), and this is perhaps the main new aspect that the present textbook offers. The contents are arranged in five chapters. Chapter 1 provides the necessary topological foundations such as manifolds, homotopy of maps, fundamental groups and coverings. Chapter 2 is entitled ``Differential geometry of Riemann surfaces''. The author gives an introduction to the concept of a Riemann surface and concentrates, in the sequel, on those surfaces that are quotients of the upper-half plane. In the course of this chapter, the foundations of two-dimensional Riemannian geometry (up to the Gauss-Bonnet theorem) are developed, the hyperbolic geometry (in the sense of Kobayashi) of Riemann surfaces is discussed, the topological classification of surfaces is explained, and a first account on conformal structures -- here on complex tori -- is laid before the reader. Chapter 3 turns to the analysis of harmonic maps. After a brief introduction to the functional-analytic aspects (Hilbert spaces and Sobolev spaces), the Dirichlet principle is thoroughly justified. Then the regularity theory for elliptic partial differential equations is developed in great generality (after Korn, Lichtenstein, and Schauder), mainly in order to get prepared for the following treatment of harmonic maps between surfaces, the basic tool for the author's approach to Teichmüller theory in Chapter 4. Starting from the concept of moduli for compact Riemann surfaces, harmonic maps are then used to construct holomorphic quadratic differentials, which are in one-to-one correspondence with the conformal structures on a given surface, yet in a way that is different from Teichmüller's original construction. Anyway, Teichmüller's theorem on the variation of conformal structures is obtained in a new and certainly easier way. As an alternative, very enlightening description of the topology of Teichmüller spaces, the author discusses the real-analytic approach via Fenchel-Nielsen coordinates. The last section of Chapter 4 is devoted to the uniformization theorem for compact Riemann surfaces. The proof given here is again based on the method of harmonic maps, showing once more the power of this concept and its methodical framework. The concluding Chapter 5 deals with the algebro-geometric aspects of compact Riemann surfaces. The author presents the usual standard material that is needed to state and to prove the Riemann-Roch theorem, without using sheaf-cohomological techniques. After introducing singular homology and De Rham cohomology for compact Riemann surfaces, and discussing holomorphic and harmonic differential forms on them, he derives the Riemann period relations and the Riemann-Roch theorem for divisors. Line bundles, projective embeddings, and compact Riemann surfaces as projective curves are also included, and the explicit computation (via Riemann-Roch) of the dimension of the space of quadratic differentials gives the dimension of the corresponding Teichmüller space (and, therefore, of the moduli space \(M_g\) of compact Riemann surfaces of genus \(g\)). The final sections are devoted to proofs of Abel's theorem. Jacobi's inversion theorem, and to an illustration of the foregoing results by means of elliptic curves. Altogether, this textbook is a valuable enrichment of the existing literature on compact Riemann surfaces. More than that, it really does justice to its ambition of representing an introduction to some important principles of contemporary mathematics. The material is displayed in a very lucid, rigorous and detailed manner, thus showing the author's admirable knowledge and methodical skills. Each section of each chapter comes with exercises closely related to the foregoing topic. In general, the prerequisites are kept to a minimum, with respect to the level of the subject, and a reader who is familiar with the principles and concepts of real analysis, one-dimensional complex function theory, and elementary topology will find the reading very profitable and enjoyable. Also, the present book is unquestionably an excellent introduction to other, more specific or advanced treatises on Riemann surfaces, Teichmüller theory, harmonic maps, or complex algebraic curves and Abelian varieties.