Jürgen Jost, the author of Compact Riemann Surfaces: An Introduction to Contemporary Mathematics, characterizes his book as, among other things, “an introduction to non-linear analysis in geometry.” This is a particularly tantalizing phrase in the wake of all the excitement about Perelman’s proof of the Poincaré conjecture, accompanied as it was by everything from high drama to low comedy. To be sure, it’s not clear which of these two dramatic categories the main events of the story belong to, particularly the controversy that arose in connection with Yau and his student and Perelman’s refusal to accept the Fields Medal and, apparently, the 106 Clay dollars. But it’s undeniable that these are incomparably exciting times for geometry, in particular for the philosophy of attacking geometrical or, rather, topological, questions by means of analysis — Ricci vindicatus.
This said, is there any better way for what Jost calls an intermediate student (roughly, the German counterpart to a beginning American graduate student) to get into the game than by studying the most elegant and fecund marriage of analysis and geometry of all, Riemann surfaces? By definition, a Riemann surface is a one-dimensional C-manifold, and the standard examples of these objects include, next to C itself, the Riemann sphere (which Jost identifies as being the most important example), the torus, and any conformal image of a given Riemann surface. Says Jost (p. vii): “I consider Riemann surfaces as an ideal meeting ground for analysis, geometry, and algebra and as ideally suited for displaying the unity of mathematics. Therefore, they are perfect for introducing intermediate students to advanced mathematics.”
Compact Riemann Surfaces: An Introduction to Contemporary Mathematics starts off with a wonderful Preface containing a good deal of history, as well as Jost’s explicit dictum that there are three foci around which the whole subject revolves, namely, uniformization (Riemann, Poincaré, Koebe — by the way, like Riemann, Jost uses the Dirichlet Principle in this connection!), the Riemann-Roch Theorem (“… we give an essentially classical proof …”), and Teichmüller’s Theorem, a comparatively much younger result. In connection with the latter, Jost uses harmonic maps, pointing out that Yau went this route in proving Calabi’s Conjecture. The book covers such themes as the fundamental group and (co)homology, from algebraic topology; metrics, curvature, geodesics, and the Gauss-Bonnet Theorem, from differential geometry; PDE and functional analysis; the calculus of variations; and obviously certain topics from algebraic geometry, Riemann-Roch already having been mentioned. Despite the manifest density of the treatment (less than 300 pages for all this good stuff ….) Jost’s presentation is quite accessible, modulo a lot of diligence on the part of the reader. It’s a very good and useful book, very well-written and thorough.
Two personal notes: The very last section of Compact Riemann Surfaces: An Introduction to Contemporary Mathematics is a short but thorough (and elegant) treatment of another topic which has recently seen an exponentiation of activity (after Wiles took care of Fermat’s Last Theorem): elliptic curves. I think this shows superb taste on Jost’s part. And I currently have occasion to do a lot of studying in the area of finite dimensional complex manifolds; I intend to put Compact Riemann Surfaces: An Introduction to Contemporary Mathematics to good use in this connection myself.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.