The main topic of this book is the theory of compact Riemann surfaces and their connections to other areas of mathematics (two-dimensional differential geometry, algebraic topology, algebraic geometry, the calculus of variations and the theory of elliptic partial differential equations). The discussion includes three fundamental theorems: the Riemann-Roch theorem, the Teichmüller theorem and the uniformization theorem. One of the main tools used throughout the book is the theory of harmonic maps. The book can be also taken as a nice introduction to nonlinear analysis applied to geometry. The first chapter contains background material from topology (e.g., fundamental group and covering spaces). In the second chapter, Riemann surfaces are studied from the point of view of two-dimensional Riemannian geometry. It includes a discussion of curvature, the Gauss-Bonnet theorem, of special Riemann surfaces which are quotients of the Poincaré upper half plane with a hyperbolic metric, and of conformal structures on tori. The third chapter is devoted to the study of the Dirichlet principle and harmonic maps. Teichmüller theory is presented in the fourth chapter, the topological structure of Teichmüller spaces is described and the uniformization theorem for compact Riemann surfaces is proved. The last chapter contains an algebraic geometry approach to Riemann surfaces; homology and cohomology groups of Riemann surfaces are introduced and their relations to forms are given. The main topic is the Riemann-Roch theorem and the Abel theorem on elliptic functions. The book gives an excellent description of the contemporary theory of Riemann surfaces; it can be recommended to all mathematicians interested in the field.

Reviewer:

jbu