Invariant Distances and Metrics in Complex Analysis

It is a hallowed and much cited result in the literature of several complex variables that the unit ball and the unit polydisk in \(\mathbb{C}^n\) are not biholomorphically equivalent. This 1906 result of Henri Poincaré has had an enormous influence on the development of the subject for the past 100 years. Knowing that there is no Riemann mapping theorem in the several variable setting, we seek substitute results. We also search for tools that will help us understand the situation.

One of the major developments that has come to the rescue is the creation of various invariant metrics. Again it was Poincaré who first invented a conformally invariant metric (the so-called Poincaré metric) on the disc. In the 1920s and 1930s, Stefan Bergman gave a construction for a biholomorphically invariant metric that works on any bounded domain. Later on, Carathéodory and Kobayashi gave metric constructions that are particularly appealing because they are based on the classical proof of the Riemann mapping theorem in one complex variable.

There are a number of other invariant metrics — due to N. Sibony, K. T. Hahn, and others. But it is safe to say the the Bergman, Carathéodory, and Kobayashi metrics are the most prominent and most widely used. There has been a flood of results on these three metrics in the past fifty years. C. Fefferman’s Fields-Medal-winning work on the boundary behavior of biholomorphic mappings was based on a remarkable study of the Bergman metric, and that gave this whole enterprise a shot in the arm.

The book under review is an encyclopedic treatment of invariant metrics and their applications in one and several complex variables. It is an ambitious project, and a successful one. Certainly there are other books that treat invariant metrics to a greater or lesser extent (including some by this reviewer), but none that is nearly so comprehensive as the present one. Jarnicki and Pflug are prominent authorities in the field, and well placed to write a book like this one. Both authors have published extensively on invariant metrics, and both are quintessentially knowledgeable.

The book has many charming and useful features. The Bibliography alone — with more than 500 entries — is a valuable resource. The Table of Notation is unusually detailed and very helpful. Following the inspiration of Bourbaki’s famous “curvy road” sign, these authors label some passages in their book with a sign to indicate that this is an open problem that the reader should think about, and other passages with a sign to tell the reader to pick up the pencil and calculate.

The book is more than 800 pages long. It would not work well as a textbook, partly because it is so extensive and partly because it is so dense. It is also not the sort of book that one just sits down and reads straight through. But it is a wonderful resource, and is written in such a way that it is very natural for the reader to dip into various portions as the need arises.

The writing in the book is exceptionally clear, and the level of detail is usually just about right. The authors are quite skilled at providing context and motivation, and they give plenty of interesting examples and ancillary results. Each chapter ends with a brief list of (research) problems, and these will give students and faculty alike much food for thought.

Certainly the book of Jarnicki and Pflug will occupy a prominent spot on my bookshelf, and will soon show the well-worn characteristics of a tome that is both respected and admired.

Steven G. Krantz has taught at UCLA, Princeton University, Penn State, and Washington University in St. Louis. He was Chair of the latter department for five years. He has won the Chauvenet Prize, the Beckenbach Book Award, and the Kemper Prize. He is currently the editor of the Notices of the American Mathematical Society.