The Theory and Practice of Conformal Geometry

by Steven G. Krantz

Overview

In this original text, prolific mathematics author Steven G. Krantz addresses conformal geometry, a subject that has occupied him for four decades and for which he helped to develop some of the modern theory. This book takes readers with a basic grounding in complex variable theory to the forefront of some of the current approaches to the topic. "Along the way," the author notes in his Preface, "the reader will be exposed to some beautiful function theory and also some of the rudiments of geometry and analysis that make this subject so vibrant and lively."
More up-to-date and accessible to advanced undergraduates than most of the other books available in this specific field, the treatment discusses the history of this active and popular branch of mathematics as well as recent developments. Topics include the Riemann mapping theorem, invariant metrics, normal families, automorphism groups, the Schwarz lemma, harmonic measure, extremal length, analytic capacity, and invariant geometry. A helpful Bibliography and Index complete the text.

Product Details

ISBN-13:	9780486810324
Publisher:	Dover Publications
Publication date:	03/17/2016
Series:	Aurora: Dover Modern Math Originals
Sold by:	Barnes & Noble
Format:	eBook
Pages:	304
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About the Author

Steven G. Krantz is Professor of Mathematics at Washington University in St. Louis. He is Editor of the Notices of the American Mathematical Society and the author of many books in mathematics, including Real Analysis and Foundations: Third Edition, The Proof is in the Pudding: The Changing Nature of Mathematical Proof, and A Mathematician Comes of Age.

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The Theory and Practice of Conformal Geometry

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By Steven G. Krantz

Dover Publications, Inc.

Copyright © 2016 Steven G. Krantz
All rights reserved.
ISBN: 978-0-486-81032-4

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CHAPTER 1

The Riemann Mapping Theorem

Prologue: There is hardly a more profound theorem from nineteenth century complex analysis than the Riemann Mapping theorem. Even to conceive of such a theorem is virtually miraculous. Although Riemann's original proof was flawed, it pointed in the right direction. Certainly a great deal of modern complex function theory has been inspired by the Riemann Mapping theorem (RMT).

Throughout this book, we shall use the term domain to mean a connected, open set. While the Riemann Mapping theorem gives us a complex-analytic classification of simply connected planar domains, a theory (in fact several theories) has developed for multiply connected domains. This includes the Ahlfors map, the canonical representation, and the uniformization theorem. We treat all of these in the present chapter. Although we do not treat the topic here, Riemann surface theory is an outgrowth of the study of conformal mappings.

Perhaps the most important modern concept in this circle of ideas is Teichmüller theory, which creates a moduli space for Riemann surfaces. It is beyond the scope of the present book, but it provides a pointer for further reading.

1.0 Introduction

Capsule: It is natural to think of the Riemann Mapping theorem in the context of simply connected domains. However, from the point of view of analysis, it is more convenient to have a different formulation of the topological condition. In this section we introduce the notion of holomorphic simple connectivity: A domain U is holomorphically simply connected if any holomorphic function on U has a holomorphic antiderivative.

It is easy to verify that any topologically simply connected domain is holomorphically simply connected. So we certainly suffer no loss of generality to use this substitute idea. It also streamlines our treatment.

In thinking about the topology of the plane, it is natural to ask which planar open sets are homeomorphic to the open unit disc. The startling answer is that the Riemann Mapping theorem tells us that any connected, simply connected open set (except the plane) is not only homeomorphic to the disc but conformally equivalent to the disc. One can verify separately, by hand, that the entire plane is also homeomorphic to the disc (but certainly not conformally equivalent).

Riemann's astonishing theorem has many different proofs, and we shall consider some of them here. Some of the proofs are "existence proofs," and some constructive. Some are geometric and some are analytic. The book [BIS] covers ideas connected to the Riemann Mapping theorem comprehensively.

We end this section with a formal enunciation of the Riemann mapping theorem:

Theorem (RMT): Let U [??] C be any simply connected domain that is not conformal to the entire plane. Then U is conformally equivalent to the unit disc.

1.1 The Proof of the Analytic Form of the Riemann Mapping Theorem

Capsule: We actually prove the Riemann Mapping theorem in an appendix. In this section, we set up the proof. We develop the idea of holomorphic simple connectivity, and establish some of its properties. We discuss some of the significance of this important result.

Classical arguments, which may be found in any complex analysis text (see for example [GRK1]), show that topological simple connectivity implies an analytic form of simple connectivity that we now define.

Definition 1.1 A connected open set U [??] C is holomorphically simply connected if, for each holomorphic function f : U [right arrow] C, there is a holomorphic antiderivative F — that is, a function satisfying F'(z) = f(z) on U.

Example 1.2 Certainly open discs and open rectangles are holomorphically simply connected. One constructs F from f with a simple line integral.

Notice that, on a holomorphically simply connected domain, a nonvanishing, holomorphic function f will have a logarithm — for we can just take an antiderivative of f'/f. Then it follows that such an f will have a square root.

Let D [??] C be the unit disc. Let U be a holomorphically simply connected open set in C that is not equal to all of C. Fix a point P [member of] U and set

F = {f : f is holomorphic on U, f : U [right arrow] D, f is one-to-one, f (P) = 0}

We shall prove the following three assertions:

(1) F is nonempty.

(2) There is a function f0 [member of] F such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3)If g is any element of F such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then, g maps U onto the unit disc D.

These three assertions taken together imply the Riemann Mapping theorem (see the discussion below). The proof of assertion (1) is by direct construction. Statement (2) is proved with a normal families argument. Statement (3) is the least obvious and will require some work: If the conclusion of (3) is assumed to be false, then we are able to construct an element [??] [member of] F such that ¦[??](P)¦ > ¦g'(P)¦

The proof of the Riemann Mapping theorem (as we know it today, not developed by Riemann but rather by Carathéodory and Koebe and others) is quite standard and can be found in most any text. For completeness we include it in an appendix to this section.

Riemann's original proof of his theorem solved a somewhat different extremal problem related to the Dirichlet problem. His argument was flawed because he, in fact, assumed that the extremal problem had a solution. He did not prove it. The proof was rescued some years later by refining the Dirichlet principle used by Riemann. Our modern proof sidesteps those difficulties, and uses Montel's theorem to show that the extremal problem has a solution.

There are a number of other approaches to proving the Riemann Mapping theorem. We shall begin the considerations of this chapter by presenting a quite modern proof based on ideas of Thurston. In fact, Thurston introduced the profound new idea of circle packing, and Rodin and Sullivan found a way to prove the Riemann Mapping theorem using circle packing. We present their proof here.

A proof that relies on the solution of the Dirichlet problem follows next. This is quite similar in spirit to Riemann's original approach to the theorem. Then we turn to Ahlfors's generalization of Riemann's theorem.

APPENDIX: Traditional Proof of the Riemann

Mapping Theorem

Proof of (1): If U is bounded, then this assertion is easy: If we simply let a = 1/(2 sup{|z| : z [member of] U}) and b = -aP, then the function f(z) = az + b is in F. If U is unbounded, we must work a bit harder. Since U ≠ C, there is a point Q [??] U. The function [empty set] (z) = z - Q is nonvanishing on U, and U is holomorphically simply connected. Therefore there exists a holomorphic function h such that h2 = [empty set]. Notice that h must be one-to-one since [empty set] is. Also there cannot be two distinct points z1, z2 [member of] U such that h(z1) = -h(z2) [otherwise [empty set](z1) = [empty set] (z2)]. Now h is a nonconstant holomorphic function; hence an open mapping. Thus the image of h contains a disc D(b, r). But then the image of h must be disjoint from the disc D(-b, r). We may therefore define the holomorphic function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since |h(z) - (-b)| ≥ r for z [member of] U, it follows that f maps U to D. Since h is one-to-one, so is f. Composing f with a suitable automorphism of D (a Möbius transformation), we obtain a function that is not only one-to-one and holomorphic with image in the disc, but also maps P to 0. Thus f [member of] F.

Proof of (2): Of course we may select a sequence of elements fj [member of] F so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We want to claim that {fj} is a normal family. But of course each |fj| is bounded by 1, so Montel's theorem applies.

We may derive the desired conclusion once it has been established that the limit derivative-maximizing function is itself one-to-one. Suppose that the fj [member of] F converge normally to a function f0, with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We want to show that f0 is one-to-one into D. The argument principle, specifically Hurwitz's theorem, will now yield this conclusion:

Fix a point b [member of] U. Consider the holomorphic functions gj(z) = fj(z) - fj(b) on the open set U\{b}. Each fj is one-to-one; hence each gj is nowhere vanishing on U \ {b}. Hurwitz's theorem guarantees that either the limit function f0(z) - f0(b) is identically zero or is nowhere vanishing. But, for a function h [member of] F, it must hold that h'(P) ≠ 0 because if h' (P) were equal to zero, then that h would not be one-to-one. Since F is nonempty, it follows that suph [member of] F ¦h'(P)¦ > 0. Thus the function f0, which satisfies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], cannot have f'0(P) = 0 and f0 cannot be constant. The only possible conclusion is that f0(z) - f0(b) is nowhere zero on U \ {b}. Since this statement holds for each b [member of] U, we conclude that f0 is one-to-one.

Proof of (3): Let g [member of] F and suppose that there is a point R [member of] D such that the image of g does not contain R. Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here we have composed g with a transformation that preserves the disc and is one-to-one. Note that [??] is nonvanishing.

The holomorphic simple connectivity of U guarantees the existence of a holomorphic function ψ: U [right arrow] C such that ψ2 = [??]. Now ψ is still one-to-one and has range contained in the unit disc. However, it cannot be in F since it is nonvanishing. We repair this by composing with another Möbius transformation. Define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then ρ(P) = 0, ρ maps U into the disc, and ρ is one-to-one. Therefore ρ [member of] F. Now we will calculate the derivative of ρ at P and show that it is actually larger in modulus than the derivative of g at P.

We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

But g(P) = 0; hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We conclude that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However 1 + |R| > 1 (since R ≠ 0) and |[psi (P)| = [square root of |R|]. It follows, since (1 + |R|)/(2[square root of |R|]) > 1, that

|[psi (P)| > |[g'(P)|

Thus, if the mapping g of statement (3) at the beginning of the section were not onto, then it could not have property (2), of maximizing the absolute value of the derivative at P.

We have completed the proofs of each of the three assertions and hence of the analytic form of the Riemann Mapping theorem. For completeness, we again enunciate the result.

Prelude: This is the celebrated Riemann Mapping theorem (RMT). Our three steps converge now rather nicely to the proof of this result.

Theorem 1.3 Let U be a planar domain which is not the entire plane and which is analytically simply connected. Then there is a conformal map φ: U [right arrow] D from U to the unit disc.

Certainly statements (2) and (3) taken together give us such a mapping which is both one-to-one and onto.

The proof of statement (3) may have seemed unmotivated. Let us have another look at it. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

S(z) = z2

Then, by our construction, with h = µ-1 º S º τ-1:

g = µ-1 º S º τ-1 º ρ

= h º ρ.

Now the chain rule tells us that

|g'(P)| = |h'(0)| • |ρ'(P).

Since h(0) = 0, and since h is not a conformal equivalence of the disc to itself, the Schwarz lemma tells us that |h'(0)| must be less than 1. But this says that |g'(P)| < |ρ'(P)|, giving us the required contradiction.

1.2 A New Proof of the RMT

Capsule: In a lecture given in the 1980s, W. Thurston proposed the idea that conformality can be understood in terms of circle packing. This idea captured the imagination of a number of mathematicians, and a new subject was quickly born.

B. Rodin and D. Sullivan were fairly quickly able to create a circle-packing proof of the Riemann Mapping theorem. This result really put Thurston's idea on the map, and also placed into focus some of the key techniques that would be widely used in this new study.

In this section, we explain the proof of Rodin and Sullivan.

There are many proofs of the Riemann Mapping theorem. Some of them are quite analytical — see, for instance, [BUR, pp. 293 ff.] Others are more geometric. We present one of those here.

In fact, this proof stems from a lecture given by W. P. Thurston in 1985. The basic idea is that we can internally approximate a simply connected planar domain with a circle packing, and then produce a corresponding packing in the unit disc. The natural mapping of the nerves of the packings then gives an approximate conformal mapping. In the limit, the Riemann mapping is produced.

Circle packing has given rise to an entire new approach to complex function theory. The elegant book [STE] is an homage to these new developments. B. Rodin and D. Sullivan [ROS] determined how to make Thurston's ideas concrete, and how to provide a rigorous circle-packing proof of the Riemann Mapping theorem. That is the proof that we present here.

1.2.1 Some Definitions

Let U be a planar domain. A circle packing in U is a collection of closed discs contained in U and having disjoint interiors. See Figures 1.1, 1.2, and 1.3 for a variety of circle packings.

The nerve of the circle packing is the embedded 1-complex whose vertices are the centers of the discs and whose edges are the geodesic segments joining the centers of tangent discs and passing through the point of tangency. We shall restrict our attention in this discussion to a circle packing whose nerve is the 1-skeleton of a triangulation of some open, connected subset in the plane or the Riemann sphere. In particular, a circle packing of the sphere is one whose associated triangulation is a triangulation of the sphere. It is easy to see that, in a circle packing of the unit disc D, the carrier of the associated triangulation is a proper submanifold of the unit disc.

A finite sequence of circles from a circle packing is called a chain if each circle except the last is tangent to its successor. The chain is a cycle if the first and last circles are tangent.

Now let c be a circle in a circle packing. The flower centered at c is the closed set consisting of c and its interior, together with all circles tangent toc and their interiors, and also the interiors of all the triangular interstices formed by these circles.

We shall not prove the next result, but instead refer the reader To [AND1], [AND2], [THU, Chapter 13], and [ROS, Appendix 2].

Prelude: This theorem is geometrically central to the theory. It tells us that any triangulation that we will encounter comes from a circle packing. It also gives an important uniqueness statement.

Theorem 1.4 Any triangulation of the sphere is isomorphic to the triangulation associated to some circle packing. The isomorphism can be required to preserve the orientation of the sphere and then this circle packing is unique up to Möbius transformations.

A topological annulus A in the plane has a modulus mod A that can be defined, without reference to conformal mapping, as the infimum of the L2 norms of all Borel measurable functions ρ on the plane such that 1 ≤ ∫ ρ(z) |dz| along all degree one curves in A. This is closely related to the idea of extremal length; see [AHL1] and also our Chapter 7.

Definition 1.5 An orientation-preserving homeomorphism f between two planar domains is called K-quasiconformal, 1 ≤ K < ∞, if

K-1 mod A ≤ mod [f(A)]≤ K mod A

for every annulus A in the domain of f.

Intuitively, a K-quasiconformal mapping does not distort the picture by more than a factor of K. Some useful facts about quasiconformal mappings are these:

(1)K-quasiconformality is a local property (see [AHL3, page 22]).

(2) A 1-quasiconformal map is conformal and conversely.

Further, we need the fact that a simplicial homeomorphism is K-quasiconformal for K depending only on the shapes of the triangles involved.

1.2.2 A Sketch of Thurston's Idea

Let U be a simply connected, proper subdomain of the plane. Now efficiently fill U with small circles from a regular, hexagonal circle packing. Surround these circles by a Jordan curve. Use Andreeev's theorem to produce a combinatorially equivalent packing of the unit disc, with the unit circle corresponding to the Jordan curve. What we hope is that the correspondence between the circles of the two packings will approximate the Riemann mapping.

1.2.3 Details of the Proof

Lemma 1.6 (The Ring Lemma) There is a constant r depending only on k such that, if k circles surround the unit disc (i.e., they form a cycle externally tangent to the unit disc), then each circle has radius at least r. See Figure 1.4.

Proof of the Lemma: Fix k. There is a uniform lower bound for the radius of the largest outer circle c1 (this occurs when all the k outer circles have equal radius). A circle c2 adjacent to c1 also has a uniform lower bound for its radius because, if c2 were extremely small, then a chain of k - 1 circles starting from c2 could not escape from the crevice between c1 and the unit circle. Repeat this reasoning for the circle c3 adjacent to c2, and so forth.

Lemma 1.7 (The Length-Area Lemma) Let c be a circle in a circle packing of the unit disc. Let S1, S2, ..., Sk be k disjoint chains which separate c from the origin and from a point of the boundary of the disc. Denote the combinatorial lengths of these chains by n1, n2, ... nk. Then

radius (c) ≤ (n-11 + n-12 + ... n- 1k)-1.

Proof: Suppose that the chain Sj consists of circles of radii rj[??]. Then, by the Schwarz inequality,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let sj = [summation][??]r[??l be the geometric length of Sj. We find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore

s [??] min {s1, ..., sk}

satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since s is greater than the diameter of c, this last inequality proves the lemma.

Lemma 1.8 (The Hexagonal Packing Lemma) There is a sequence of real numbers sj, decreasing to zero, with the following property. Let c0be a circle in a finite packing P of circles in the plane, and suppose that the packing P around c0is combinatorially equivalent to n generations of the regular hexagonal circle packing about one of its circles. Then the ratio of radii of any two circles in the flower around c differs from unity by less than sn.

Corollary 1.9 A circle packing in the plane with the hexagonal pattern is the regular hexagonal packing.

(Continues...)

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Table of Contents

Contents

Preface,
1 The Riemann Mapping Theorem,
2 Invariant Metrics,
3 Normal Families,
4 Automorphism Groups,
5 The Schwarz Lemma,
6 Harmonic Measure,
7 Extremal Length,
8 Analytic Capacity,
9 Invariant Geometry,
10 A New Look at the Schwarz Lemma,
Bibliography,
Index,