Stephen Krantz's Complex Analysis: The Geometric Viewpoint was first published in 1992 in the MAA's Carus Mathematical Monographs series. Since the book was very well received and has remained popular, it is not surprising, eleven years later, to see a second edition.
The most important achievement of Krantz's monograph is the unified, accessible and readable presentation of important mathematical ideas in a volume that can be covered in one term by an advanced undergraduate student, under appropriate guidance. In this sense, I think the book achieves the goal stated by the author in his preface to the first edition: "we intend to introduce the reader with a standard one semester background in complex analysis to the geometric method. All geometric ideas will be developed from first principles, and only to the extent needed here" (p. xiii). Thus, this book can be used as a supplement to a course in complex analysis, as the author suggests.
I also suggest it could be used as a main reference for an individual research project or a reading course at the undergraduate level. I believe a student who reads for the first time the important results in this presentation (e.g. the Schwarz's lemma, Liouville's theorem or Montel's theorem) has a very good chance to understand standard complex analysis as it is studied in graduate school. The author definitely has in mind that many readers may be seeing some of these results for the first time here (cf. p. xiv).
In the introductory chapter, the author presents the main results in standard complex analysis: the Maximum Principle, the Schwarz lemma (and the Schwarz-Pick lemma), the Riemann Mapping theorem and the theorems of Picard. The next chapter covers a few basic notions of differential geometry, with emphasis on complex domains and the complex geometric viewpoint. A readable detour into non-Euclidean geometry, with some historical details, reminds us Steven Krantz's interest in the historical developments of ideas, as well as of his collection of anecdotes and relevant details. (See his Mathematical Apocrypha) At the end of this section, the author emphasizes: "conformality, and the computations that we have already performed in another context, are sufficient to see that Euclid's 2000-year-old axiom is independent of the other four axioms of Euclid's geometry."
Chapter 2 covers the relation between curvature and the Schwarz Lemma, and here Krantz underlines that "it was Ahlfors (see Trans. Am. Math. Soc. 43, 359-364 (1938)) who first realized that the Schwarz Lemma is really an inequality about curvature" (p. 70). This approach is the main motivation of this book's architecture, as the author suggests in the preface to the first edition: "In [Ahlfors'] work it was demonstrated that the Schwarz Lemma can be viewed as an inequality of certain differential geometric quantities on the disc (curvatures). This point of view — that substantive analytic facts can be interpreted in the language of Riemannian geometry — has developed considerably in the last fifty years. It provides new proofs of many classical results in complex analysis, and has led to new insights as well."
Chapter 2 also includes Liouville's theorem, normal families, a generalization of Montel's theorem and the Great Picard Theorem. However, the chapter that I think is the best example of Krantz's clear mathematical writing is the one on the new invariant metrics. It covers the Carathéodory and Kobayashi metrics, their completeness, and the connection between curvature and the property of nondegeneracy of the Kobayashi metric. A student exposed to this material may be motivated to pursue a deeper study in this subject, including perhaps the seminal work, cited several times in Krantz's presentation, Hyperbolic Complex Spaces, by S. Kobayashi (Springer-Verlag, 1998).
The book also includes an introduction to Bergman theory and basic concepts in several complex variables. The appendix of Krantz's book includes a proof that the curvature defined in chapter 2 in the context of complex domains is the same as the curvature arising in Cartan's structural equations. The author connects in this chapter his presentation to the standard differential geometry background, as it is presented for example in Elementary Differential Geometry, by B. O'Neill, (Academic Press, 1966). The reader also finds topics of current research interest: "While it is still believed that a biholomorphic mapping of any two smoothly bounded domains must extend smoothly to their respective boundaries, we are far from being able to prove this assertion" (p. 187). Furthermore, in the Epilogue, it is shown that "very little is known about explicitly calculating and estimating the differential invariants described in the present monograph. It is hoped that this book will spark some new interest into these matters" (p.189).
Throughout the book, the author avoids unnecessary abstract approaches, technicalities, formalisms and calculations. As a geometer, I found one of Krantz's viewpoints particularly interesting; he writes that "the techniques of geometry have not proliferated as much as they might have because of the complexity of language" (p.29). Thus, in his presentation he tries to eliminate the technicalities that obscure the ideas, both geometric and analytic. The lucid and highly readable presentation is a result of this educational principle, most clearly described perhaps in this paragraph (p.29): "The best way to learn new mathematics is in the context of what is already familiar. In the milieu of one complex variable, the notions of Riemannian metric, of geodesic, and of curvature become rather simple. The standard geometric device of tensors and bundles is not necessary. The result is that one can learn the flavor and some of the methodology of differential geometry without being encumbered by its notation and machinery." We can see how this principle works in writing by reading the proof of K. T. Hahn's theorem, where Krantz chooses the most useful (and visually effective) viewpoint (p. 157): "We will use the language of geodesics. Going by the book, a geodesic is defined by a differential equation. For our purposes here, we may think of a geodesic as a locally length-minimizing curve."
As I started to read this book, my first reaction was to think that for anyone who later will do research on in Kobayashi hyperbolic manifolds, this is probably the best introduction ever written. But Krantz's potential audience is much wider; his approach is just right for a book at the introductory level. The result is not only a comprehensive overview of the area itself, but also a very informative and inspiring monograph.
Bogdan D. Suceavă (firstname.lastname@example.org) has been assistant professor at California State University at Fullerton since 2002. He received his master's degree from University of Bucharest, in Romania (in 1995) and his Ph.D. from Michigan State University (in 2002), with a thesis entitled New Riemannian and Kaehlerian curvature invariants and strongly minimal submanifolds. His research area is differential geometry, especially the geometry of submanifolds.