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Conformal Mapping on Riemann Surfaces

Harvey Cohn
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Wood
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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.

PART ONE Review of Complex Analysis
Introductory Survey
Chapter 1. Analytic Behavior
Differentiation and Integration
1-1. Analyticity
1-2. Integration on curves and chains
1-3. Cauchy integral theorem
Topological Considerations
1-4. Jordan curve theorem
1-5. Other manifolds
1-6. Homologous chains
Chapter 2. Riemann Sphere
Treatment of Infinity
2-1. Ideal point
2-2. Stereographic projection
2-3. Rational functions
2-4. Unique specification theorems
Transformation of the Sphere
2-5. Invariant properties
2-6. Möbius geometry
2-7. Fixed-point classification
Chapter 3. Geometric Constructions
Analytic Continuation
3-1. Multivalued functions
3-2. Implicit functions
3-3. Cyclic neighborhoods
Conformal Mapping
3-4. Local and global results
3-5. Special elementary mappings
PART TWO Riemann Manifolds
Definition of Riemann Manifold through Generalization
Chapter 4. Elliptic Functions
Abel's Double-period Structure
4-1. Trigonometric uniformization
4-2. Periods of elliptic integrals
4-3. Physical and topological models
Weierstrass' Direct Construction
4-4. Elliptic functions
4-5. Weierstrass' Ã function
4-6. The elliptic modular function
Euler's Addition Theorem
4-7. Evolution of addition process
4-8. Representation theorems
Chapter 5. Manifolds over the z Sphere
Formal Definitions
5-1. Neighborhood Structure
5-2. Functions and differentials
Triangulated Manifolds
5-3. Triangulation structure
5-4. Algebraic Riemann manifolds
Chapter 6. Abstract Manifolds
6-1. Punction field on M
6-2. Compact manifolds are algebraic
6-3. Modular functions
PART THREE Derivation of Existence Theorems
Return to Real Variables
Chapter 7. Topological Considerations
The Two Canonical Models
7-1. Orientability
7-2. Canonical subdivisions
7-3. The Euler-Poincaré theorem
7-4. Proof of models
Homology and Abelian Differentials
7-5. Boundaries and cy
7-6. Complex existence theorem
Chapter 8. Harmonic Differentials
Real Differentials
8-1. Cohomology
8-2. Stokes' theorem
8-3. Conjugate forms
Dirichlet Problems
8-4. The two existence theorems
8-5. The two uniqueness proofs
Chapter 9. Physical Intuition
9-1. Electrostatics and hydrodynamics
9-2. Special solutions
9-3. Canonical mappings
PART FOUR Real Existence Proofs
Evolution of Some Intuitive Theorems
Chapter 10. Conformal Mapping
10-1. Poisson's integral
10-2. Riemann' s theorem for the disk
Chapter 11. Boundary Behavior
11-1. Continuity
11-2. Analyticity
11-3. Schottky double
Chapter 12. Alternating Procedures
12-1. Ordinary Dirichlet problem
12-2. Nonsingular noncompact problem
12-3. Planting of singularities
PART FIVE Algebraic Applications
Resurgence of Finite Structures
Chapter 13. Riemann's Existence Theorem
13-1. Normal integrals
13-2. Construction of the function field
Chapter 14. Advanced Results
14-1. Riemann-Roch theorem
14-2. Abel's theorem
Appendix A. Minimal Principles
Appendix B. Infinite Manifolds
Table 1: Summary of Existence and Uniqueness Proofs
Bibliography and Special Source Material