Conformal Mapping on Riemann Surfaces

One early observation in a standard course in complex variables is that the square root function is multi-valued on the complex plane. We knew this already, of course, but in the complex setting we see that this an essential feature of how the function works and is no longer to be dismissed as a minor technicality as is typical in calculus. Indeed, we discover that the real problem is that we are looking at the wrong domain — what we really want to do is slit the complex plane from 0 to infinity, and then paste two copies of this object along their slits to form our first Riemann surface, defined more or less as the natural object on which the square root function is single-valued and analytic.

The big news is that this connection can be pushed to the highest level of generality possible: any Riemann surface can be described as the natural domain of some complex function, and vice versa. Establishing this link between functions and surfaces is the primary objective of Harvey Cohn’s Conformal Mapping on Riemann Surfaces. The objective statement (page 121) is:

Every irreducible algebraic function determines (under analytic continuation) some compact Riemann manifold and, conversely, every compact Riemann manifold is equivalent to the Riemann surface determined by an appropriate algebraic function.

The journey to this result is complicated and Cohn recognizes it as an opportunity to tour the development of a broad swath of modern mathematics. After a substantial review of complex variables, wherein standard results are brought into the necessary context, the reader is brought through the development of topics such as elliptic functions, homology and cohomology, and harmonic analysis.

What is enjoyable about the approach is the attention Cohn pays to motivation. We see how the techniques of complex analysis are variously employed to build toward this deeper understanding of what analytic continuation really can buy. This motivation is largely generated by careful attention to history. Nearly as interesting as Riemann’s original ideas is the machinery developed in the subsequent decades to complete his proof. Somewhat incongruously, this historical reverence helps the book maintain significance more than fifty years after its publication. (Although the title is a bit anachronistic: conformal mapping has become far more practical in those fifty years, and this title could wrongly suggest a more applied text.)

The writing is lucid, good pictures are plentiful, and well-chosen and relevant exercises (without solutions) follow each section. Many compromises need to be made with so many ideas coming together and it would be very easy to get bogged down in technical or historical details, but this author recognizes when it is appropriate to appeal to mathematical or physical intuition (e.g., many of the mechanics of triangulating surfaces), when to refer the reader elsewhere (e.g. the usual deferment of the Jordan curve theorem), and how to sort real technicalities from important but subtle complications (where much of the real work is done).

This is a fine text on which to build a study of complex structures. The thorough review of the introductory material allows for experience rather than expertise in complex analysis to be the prerequisite. A beginning graduate student or advanced undergraduate could get quite a lot from it; indeed, the author advocates this material is best studied before pursuing coursework in algebra or topology (an interesting thesis, but probably not practical given the way most curricula are organized). It is appropriate for self-study, a standalone text, or as a supplement to other treatments such as Jones and Singerman’s Complex Functions: An Algebraic and Geometric Viewpoint. It is appropriate for libraries at undergraduate and graduate institutions. This reviewer plans to keep it around for efficient reminders of how the general theory fits together.

It is no exaggeration to say that the content is important to the development of contemporary mathematics as we know it, and Cohn’s development has a place among the many other available treatments of the topic, the breadth of which supports a varied literature. Thanks to Dover Publications for making an inexpensive edition of this book available.

Bill Wood is an Assistant Professor of Mathematics at the University of Northern Iowa.