# Complex Analysis: The Geometric Viewpoint

Steven G. Krantz
Volume: 23
Edition: 2
Pages: 238
https://www.jstor.org/stable/10.4169/j.ctt5hh83k

1. Front Matter
(pp. i-viii)
2. Acknowledgments
(pp. ix-x)
S.G.K.
3. Preface to the Second Edition
(pp. xi-xii)
4. Preface to the First Edition
(pp. xiii-xiv)
(pp. xv-xviii)
6. CHAPTER 0 Principal Ideas of Classical Function Theory
(pp. 1-28)

The purpose of this book is to explain how various aspects of complex analysis can be understood both naturally and elegantly from the point of view of metric geometry. Thus, in order to set the stage for our work, we begin with a review of some of the principal ideas in complex analysis. A good companion volume for this introductory material is [GRK]. See also [BOAS] and [KR3].

Central to the subject are the Cauchy integral theorem and the Cauchy integral formula. From these follow the Cauchy estimates, Liouville’s theorem, the maximum principle, Schwarz’s lemma, the argument principle, Montel’s theorem,...

7. CHAPTER 1 Basic Notions of Differential Geometry
(pp. 29-66)

Differential geometry has developed into one of the most powerful mathematical tools of modern mathematics. It has become an integral part of the theories of differential equations, harmonic analysis, and complex analysis, to name just a few examples.

In spite of their importance, the techniques of geometry have not proliferated as much as they might have because of thecomplexity of the language. The characterization of differential geometry as “that portion of mathematics which is invariant under change of notation” is unfortunately rather accurate.

The best way to learn new mathematics is in the context of what is already familiar....

8. CHAPTER 2 Curvature and Applications
(pp. 67-88)

If$U \subseteq \mathbb{C}$is a planar domain and ρ is a metric onU, then thecurvatureof the metric ρ at a point$z \in U$(at which$\rho (z) \ne 0$) is defined to be

$KU,\rho (z) = k(z) \equiv \frac{{ - \Delta \log \rho (z)}} {{\rho {{(z)}^2}}}$. (*)

(Here zeros of${\rho (z)}$will result in singularities of the curvature function—kis undefined at such points.) We will study this quantity for a bit, and then summarize and provide motivation afterward.

Since ρ is twice continuously differentiable, this definition makes sense. It assigns to each$z \in U$a numerical quantity. The most important preliminary fact aboutκis its conformal invariance:

Let${U_1}$and${U_2}$be...

9. CHAPTER 3 Some New Invariant Metrics
(pp. 89-136)

Refer to Section 0.3 for the statement and sketch of the proof of the Riemann mapping theorem. The Riemann mapping function is the solution to a certain extremal problem: to find a map of the given domainUinto the discDwhich is one-to-one, maps a given pointPto 0, and hasderivative of greatest possible modulusλpatP. The existence of the extremal function, which also turns out to be one-to-one, is established by normal families arguments; the fact that the extremal function is onto is established by an extra argument which is in fact the...

10. CHAPTER 4 Introduction to the Bergman Theory
(pp. 137-160)

It is a remarkable fact—discovered by Stefan Bergman in 1927—that a bounded domain Ω in$\mathbb{C}$or in${\mathbb{C}^n}$can be equipped with a “canonical” reproducing kernel. [Here we use the phrase “reproducing kernel” to mean a function$k(z,\zeta )$of two variables—like the familiar Cauchy kernel—with the property that integration against a holomorphic functionfproduces the value offatz. In one complex variable such a formula could take the form

$f(z) = \int_{\partial \Omega } {k(z,\zeta )} f(\zeta )d\zeta$

or

$f(z) = \iint_\Omega {k(z,\zeta )f(\zeta )d\xi dn}$

when$\zeta = \xi + i\eta$. Such a formula means, in effect, that the kernelkcontains valuable information about holomorphic functions on Ω.]...

11. CHAPTER 5 A Glimpse of Several Complex Variables
(pp. 161-188)

At a naive level, the analysis of several real variables is much like the real analysis of one variable; the main difference is that one deals withn-tuples of reals instead of scalars and one needs matrices to keep track of information. Of course deeper study reveals much complexity and richness in analysis of several real variables. It is noteworthy that, for the most part, this richness was discovered rather late in the history of the subject—mostly in the last fifty years.

The history of analysis of several complex variables is quite different. Early in the subject, in the...

12. Epilogue
(pp. 189-190)

In complex analysis, geometric methods provide both a natural language for analyzing and recasting classical problems and also a rubric for posing new problems. The interaction between the classical and the modern techniques is both rich and rewarding.

Many facets of this symbiosis have yet to be explored. In particular, 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 in these matters....

13. Appendix on the Structure Equations and Curvature
(pp. 191-204)
14. Table of Symbols
(pp. 205-208)
15. References
(pp. 209-212)
16. Index
(pp. 213-219)