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Overview
Starting with an exploration of the subject in terms of holomorphic and subharmonic functions, the treatment proves a Schwarz lemma for plurisubharmonic functions and discusses the basic properties of the Poincaré distance and the Schwarz-Pick systems of pseudodistances. Additional topics include hyperbolic manifolds, special domains, pseudometrics defined using the (complex) Green function, holomorphic curvature, and the algebraic metric of Harris. The second part explores fixed point theorems and the analytic Radon-Nikodym property.
Product Details
| ISBN-13: | 9780486810973 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 04/06/2016 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 256 |
| File size: | 39 MB |
| Note: | This product may take a few minutes to download. |
About the Author
Read an Excerpt
The Schwarz Lemma
By Seán Dineen
Dover Publications, Inc.
Copyright © 1989 Sean DineenAll rights reserved.
ISBN: 978-0-486-81097-3
CHAPTER 1
THE CLASSICAL SCHWARZ LEMMA
In this course we discuss intrinsic metrics and distances on complex manifolds. The underlying areas are complex analysis (of one, several, and infinitely many variables), functional analysis (Banach space theory), differential geometry, and potential theory. We assume some familiarity with complex analysis and Banach space theory but assume no knowledge of differential geometry or potential theory. It is our hope that this course will lead to a better understanding of the interconnections between some of the many different aspects of complex analysis and stimulate further research.
N will denote the set of natural numbers, D the open unit disc in the field of complex numbers C, [??] the closed unit disc in C, T = [??]\D is the unit circle, and R and R+ are the set of real numbers and positive real numbers respectively. A domain is a connected open subset of a Banach space and we assume, unless otherwise stated, that all complex manifolds are connected. X and Y will denote complex Banach spaces and, for D1 and D2 complex manifolds modelled on X and Y respectively, H(D1, D2) will denote the set of all holomorphic mappings from D1 into D2. If D2 = C we use the notation H(D1) in place of H(D1, C). We let L(X,Y) denote the vector space of all continuous linear mappings from the Banach space X into the Banach space Y. If Y = C we write X' in place of L(X, C) and if X = Y we write C(X) in place of L(X, X).
In many cases we shall assume that D1 and D2 are domains in X and Y. In this case f [member of] H(D1, D2) if and only if f is continuous and for each one-dimensional affine subspace E of X and each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a holomorphic function of one complex variable. There are many other equivalent definitions (e.g. use Taylor series expansions or Cauchy-Riemann equations) of holomorphic mapping between Banach spaces. To the reader unfamiliar with infinite dimensional holomorphy (i.e. the study of holomorphic mappings between locally convex spaces and, in particular, Banach spaces) we suggest, at least on a first reading, that you treat everything in Part I as finite dimensional but keep in mind that in moving to infinite dimensions we lose local compactness and Lebesgue measure.
1.1 The Schwarz lemma and the Schwarz–Pick lemma
We begin our study by discussing the classical Schwarz lemma of one complex variable. We shall use the Schwarz lemma to define intrinsic distances and these distances will in turn lead us to new versions of the Schwarz lemma. The Schwarz lemma itself consists of four hypotheses and three conclusions, each of which will be modified in later chapters.
If D is a domain in a Banach space or more generally a complex manifold modelled on a Banach space (we use the term complex Banach manifold) we let Aut(D) denote the set of all biholomorphic automorphisms on D, i.e. f [member of] Aut(D) if
(a) f : D ->D if bijective and holomorphic,
(b) f–1 is holomorphic.
For finite dimensional spaces (a) [??] (b), but it is not known if the implication is true in infinite dimensions (Suffridge).
Classical Schwarz Lemma.If f [member of] H(D; D) and f 0) = 0 then
(i) [absolute value of f(z)] ≤ [absolute value of z] for all z [member of] D,
(ii) [absolute value of f'(0)] ≤ 1.
Moreover, if [absolute value of f(z)] = [absolute value of z] for some non-zero z [member of] D or [absolute value of f'(0)] = 1 then f(z) = eiθz for some θ [member of] R and all z in D.
Proof. Since f(0) = 0 we have f(z) = [∞.summation over (n=1)] anzn for z [member of] D. Let h(z) = [∞.summation over (n=1)] anzn–1 for z [member of] D. Then h [member of] H(D) and f(z) = zh(z) for all z [member of] D. By the maximum modulus theorem
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
for all r, 0 < r< 1. Since [absolute value of f(z)] ≤ 1 for all z [member of] D we get, on letting r -> 1, that sup{[absolute value of h(z)]; z [member of] D ≤ 1. Hence [absolute value of f(z)] ≤ [absolute value of z] for all z [member of] D. This completes the proof of (i). Moreover, if [absolute value of f(z0)] = [absolute value of z0] for some z0 [member of] D\{0} then [absolute value of h(z0)] = 1 and, by the maximum modulus theorem, h is a constant function of modulus 1, i.e. there exists θ [member of] R such that f(z) = zh(z) = eiθz for all z [member of] D.
Since f(z)/z = h(z) for z [member of] D\{0} and f(0) = 0 it follows that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
If [absolute value of f'(0)] = [absolute value of h(0)] = 1 then, by the maximum modulus theorem, h is a constant function of modulus 1 and, as before, this implies f(z) = eiθz for some θ [member of] R.
We next prove a generalization of this lemma — the Schwarz–Pick lemma — but in order to do this we require the form of the biholomorphic automorphisms of D. We first define the Möbius transformations φa. For a [member of] C let φa(z) = z – a/1 – [??]z. We have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and φ–a o φa(z) = z for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For θ [member of] R and a [member of] D,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Hence φa(D) = D and φa is a biholomorphic automorphism of D whenever [absolute value of a] < 1.
If f [member of] Aut(D) then ψ := φf(0) o f [member of] Aut(D) and ψ(0) = 0. By the Schwarz lemma [absolute value of ψ'(0)] ≤ 1. Since ψ'–1 [member of] Aut(D) we also have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Hence [absolute value of ψ'(0)] = 1 and by the Schwarz lemma ψ(z) = eiθz for some θ [member of] R and all z [member of] D. Hence f = φ–f(0) o and all elements in Aut(D) are the composition of a rotation and a Möbius transformation.
Schwarz– Pick Lemma.If f [member of] H(D; D) then
(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
We have equality in (i) and (ii) if f [member of] Aut(D). If equality holds in (i) for one pair of points z ≠ w or if equality holds in (ii) at one point z then f [member of] Aut(D).
Proof. Let g = φf(w) o f o φ–w. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all z [member of] D and g [member of] H(D; D). In particular, g(0) = φf(w)(f(w)) = 0. By Schwarz's lemma [absolute value of g([xi]) ≤ [absolute value of [xi]] for all [xi] [member of] D and hence
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and thus
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
This proves (i). If equality holds for some pair z ≠ w then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and by the Schwarz lemma g(z) = eiθz for some θ [member of] R and all z [member of] D. Hence g [member of] Aut(D) and f = φ–f(w) o g o φw [member of] Aut(D).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
This proves (ii). Now suppose we have equality in (ii). Letting z = w and using the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we see that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Hence g [member of] Aut(D) and f = φ–f(w) o g o φw [member of] Aut(D).
Finally, if f [member of] Aut(D) then (i) applied to f and f–1 gives
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and we have equality. Similarly for (ii) we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and again we have equality for f [member of] Aut(D).
Remark. If f(0) = 0 then part (i) of the Schwarz–Pick lemma with w = 0 gives the Schwarz lemma.
The move from one to several complex variables frequently leads to fundamental problems and more technical proofs, and naturally one expects that a further generalization to infinite dimensions should lead to even more difficulties. This is not always the case. Perhaps it is because, in finite dimensions, a first approach is usually via coordinates while, in infinite dimensions, a coordinate-free approach is more natural and the coordinate approach — which assumes the existence of a Schauder basis — is tried later. When the coordinate-free approach is successful, the final proofs are often less technical and the estimates sharper (in the sense that we know they are dimension free). Without coordinates the Hahn-Banach theorem and its corollaries play a more central role (for instance, the Hahn-Banach theorem is used to prove the Cauchy inequalities which are used in the next proposition). We illustrate these points by moving directly to infinite dimensions and giving some generalizations of the results already proved.
First we recall some versions of the Hahn-Banach theorem and introduce some notation.
Hahn–Banach Theorem.Let X be a Banach space.
(a) If C is a closed convex subset of X and x0 [not member of] C then there exists φ [member of] X' such that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(b) If x [member of] X then there exists φ [member of] X', [parallel]φ[parallel] = 1, such that φ (x) = [parallel]x]parallel]
(c) If x [member of] X then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
If f : A -> (Y, [parallel]•[parallel]) is a mapping we let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If A is a Banach space we let A0 denote the open unit ball of A and we write [parallel]f]parallel] in place of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If X and Y are Banach spaces and φ : X ->Y is a linear mapping then we say φ is an isometry if φ(X) = Y and [parallel]φ(x) [parallel] = [parallel]x]parallel] for all x [member of] X. If X0 and Y0 are the open unit balls of X and Y respectively then it is easily seen that an injective linear mapping φ: X -> Y is an isometry if and only if φ(X0) = Y0.
Proposition 1.1 (Harris, Phillips.). If X is a Banach space with open unit ball X0, h [member of] Aut(X0), and h(0) = 0 then h is the restriction to X0 of a linear isometry of X onto X.
Proof. By the Cauchy inequalities [parallel]h'(0)[parallel] ≤ 1 and [parallel](h–1)'(0)[parallel] ≤ 1. Since (h–1)'(0) = (h'(0))–1 we have for all x [member of] X
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and hence [parallel]h'(0)(x)[parallel] = [parallel]x]parallel]. Since h is invertible we have h'(0)(X) = X and hence h'(0) is a linear isometry of X onto X. Let f = (h'(0))–1 o h. Then f [member of] Aut(X0), f(0) = 0, and f'(0)x = x all x [member of] X. Let f'(0) + Pk + ... be the Taylor series expansion of f at 0 with Pk denoting the first non-zero term of degree ≥ 2. Then
f'(0) + nPk + ...
is the Taylor series expansion of
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
at 0. Since fn [member of] H(X0, X0) the Cauchy inequalities imply that [parallel]nPk]parallel] ≤ 1 for all n and hence Pk = 0. Hence f(x) = x for all x [member of] X0 and h(x) = h'(0) o f(x) = h'(0)(x) for all x [member of] X0.?
The technique used above is known as the Cartan iteration trick and can also be used to prove the following:
If D is a bounded domain in a Banach space and f, g [member of] Aut(D) satisfy a = f(a) = [g(a) and f'(a) = g'(a) for some point a in D then f [equivalent to] g.
This, in turn, is crucial in proving, for T> bounded symmetric, that Aut(D) can be endowed with the structure of a Lie group and that complete holomorphic vector fields on arbitrary bounded circular domains are polynomials of degree less than or equal to two (see chapter 9).
The proof given above is due to Harris but it is not his original proof. The original proof used a different one-dimensional extension of the Schwarz lemma and complex extreme points and, although more technical, had several other consequences including a simple proof of the strong maximum modulus theorem (proposition 6.19).
In the proof of proposition 1.1 we used the fact that h [member of] Aut(X0) to show that h'(0) was a linear isometry from X onto X. Conversely if h [member of] H(X0, X0) and h'(0) is a linear isometry of X onto X then h [member of] Aut(D). This result, when h(0) = 0, is a consequence of the following continuous form of the Schwarz lemma which we state without proof and which is also due to Harris.
(Continues...)
Excerpted from The Schwarz Lemma by Seán Dineen. Copyright © 1989 Sean Dineen. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
The Schwarz lemma is among the simplest results in complex analysis that capture the rigidity of holomorphic functions. This self-contained volume provides a thorough overview of the subject; it assumes no knowledge of intrinsic metrics and aims for the main results, introducing notation, secondary concepts, and techniques as necessary. Suitable for advanced undergraduates and graduate students of mathematics, the two-part treatment covers basic theory and applications.
Starting with an exploration of the subject in terms of holomorphic and subharmonic functions, the treatment proves a Schwarz lemma for plurisubharmonic functions and discusses the basic properties of the Poincaré distance and the Schwarz-Pick systems of pseudodistances. Additional topics include hyperbolic manifolds, special domains, pseudometrics defined using the (complex) Green function, holomorphic curvature, and the algebraic metric of Harris. The second part explores fixed point theorems and the analytic Radon-Nikodym property.
Dover (2016) republication of the edition originally published by the Clarendon Press, Oxford, 1989.
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