When going through school, I, like so many of my generation (and the abutting ones, I’m sure), became more and more aware of the fact that the collection of complex analysis textbooks was bounded above, so to speak: Ahlfors’ book on the subject possessed nigh on mythological status as the standard against which all others must measure themselves. This said, I must own that I did not get around to working seriously with the book until a few years into my life as an academic, when an extremely gifted (and graduate school bound) senior asked me for a reading course that would prepare her best for what lay ahead in complex analysis. I gave her Ahlfors’ book to work through under my supervision, and, to be sure, it soon became clear why the book enjoyed the high reputation it did.
Of course, the Ahlfors myth was also informed by the fact that he belonged to a constellation of Scandinavian analysts known for slaying mathematical dragons left and right. Already as a rookie I’d heard of Abel, of course, and soon I had learned of Nevanlinna, and presently Mittag-Leffler, Lindelöf, Carleson, Hörmander, Beurling, and eventually Selberg. Ahlfors’ name was included pretty early on in this sequence: many, and then many more, of my professors had connections with Ahlfors. Manifestly he resided at the hub of a web of analysts across the country: truly a major player.
The book under review, Conformal Invariants: Topics in Geometric Function Theory, is a correspondingly major work in the field. Now an AMS Chelsea publication, the book first appeared in 1973 and is a masterpiece. Derived from lectures given at Harvard “over many years, … the topics [dealt with] would now be considered quite classical … Nevertheless, the mathematics feels very much alive and still exciting.” Then, as regards the mathematics proper: “The opening chapter on Schwarz’ lemma contains Ahlfors’ celebrated [and Fields Medal winning] discovery, from 1938, of the connection between that very classical result and conformal metrics of negative curvature.” The authors of the Foreword, Peter Duren, F. W. Gehring, and Brad Osgood, go on to note subsequently that “[i]t would be hard to overestimate the impact of that method, but until [this] book’s publication there were very few places to find a coherent exposition of the main ideas and applications.” This by itself is a sufficient raison d’être both for the original publication of the book and its present re-issue.
Next, in the book’s Preface, Ahlfors starts out by noting that his book is “primarily intended for students with approximately a year’s background in complex variable theory.” According to the letter of the law this is true, but according to its spirit, well, let’s just say that the indicated year’s background had better be of a pretty sporty variety: I’d like to recommend a comfort zone at the level of, say, Saks-Zygmund’s Analytic Functions (a truly gorgeous book, despite the authors’ use of rectangles as natural contours for Cauchy integrals, instead of circles), or maybe (both volumes of) Einar Hille’s Analytic Function Theory, or, of course, the aforementioned Complex Analysis, by Ahlfors himself.
The book under review gets off the ground very fast. For example, p.3 sports the statement of the Schwarz Lemma, immediately followed by the statement that
[t]here is no need to reproduce the well-known proof. It was noted by Pick that the result can be expressed in invariant form [:] An analytic mapping of the unit disc into itself decreases the non-euclidean distance between two points, the non-euclidean length of an arc, and the non-euclidean area of a set ... [And then:] Nontrivial equality holds only when [the mapping] is a fractional linear transformation [of a certain type].
Yes, this is unquestionably a beautiful and evocative way to interpret Schwarz’ Lemma, but it does assume a certain readiness on the reader’s part to accompany Ahlfors on the “dangerous” trip he has charted. Indeed, Ahlfors immediately proceeds to Pick’s proof of a generalization “which deserves to be better known,” involving positive definite, or semidefinite, Hermitian forms.
These passages suffice to illustrate the relative austerity of Conformal Invariants: Topics in Geometric Function Theory, even as they disclose ab initio that we’re in for some very exciting analysis and geometry. Ahlfors spends over 20 pages on the initial chapter on Schwarz’ Lemma and its applications (including Bloch’s Theorem and both Picard Theorems), after which, in even more compact fashion, it’s on to the topics of capacity, harmonic measure, extremal length (this is an appropriately beefy chapter), univalent functions, material by Löwner and by Schiffer, and finally the trio: “Properties of the extremal functions,” “Riemann surfaces,” and “The uniformization theorem.” The last two chapters, covering only about 30 pages, should be read by every one about to become involved with Riemann surfaces, modulo a decent independent preparation: it’s a wonderful presentation replete with beautiful proofs.
Conformal Invariants: Topics in Geometric Function Theory being a textbook, there are a good number of exercises provided by the author, and, indeed, the truly committed reader has pencil and paper ready to go to battle. But this book is of course a lot more than a textbook: Ahlfors’ passion for the material, and his high mastery of it, shines through on every page. This book is truly a beautiful work of mathematical art.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.