Invariant Distances and Metrics in Complex Analysis

DE GRUYTER EXPOSITIONS

Marek Jarnicki Peter Pflug

IN MATHEMATICS

Invariant Distances and Metrics in Complex Analysis

w DE

G

M

de Gruyter Expositions in Mathematics 9

Editors

O. H. Kegel, Albert-Ludwigs-Universitat, Freiburg V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, Ohio State University, Columbus R. O. Wells, Jr., Rice University, Houston

de Gruyter Expositions in Mathematics

I

The Analytical and Topological Theory of Semigroups, K. H. Hofmann, J. D. Lawson, J. S. Pym (Eds.)

2

Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues

3

The Stefan Problem, A. M. Meirmanov

4

Finite Soluble Groups, K. Doerk, T 0. Hawkes

5

The Riemann Zeta-Function, A. A. Karatsuba, S. M. Voronin

6

Contact Geometry and Linear Differential Equations, V. R. Nazaikinskii, V. E. Shatalov, B. Yu. Sternin

7

Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, M. V. Zaicev

8

Nilpotent Groups and their Automorphisms, E. I. Khukhro

Invariant Distances and Metrics in Complex Analysis by

Marek Jarnicki Peter Pflug

W DE

G Walter de Gruyter Berlin New York 1993

Authors

Marek Jarnicki Instytut Matematyki Uniwersytet Jagiellotiski 30-059 Krakow, Poland e-mail: [email protected]

Peter Pflug Fachbereich Mathematik

Universiti t Osnabruck - Standort Vechta D-2848 Vechta, Germany e-mail: pflugvec(a)dosunil.rz.uni-osnabrueck.de

1991 Mathematics Subject Classification: Primary: 32-02 Secondary: 32 Hxx; 32 Exx

U Punted on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Jarnicki, Marek.

Invariant distances and metrics in complex analysis / by Marek Jarnicki, Peter Pflug. cm. - (Dc Gruyter expositions in mathematics ; 9) p. Includes bibliographical references and index. ISBN 3-11-013251-6 1. Metric spaces. 2. Pseudodistances. 3. Invariant%. 4. Functions of complex variables. I. Pflug. Peter. 1948-. Ii. Title. III. Series. QA611.28.J 37 514'.32-dc2O

1993 93-17545

CIP

Die Deutsche Bibliothek - Cataloging-in-Publication Data Jarnicki, Marek:

Invariant distances and metrics in complex analysis / by Marek Jarnicki ; Peter Pflug. - Berlin ; New York : de Gruyter, 1993 (De Gruyter expositions in mathematics ; 9) ISBN 3-11-013251-6 NE: Pflug. Peter:; GT

.C, Copyright 1993 by Walter de Gruyter & Co., D-10785 Berlin.

All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Disk Conversion: D. L. Lewis, Berlin. Printing: Gerike GmbH, Berlin. Binding: Luderitz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.

To Mariola and Rosel

Preface

One of the most beautiful results in the classical complex analysis is the Riemann mapping theorem which says that, except the whole complex plane, every simply connected plane domain is biholomorphically equivalent to the unit disc. Thus, the topological property "simply connected" is already sufficient to describe, up to biholomorphisms, a large class of plane domains. On the other hand, the Euclidean ball and the bidisc in C2 are topologically equivalent simply connected domains but they are not biholomorphic. This observation, which was made by H. Poincare as early as at the end of the last century, shows that even inside the class of bounded simply connected domains there is no single model (up to biholomorphisms) as it is in the plane case. Therefore, it seems to be important to associate with domains in C" tractable objects that are invariant under biholomorphic mappings. Provided that these objects are sufficiently concrete, one can hope to be able to decide, at least in principle, whether two given domains are biholomorphically distinct. An object of this kind was introduced, for example, by C. Caratheodory in the thirties. His main idea was to use the set of bounded holomorphic functions as an invariant. More precisely, he defined pseudodistances on domains via a "generalized" Schwarz Lemma. A specific property of these pseudodistances is that holomorphic mappings act as contractions. Thus, in particular. biholomorphic mappings operate as isometrics. For such objects the name "invariant pseudodistances" has become

very popular. This is where the title of our book comes from, although in the text we prefer to talk about holomorphically contractible pseudodistances. Apart from the class of bounded holomorphic functions, other classes of functions are used to obtain, via extremal problems, new objects contractible with respect to certain families of holomorphic mappings. For example, the class of square integrable holomorphic functions was used by S. Bergman. Moreover, all these objects admit infinitesimal versions associating to any "tangent vector" a specific length contractible under holomorphic mappings. Besides using families of functions to associate (via an extremal problem) tractable objects with domains in C", one can consider sets of analytic discs as new biholomorphic invariants. This idea is due to S. Kobayashi.

The main goal of our book is to present a systematic study of invariant pseudodistances and their infinitesimal counterparts, the invariant pseudometrics. To illustrate various aspects of the theory we add a lot of concrete examples and applications. Although we have tried to make the book as complete as possible, the choice of topics we present obviously reflects our personal preferences.

viii

Preface

Our interest in this area started in the middle of the eighties when we, somehow accidentally, came across the "Schwarz Lemma on Cartesian products" (in the terminology of the book, the "product property of the Carath6odory distance"). This result was stated in the 1976 survey article Intrinsic distances, measures and geometric function theory of S. Kobayashi but no proof was given there (in fact, as it turned out later, no proof did exist at that time). In our attempts to find a proof of this theorem we have gone deeper into the field of invariant distances. For instance, we have learned that a lot of seemingly simple questions were still waiting for solutions. We were able to solve some of them but most remain still without answer. We have put many of these problems into the text (marking them by B. The reader is encouraged to work on some of them. According to our experience over the last ten years, we feel that we should refrain from discussing manifolds and complex spaces. So we only deal with domains in C1. Even here, of course, plenty of results are beyond the scope of our book. For the convenience of the reader who would like to go further, we collect (without proofs) a part of this material in a supplementary chapter (Miscellanea). We mention that although many of the results in the book are stated in the domain case, they can be almost literally transformed to the manifold case; see, for instance, [Aba 3J, [Din], [Fra-Ves], [Kob 4J, [Lang], [Nog-Och].

During the preparation of this book we had to decide what kind of knowledge the reader is supposed to have. We have assumed that he is familiar with standard complex analysis of several variables. Of course, what we mean by "standard" reflects our academic education. As a form of a compromise, we add an appendix in which we collect results we assume known (or which are not easy to find in the literature). Moreover, chapters conclude with rough Notes and some Exercises. In the text we often use certain standard symbols and notation without explicit

definitions and the reader is referred to the "List of symbols" at the end of the book. Moreover, abbreviations HF, PSH, PSC, AUT, OR, MA, and H refer to the sections of the Appendix. It is our deep pleasure to be able to record a debt of gratitude to our teachers: Professors Hans Grauert and J6zef Siciak, who have led our first steps in complex analysis. Next, we would like to thank our colleagues for stimulating discussions and help during writing this book. Especially, thanks are due to M. Capinski, H: J. Reiffen, R. Zeinstra, and W. Zwonek, who also helped us in corrections of the text. We express our gratitude to Mrs. H. Bdske who spent a lot of time typing and retyping our notes. We thank both our universities for support before and during the preparation of the book. Finally, we are deeply indebted to the Walter de Gruyter Publishing Company, especially to Dr. M. Karbe, for having encouraged us to write this book.

Krak6w - Vechta, December 1992.

Marek Jarnicki Peter Pflug

Contents

Preface I

11

III

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Hyperbolic geometry of the unit disc . . . . . . Exercises . . . .

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vii

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1

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14

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15

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16

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27 29

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33

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The Carathr odory pseudodistance and the Carathdodory-Reiffen pseudometric . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definitions. General Schwarz-Pick Lemma . . . . . . . . . . . . . . . . 2.2 Balanced domains . . . . . . . . . . . . . 2.3 Caratheodory hyperbolicity 2.4 The Caratheodory topology . . . . . . . . . . . . 2.5 Properties of c('f and y. Length of curve. Inner Caratheodory pseudodistance . . . . . . . . . . . . . . . . . .

2.6 Two applications . . . . 2.7 A class of n-circled domains

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Notes . Exercises

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48

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53 65 66

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The Kobayashi pseudodistance and the Kobayashi-Royden . . . . . . . . . . . . . . . . . . pseudometric 3.1 The Lempert function and the Kobayashi pseudodistance . . . . . . . . . . . . . . . . . . 3.2 Tautness . . . . . . . . . . . . 3.3 General properties of k . . . . . . 3.4 An extension theorem 3.5 The Kobayashi-Royden pseudometric 3.6 The Kobayashi-Buseman pseudometric

IV

3.7 Product-formula Notes . . . . . .

Exercises

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71 71

77 82 87 90 99

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106 108 109

. . . . . . . . . . . . . . Contractible systems 4.1 Abstract point of view . . . . . . . . . . . 4.2 Extremal problems for plurisubharmonic functions 4.3 Inner pseudodistances. Integrated forms. Derivatives. . . Buseman pseudometrics. C' -pseudodistances

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111

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111

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115

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139

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Contents

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4.4 Example - elementary n-circled domains

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149 152 153

Contractible functions and metrics for the annulus Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . .

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154

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165 166

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Notes . Exercises V

VI

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. . The Bergman metric . . . . . . 6.1 The Bergman kernel 6.2 The Bergman pseudometric 6.3 Comparison and localization 6.4 The Skwarczynski pseudometric Notes . . . . . . . . . . . . . . . . . . . . . . Exercises

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169 169 185 190 195 198

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200

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202 202 207 213 216 223 230 234 235

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VII Hyperbolicity and completeness . . . . . . 7.1 Global hyperbolicity 7.2 Local hyperbolicity . . . . . 7.3 Completeness - general discussion 7.4 Carathdodory completeness . . . . . 7.5 Kobayashi completeness . . . 7.6 Bergman completeness Notes . Exercises

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237 237 243 255 264 278

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VIII Complex geodesics. Lempert's theorem . . . . . . . . . . 8.1 Complex geodesics . . . . . . . . 8.2 Lempert's theorem 8.3 Uniqueness of complex geodesics . . 8.4 Geodesics in convex complex ellipsoids 8.5 Biholomorphisms of complex ellipsoids

8.6 Schwarz Lemma - the case of equality

IX

X

8.7 Criteria for biholomorphicity Notes . . . . . . . . . . .

Exercises

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281

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285 288 290

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Comparison on strongly pseudoconvex domains . . . 10.1 Strongly pseudoconvex domains

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Product-property . . Exercises

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296 309

310 310

Contents

xi

10.2 The boundary behavior of the Carathdodory and the Kobayashi distances . . . . . . . . . . . . . . 10.3 Localization . . . . . . . . . . . . . . . . . 10.4 Boundary behavior of the Caratheodory-Reiffen and the Kobayashi-Royden metrics . . . . . . . . . 10.5 A comparison of distances . . . . . . . . . . . . . 10.6 Characterization of the unit ball by its automorphism group Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises Miscellanea . . . . . . . . . . . . . . . . . A The automorphism group of bounded domains Holomorphic curvature B . . . . . . . . C Complex geodesics . . . . . . . . . . D Criteria for biholomorphicity . . . . . . E Boundary behavior of contractible metrics on weakly pseudoconvex domains . . . .

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316 326

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331

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342 344 352 353

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355 355 356 359

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361

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363

Appendix . . . . . . . . . . . . . . . . . . . . . HF Holomorphic functions . . . . . . . . . . . PSH Subharmonic and plurisubharmonic functions . . PSC Domains of holomorphy and pseudoconvex domains AUT Automorphisms . . . . . . . . . . . . . . Automorphisms of the unit disc . . . . . . . Automorphisms of the unit polydisc . . . . . . Automorphisms of the unit Euclidean ball . . .

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GR Green function and Dirichlet problem MA Monge-Ampere operator . . . . Hardy spaces

H

References

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List of symbols Index

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367 367 370 375 379 379 379 380 380 383 384 387 400 405

Chapter I

Hyperbolic geometry of the unit disc

The main concept of the theory that will be developed in this book is to describe the holomorphic structure of a given domain G C C" in terms of geometric properties

of the space (G. dc; ), where dG is a suitable pseudodistance on G. We will see that there are various systems (G, dG)G which may be useful for this purpose. We will always assume that any such a system is holomorphically coherent in the sense that, for two arbitrary domains G C C", D c C', any holomorphic mapping F: G -+ D is a contraction of (G, dG) into (D, dD), i.e. dp(F(z"), F(z )) < dG(z', z"),

z', z" E G.

(*)

In particular, any biholomorphic mapping F: G -+ D has to be an isomerry between (G, dG) and (D, do). Obviously, if the system (G, dG)G is too poor we cannot expect any essential influence of geometric properties of (G, dG) on holomorphic properties of G. Therefore, one must exclude these trivial systems. We can reach this goal by different methods; the simplest way seems to be the following one. Assume that dE is "very good" (from holomorphic and geometric points of view), where E denotes the open unit disc in C. Then we can hope that, in view of (*), at least some of the "very good" properties of dE propagate via holomorphic mappings to almost all (G, dG). The aim of this chapter is to explain what we mean by "very good" properties of dE. Let m(A

A'A" E E,

") 1

Y(A) := 1 - 1A12'

A E E.

Note that the definition of m may be extended to C x C \ ((A', V): A'A" = 1). Using the above notation, the Schwarz.-Pick Lemma may be formulated as follows (cf. [Rud 1)):

1. Hyperbolic geometry of the unit disc

2

Lemma 1.1 (Schwarz-Pick Lemma). Let f E O(E, E). Then

(a) m(f(A'), f

m(A', A"), ),,, A"I E E;

(b) Y(f (A))If'(A)I < Y(A), A E E; (c) the following statements are equivalent:

(i) f E Aut(E), (ii) in (f (A'), f

in

A"), A', A" E E,

(iii) in (f (A'0), f (),o)) = m(A'0, 1 0 " ) f o r some A , 10" E E with Ao (tv) Y(f (A))I f'(A)I = Y(A), A E E, (v) Y(f (A0))If'(Ao)I = Y(A0) for Some Ao E E.

Roughly speaking, any holomorphic function f : E -> E is an m- and a ycontraction. Moreover, the only holomorphic m- or y-isometries are the automorphisms of E. Now, we list the elementary properties of the functions in and y that will be useful in the sequel:

1.2. M E C°O(E x E \ ((A, X): A E E)), m2 E C°°(E x E), Y E C°D(E).

1.3. For any a E E, m(., a) = IhaI, where ha(A) := (A-a)/(1-aX); cf. Appendix AUT 1. In particular, in (., a) = 1 on a E, log m (,a) is subharmonic on E and harmonic in E \ (a). Since in is symmetric, the same is true for m(a, ); see also Remark 1.19. The function log y is subharmonic on E.

1.4.

slim

AIW a IA'

1111

Y(a),

a E E.

1.5. The function - log in (a, .) is the classical Green function for E with pole at a; cf. Appendix OR. If we put u := m2(a, ), then

y2(a) = 4(Au) (a) = (Lu)(a;1), where Cu is the Levi form of u; cf. Appendix PSH.

Lemma 1.6. For any a, b, c E E, a 0 b # c 96 a, we have in (a, b) < in (a, c) + in (c, b).

In particular, m : E x E -+ [0, 1) is a distance.

I. Hyperbolic geometry of the unit disc

3

The function in is called the Mobius distance.

Proof. Observe that for any a, b E E, a 5L b, there exists a unique automorphism

It = h",b E Aut(E) such that h(a) = 0 and h(b) E (0, 1). The function in is invariant under Aut(E), and therefore without loss of generality we may assume that a = 0, b E (0, 1). Then the inequality (1.1) reduces to the following one:

b < Jel +

c-b 1 - cb

,

CEE\{0,b}.

0

r)), Remark 1.7. Since in is invariant under Aut(E), B,,,(a, r) = a E E, 0 < r < 1, where B,,, denotes the ball w. r. t. the distance in: cf. Exercise I.I. In particular: - the topology generated by in coincides with the Euclidean topology of E, - the space (E. m) is complete. Remark 1.8. The strict triangle inequality (1.1) says that the m-segment [a, b]", := {A E E: m(a, A) + m(A, b) = m(a, b)} consists only of the ends. Thus, from the geometric point of view, the space (E. in) is trivial. Therefore we have to look for a new candidate to be a "good" distance. Property 1.4 suggests the following way which has its roots in differential geometry.

Let a: [0, 1]

E be a piecewise Cl-curve. We define its y-length by the

formula:

y(a(t))la'(t)I dt.

Ly(a) 0

Remark 1.9. In each case where we assign an object (like L. (a) above) to a (continuous) curve a : [a, b] -> C", the reader should verify whether the definition of this object is independent of the following standard identifications: [a, b] is an increasing - change of parametrization: a ^_- a o cp, where w : [a', b']

bijection which is assumed to be of the same class as a. e.g. piecewise C' if a is piecewise C, U - cancellation of constant parts: if a = const on [h, 121 c [a, h], then a a I aIit,,b].

Notice that in most cases the objects associated with curves will be also independent of the orientation of a.

4

1. Hyperbolic geometry of the unit disc

Remark 1.10. For any f E O(E, E) we have Ly(f o a) < Ly(a). In particular, the y-length is invariant under Aut(E). Define

p(A', A") := inf(Ly(a): a: [0, 1]

E,

a is a piecewise C1-curve, A' = a(0), X" = a(1)}, A'. A" E E.

Remark 1.11. It is clear that p : E x E -* R+ is a pseudodistance dominating the Euclidean distance and that for any holomorphic function f : E -+ E we have (A'), f(A"))

p(f

< p(A', I").

,l', A" E E.

In particular, p is invariant under Aut(E).

It is natural to ask whether for given a, b E E, a ,- b, there exists a C'-curve a joining a and b in E for which p(a, b) = Ly(a); in differential geometry such a curve is called a geodesic. If the answer is positive, then the next problem is to decide whether geodesics are uniquely determined up to the above identifications.

For 0 < s < I let a, (t) := Is, 0 < t < 1, i.e. a, denotes the interval [0, s] regarded as a curve. For a, b E E, a 0 b, let a,,.h := h-1 o at, (b), where h = hd,b is the automorphism defined in the proof of Lemma 1.6. Note that the image I..b of the curve a0,b lies on the unique circle C0,b that passes through a and b and is orthogonal to 8E; C0.h = hu,h(R).

Lemma 1.12. For any a, b E E, a 0 b, we have

p(a.b) = Ly(a0,b) = tank-'(m (a, b)).

(1.2)

Moreover, au.h is the unique geodesic joining a and b.

Recall that tanh-1 (t) =

;

log L+f and (tanh-1)'(t)

0 < t < 1.

Proof. All the objects involved in (1.2) are invariant under Aut(E) and so we may assume that a = 0, b E (0, 1), and consequently a,,,b = ab. First, observe that ph

P(0, b) < Lr(ah)

J

0

l

dtt` =

I

log

1 +h = tanh-'(m(0, b)).

1. Hyperbolic geometry of the unit disc

5

On the other hand, if a = u + iv: [0, 1] -* E is a piecewise C'-curve joining 0 and b, then

Ly(a) >

f' 0

u4(t) dt = 2 log 1 +b.

(1.3)

1

Thus (1.2) is satisfied and, moreover, if p(0, b) = Ly(a), then in (1.3) we have equality. This implies that v = 0, u : [0, 1] -> [0, b], and u is increasing (remember the identifications of Remark 1.9). Finally a as,. 0

Corollary 1.13. (a) p is a distance with m < p. (b) For any f E O(E, E) if p(f (Ao), f (),o)) = p(Ao, A) for some Ao, Aa E E, 40'4 then f E Aut(E). Xo (c) Bp (a, r) = B,,,(a, tanh(r)), a E E, r > 0. In particular, - the topology generated by p coincides with the standard topology of E, - (E, p) is complete. y(a), a E E.

(d) limx, A -a

(e) [a, b]P = la,b, i.e. the p-segments coincide with the images of geodesics. In

particular, p(0, s) = p(0, t) + p(t. s), 0 < t < s < 1. The distance p is called the Poincare (hyperbolic) distance. Note that, in view of (e), (E, p) is a model of a non-Euclidean geometry (the Poincare model).

We are going to justify that p is just an example of a "very good" distance we are looking for. We have already seen that from holomorphic, topological, and geometrical points of view the distance p behaves very regularly. Now, we like to point out other useful properties of p. Let a : [0, 11 -* E be a (continuous) curve. Put N

Lp(a) := sup(E p(a(tj_1). a(tj)): N E N.0 = to < ... < tN = 1}. j=1

The number Lp(a) E [0, +00] is called the p-length of a. If Lr(a) < +oo, then we say that a is p-rectifiable. Note that Lr(a) > p(a(0), a(1)). Remark 1.14. (a) For any f E O(E, E) we have L p(f oa) < L,,(a). In particular, LP is invariant under Aut(E). (b) By Corollary 1.13(e) we get Lp(a..b) = p(a, b).

Corollary 1.15. We have p = p', where

p`(a, b) := inf(LP(a): a: [0, 1] -> E, a is a curve joining a and h},

a, b E E.

1. Hyperbolic geometry of the unit disc

6

The above corollary shows that p is an inner distance. It is clear that we can repeat the same procedure for the distance m: first we define L. (a) (observe that Remark 1.14(a) remains true) and next we put

m' (a, b) := inf{L(a) : a :

[0, 1]

E, a is a curve joining a and b},

Lemma 1.16. (a) For any curve a: [0, 1] -* E we have

a, b E E.

LP(a). In

particular,

m' = p.

(1.4)

Moreover, a is m- or p-rectifiable iff a is rectifiable in the Euclidean sense. (b) For any piecewise C' -curve a : [0, 1] -* E we have

L,(a) = L,(a).

(1.5)

Notice that (1.4) may be used as an alternative way to define p. Moreover, condition (1.4) shows that m is not an inner distance.

Proof. (a) First observe that for any compact K C E there exists M > 0 such that MIX'-X"I <m(A'

x") < p(A',A") <MIA'-A"I,

A'.A"E K.

Hence for any curve a : [0, 11 --> K one gets

MLu I(a) < Lm(a) < L,(a) < MLII 11(a), where L IlI,(a) denotes the length of a in the Euclidean sense. Consequently, all the notions of rectifiability coincide. By Property 1.4 and Corollary 1.13(d), for any compact K C E and for any e > 0 there exists 3 > 0 such that 0

A',A"E K, IA' -A"I < 6,

which directly implies that Lm(a) = Lp(a); see also Lemma 2.5.2(a). (b) Without loss of generality, we may assume that a is of class C' I. By the same argument as above (Corollary 1.13(d)), for any e > 0 there exists n > 0 such that

p(a(t'), a(t")) _ Y(«(t'))I«'(t')I I < s. It' - t"I

-

0 < r'. t" < 1,

I. Hyperbolic geometry of the unit disc

7

which gives (1.5); see also Lemma 2.5.2(c).

One may also ask how close is the Poincare geometry to the holomorphic one, that is, what are the relations between the set Isom(p) of all p-isometries of E and the group Aut(E). Observe that Isom(p) = Isom(m). We can also study the

set Isom(y) of all y-isometries of E, i.e. the set of all C'-mappings f : E -* E such that

Y(f(A))I(dzf)(X)I = Y(A)IXI.

A E E, X E C,

where dx f : C -+ C denotes the R- differential of f at A. The full answer to this question is given by the following Proposition 1.17. For any mapping f : E -+ E the following conditions are equivalent:

(i) f E Isom(p), (ii) f E C1 and f E Isom(y),

(iii) either f E Aut(E) or f E Aut(E). In other words, Isom(p) = Isom(y) = Aut(E) U Aut(E).

Proof. It is clear that (iii) -+ (1) and (iii) 4 (ii). (i) -+ (iii). Taking e'oh f(o) o f in place of f we may assume that f (0) = 0 and that f (xo) = xo for some 0 < xo < 1. Then we have If(A)I = IAI and I

f (A) -xo I - f (A)xo

AEE.

Hence Ref (A) = ReX, A E E, and consequently either f (A) = A or f (A) (ii) --> (iii). Since f is a y-isometry, we have If.'(A)a + f,()L),el = C(A)la + if 1,

A E E, a, P E R,

where C(X) := Y f)a)) > 0. Hence for each A E E there exists e(A) E (-1, 1) such that fx(A) = e(A)if).(A) 0 0. Since the partial derivatives are continuous, the function a has to be constant, and consequently f is either holomorphic or antiholomorphic. Hence, by the Schwarz-Pick Lemma, f E Aut(E) UAut(E). We add two more properties of p. Proposition 1.18 (cf. [Fad 1), [Ves 2]). The function log p is strictly plurisubharmonic on E x E \ ((A, A): A E E). In particular, log p c- PSH(E x E).

1. Hyperbolic geometry of the unit disc

8

Proof Put u := log p, fix a, b E E, a 54 b, and let h := h.,b. Then

(Cu) ((a, b); (a, fl)) = (Gu)((0, h(b)); (h'(a)a h'(b)f )),

a, fl E C.

Thus, it suffices to prove that

(Lu) ((0, t); (a, fi)) > 0,

0 < t < 1, (a, P) E (C2),.

Elementary (but tedious) calculations give

(Cu)((0,t);(a,#)) = [4t(I -t2)2T2(t)]-1 x [t2T(t)I(1 -t2)a+f12

+(T(t)-t)l(l -t2)a-#12], 0 t for

0
0

Remark 1.19 (cf. [Ves 2]). We have m 1 PSH(E x E). In fact, (C,n)((0, t); (Cr. li)) = 4t

.

[(I - 12)21a12 + 2(312 - 1) Re(aO) + I#I2],

0
Proposition 1.20. Let d : E x E --> R be a function such that (i) d is invariant under Aut(E), (ii) d(0, s) = d(0, t) + d(t, s), 0 < t < s < 1,

(iii) lim"o+

d o,r

=

1.

Proof. Let V(t) := d(0, t), 0 < t < 1. In view of (ii) and (iii), W(O) = 0 and cp'(0) = 1. We will show that

(P (1) = 1

1

t2 = Y(t),

0 < t < 1.

(1.6)

I. Hyperbolic geometry of the unit disc

9

Suppose for a moment that (1.6) is true. Then S

GP(s) =

Jo

+

dt

qP'(t) dt = f s

1

t2 = 2 log

s

1

= P(0, s),

0<s<

and hence by (i), d - p. To prove (1.6), fix 0 < to < 1 and let t > 0 be such that to + t < 1. Because of (ii), we get

co(to + t) - (p(to) = d(to, to + t). On the other hand, by (i) we have

d(to,to+t) =d(hro(to),hro(to+t)) =d(0,

t 1 - (to + t )to

Finally,

lim

WP(to + t) - ((to)

=

1

1 - to

The proof for the left derivative is analogous.

For another axiomatic description of p see [Eis]. At the end of this introductory chapter we show how to establish the theorems of Picard in an elementary fashion, i.e. without using elliptic modular functions (cf. [Min-Sch]). The argument here is based on the following characterization of the function y (cf. [Ahl], [Min]).

Proposition 1.21. (a) 0(logy) = 4y2 on E. (b) If f : E -> [0, oo) is a continuous function which is C2 on (A E E: SB(A) > 0) and which satisfies

2 (log f)(A) ?

4f2()')

for A E E with j6 (X) > 0, then either 8(A) < y(A) for every A E E or p = Y on E. Proof. (a) Compute ! (b) First we will prove that IB < y.

For0
rIAI2,

yr()')

r2

AE

rE.

I. Hyperbolic geometry of the unit disc

10

5'r i i) = 0. Thus. if g, is not Since 6 is bounded on rE, it follows that identically zero, then g, takes its positive maximum on rE at a point i., E rE. Then by assumption gr is a C2-function near Ar and .o we obtain

A(loggr)(A,) < 0, i.e. 4(022(Xr)

- Yr(X,,)) < p(1og18)(Ar) -A

y.;ki.,

Thus gr(A) < 1 for every k E rE which finally gives

g(A) = limgr(A) < 1,

r/l

A E E,

i.e.f
such that

B(A1, RI) CC A

and

Aa E B(A1, R2) CC B(0, R3).

On B(0. R3) we define

u(A) := log(fi(A)/Y(A))

A E B(0, R3).

Observe that u is upper semicontinuous, u < 0, and u is C2 where A E B(0, R3) with 6(A) > 0. Then a calculation gives

0. Fix

AU P.) _ A(logfi)(A) - 0(logY)(A) 4($(),) + Y(A))(fl(A) - Y(A)) > 8M(fi(),) - Y(,X)),

where M := (l - R3)-l. The elementary inequality M log(t/s) > t - s, 0 < s < t < M, finally leads to Au(A) > 8M2u(A) whenever JAl < R3 and fi(A) > 0. Now, we introduce the following auxiliary function tr: B0.1. R,, ---

with

w(A) := u(A) + Ev(A), where v(A) := exp(-aIA - Al12) - ex p{ -uR i.

The positive numbers a and E are chosen such that the following to conditions are satisfied: (i)

4

Av(A) - 8M2v(A) > (4a2IA - Al I2 - 4a - 8M2) exp(-aIA - ? 12;

0

I. Hyperbolic geometry of the unit disc

1l

whenever R, <- IA - A, I < R2; (ii)

w(A) < 0 on B(A,, R,).

Since u(Ao) = 0 < v(Ao) and vlaB(A,.R,) = 0, the function w attains its positive maximum inside the annulus B(A,, RZ) \ B(X1, R,) at a point A2. Then we have 0 > Ow(A2) > 8M2w(A2), which contradicts w(A2) > 0.

Corollary 1.22 (Ahlfors-Schwarz Lemma). Let G C C be any domain and let

0: G -> [0, oo) be a continuous function that is C2 on 11 E G: f(A) > 0}. Moreover, assume that A(log P)(A) > Cf2(A) for every A E G with f (A) > 0; here C denotes a fixed positive number. Then for every f E O(E, G) either P (f (),))I f'(A)I <

4/Cy(A) for all A E E

(1.7a)

or

41Cy on E.

(1.7b)

Proof. Set 16 (A) :=f (f (),))I f'(A)I, A E E. and apply Proposition 1.21.

Remark. If G = E and i6 = y, then Corollary 1.22 is nothing else than the classical Schwarz-Pick Lemma.

Corollary 1.23 (Liouville Theorem). Under the conditions of Corollary 1.22 with fi(A) > O for every A E G. any f E O(C. G) is necessarily constant.

Proof Suppose that f E O(C, G) with f'(Ao) 0 0. Put g(A) := f (Ao+A), A E C. Then (1.7) with gR(A) := g(RA), A E E, implies that $(gR(0))Ig'R(0)I < (4/C)112,

R > 0.

Thus

g'(0) = f'(Ao) = 0, a contradiction. Observe that with G = E and f = y, Corollary 1.23 is just the classical Liouville Theorem. Moreover, it also contains the Little Picard Theorem. For it suffices to find a suitable function iB > 0 on G := C \ (0, 1), which satisfies the assumptions of Corollary 1.22.

1. Hyperbolic geometry of the unit disc

12

For k E C \ 10, 11 =: G, we define (cf. [Min-Sch]) (I + IAII/3)I/2

0(1)

(1 + 11 -11/3)1/2 11

1A15/3

(1.8)

- x.15/3

An easy calculation then gives

t = (log#)0)

1

1

1A15/3(1 + 1A11/3)2 + I I - kl5/3(l + 11 - A11/3)2)

18

and

lim

1XI-a

1

0(log 0)(A) l2(X)

- { +oo

if a = 00

1/32 if a = 0 or a = 1

Thus there exists a positive C with

A(logfl()L)) ? Cl2(),),

A E G,

and Corollary 1.23 can be applied. Finally, we give a short sketch how to establish the Big Picard Theorem.

Theorem 1.24 (Big Picard Theorem). Let f : E. -+ C be a holomorphic function that has an essential singularity at 0. Then f (E.) contains the whole C except at most one point.

Proof. We argue by contradiction. So, we suppose that f (E.) C C \ (0, 1).

Set fk(,k) := f (1/2 k), A E E.. Assume for a moment that (fk) is a normal family. Then it is easy to show that the origin is either a removable singularity or a pole for f ; a contradiction. Therefore, it suffices to indicate how to prove the so called Montel Theorem which says that any family F C O(E., C\(0, 1)) is normal. Observe that normality is a local property. Thus it is enough to verify that O(E, C \ (0, 1)) is a normal family.

By f we denote the function given by (1.8). Then using Corollary 1.22 we have

f(f(A))1f'(),)I

C1Y(A),

f E O(E, C \ (0, 1}), A E E,

where C, is a suitably chosen positive constant independent of f. Moreover, because of the growth of f we can find another constant C2 > 0 such that 21f'(X)I

< C2fi(f(A))If'(A)I

1+If(A)I2 -

CIC2Y(A),

A E E.

I. Hyperbolic geometry of the unit disc

13

Thus the spherical derivatives of f E O(E, C \ (0, 1)) are locally uniformly bounded from above. Hence Marty's normality criterion (see [Con], Th. 3.4) applies and it follows that O(E, C \ (0, 1)) is normal. Observe that Theorem 1.24 is equivalent to the fact that any holomorphic function f : E. -+ C which omits at least two complex values extends either holomorphically or meromorphically to E.

Remark. The theorems of Landau and Schottky can be also derived along the above ideas (see [Min-Sch], also [Hah 4]); cf. Exercise 1.4.

I. Hyperbolic geometry of the unit disc

14

Exercises 1.1. Prove that

Bm(a,r) =

B(a(l - r2)

r(1 - 1a12)

1-r21a12' 1-r21aI2)'

aEE,O
1.2. Let µ denote a measure on E defined by

µ(A) =

JA

y2(l)dm2(A),

where m2 denotes two dimensional Lebesgue measure on C and A C E is any Lebesgue measurable set. Prove that µ is invariant under Aut(E). Let T denote the p-triangle with vertices at a, b, c E E, i.e. T is determined by Ca,b, C,, and Cc,a. Prove that µ (T) = [jr - (a + 0 + y) 1, where a, P, y denote a the angles of T. 1.3. (a) Verify the formula for (L u) ((0, r ): (a, f)) in the proof of Proposition 1.18. (b) Verify the formula for (,Cm) ((0, t); (a, f4)) in Remark 1.19. 1.4. Prove the following theorems. Theorem of Landau. For any (ao, a1) E C X C. there exists a positive number

R = R(ao, ai) such that, whenever f E O(rE, C\ (0, 1)) with f (0) = ao, f`(0) _ a,, then r :5 R. Theorem of Schottky. For any (ao, 9, r) E C x (0, 1) x (0, +oa) one can find M = M(ao, 9, r) E (0, +oo) such that if f E O(rE, C \ (0, 11) with f(0) = ao, then If (z)I < M, IzI < Or.

Chapter II

The Caratheodory pseudodistance and the Caratheodory-Reiffen pseudometric

In Chapter I we introduced two continuous distances m : E x E --> 10, 1), p : E x E - 1R+ (p = tanh-1(m )) and the C°`-function y : E -+ [1, oo) such that:

m, p, and y satisfy the Schwarz-Pick Lemma, cf. Lemma 1.1, Remark 1.11, Corollary 1.13(b);

2° topm = top p = top E and the spaces (E, m), (E, p) are complete, cf. Remark 1.7, Corollary 1.13(c): Vi'6',

3°

,'Ax"

y(a), a E E, cf. Property 1.4, Corol-

lary 1.13(d);

4° the notions of m-and p-rectifiability of curves coincide with the Euclidean rectifiability and moreover

Lm = Lp, cf. Lemma 1.16(a),

L, = LP = Ly for piecewise C1-curves, cf. Lemma 1.16,

m' = p' = p = inf(Ly(a):... ), cf. Corollary 1.15, Lemma 1.16(a); 5` Isom(m) = Isom(p) = Isom(y) = Aut(E) U Aut(E), cf. Proposition 1.17: 60 10g p E P S H (E x E ), log m is separately subharmonic in E x E, cf. Proposition 1.18, Property 1.3.

In the sequel we will discuss generalizations of the functions m, p, and y for arbitrary domains G C C". There are several ways to proceed. In this chapter we present the simplest one dealing with bounded holomorphic functions (this way was the first also from the historical point of view; cf. [Carl).

16

Il. The Carathdodory pseudodistance and the Carathdodory-Reifen pseudometric

2.1 Definitions. General Schwarz-Pick Lemma For any domain G C C", n > 1, put CG* (Z', z") := sup{m(f(z'). f (Z")): f E O(G, E)), cG (z', z") := sup(p(f (z'), f (Z")): f E O(G, E)}, YG(z; X) := SUP{Y(f (z))If'(z)X I: f E O(G, E)),

z', z" E G, z'. Z" E G, Z E G, X E C",

where f'(z): C" --> C denotes the C-differential of f at z. It is clear that CG =

cG,

(2.1.1)

and (by the Schwarz-Pick Lemma) CE = m, CE = P, YE(-;1) = y. 0, cC. = 0, and yC. = 0. In view of (2.1.1), we can always pass from c to CG or conversely. Nevertheless,

Observe that

in the sequel we will use both c* and cG; c is less regular but more handy in calculations (as in the case of the unit disc). We write cG*) in all the cases where one can take c as well as CGSince m, p, and y are invariant under Aut(E), we get (z") : f E O(G, E), f (z') = 0), cc (z', z") = sup( if cG (z', z") = sup{p(o, f (Z")): f E O(G, E), f (z') = 0), I

YG(z; X) = SUP( I f'(z)XI:

f E O(G, E), f(z) = 0),

z', z" E G, z', z" E G, Z E G, X E C".

Now, applying Montel's theorem, we find that for any z', z" E G (resp. Z E G,

X E C") there exists f E O(G, E) such that f (z') = 0, If (z") I = c*G W. Z") (resp. f (z) = 0, I f'(z)X I = yG (z; X)). Any such a function f will be called an extremal function for cG*)(z', z") (resp. yG(z; X)). In particular,

CG:GxG ->[0,1),

CG:GxG R+, YG:GxC"-->R+. Since m and p are distances, the function cG*) is a pseudodistance; c is called the Mobius pseudodistance for G; cG is the Caratheodory pseudodistance for G. The

2.1 Definitions. General Schwarz-Pick Lemma

17

function yG is called the Caratheodory-Reiffen pseudometric for G. Note that for any a E G, the function yG (a;-): C" -+ R+ is a complex seminorm.

The definitions of cG and yG may be formally extended to the case where G = SZ is an arbitrary open set in C" (not necessarily connected), but then the Montel argument does not work and, for instance, ccz(z', z") = +oo if z' and z" lie in different connected components of Q. Therefore we will only consider the connected case. On the other hand, there are no difficulties in generalizing cG and yG to the case

where G = M is a connected complex manifold. Even more, the notions of cG and yG may be extended to the case of connected complex spaces. We do not intend to develop the theory into these directions. We recommend the interested reader to consult [Rei 1,2,3] for instance. Except for §2.4, we deal only with domains in C". As a direct consequence of the definitions, we get

Theorem 2.1.1 (General Schwarz-Pick Lemma). For arbitrary domains G C C", D C C' and for any holomorphic mapping F: G -+ D we have cD*)(F(z,), F(z"))

<

c(G*)(z'.

z"),

YD(F(Z); F'(Z)X) < YG(z; X),

z', z" E G, Z E G, X E C".

In particular, if F is biholomorphic, then equalities hold.

In other words, the systems (c ))G and (Va)G are holomorphicallycontraaible.

Remark 2.1.2 (cf. §4.1). Observe that from the point of view of the General Schwarz-Pick Lemma the Carath6odory pseudodistance and the CaratheodoryReiffen pseudometric are minimal in the following sense: if (dG )G is any system of functions dG : G x G R (resp. if (SG )G is any system of functions SG : G x C" -> R), where G runs on all domains in all C"'s, such that

dE(F(z'), F(z")) < dG(z', z"), (resp. SE(F(z): F'(z)X) < SG (Z; X),

z'. Z" E G, F E O(G, E) Z E G. X E C". F E O(G. E))

and dE = p (resp. SE = YE), then cG < dG (resp. YG < SG)-

Note that in Th. 2.1.1 we do not claim that co*)(F(za), F(zo)) = c(*)(zo, zo) for some zo, zo' E G, Zo 4, (resp. yD(F(zo) ; F'(zo)Xo) = yG(zo; Xo) for some Zo E G, Xo E C", Xo 0) implies that F is biholomorphic (cf. Lemma 1.1(c); as

a counterexample we have C2 9 (zi, z2) -+ (zi.0) E C2). This is not true even for D = G (; C" and even under more restrictive assumptions on 4, za' (resp. zo, X0).

18

11. The Caratheodory pseudodistance and the Carathcodur)-Reiffen p
Remark 2.1.3. Let Ge, G, G be domains in C" such that to , - G c G n (,.

Assume that for any f E H°°(G) there exists f E 0(G) with j = I on G,,. (Note that f is uniquely determined by f .) Then for any f c H" (G 1 the function

f belongs to Ho(G) and IIIIIa < IlfJIG In particular. if G = G C G. then G is an Ho`-extension of G if H°(G) C O(G )I,; iff c)((;. E) = O(G, E)1(;. Directly from the definitions and Remark 2.1.3 we get

Remark 2.1.4. If d is an H"-extension of G, G C G, then (*)

(*)

cG = CG IGxG, YG = YcIGxC'

In particular, (*) 'E. _ CE(*)IE.xE,

and hence the space (E*, c *)) is not complete.

2.2 Balanced domains Let G be a balanced domain in C", i.e. E G = G. Then

G = (zEC":h(z)<1), where

h(z) := inf{t > O : z/t E G),

Z E 0";

h is the Minkowski function of G and we write G = Gh. It is easilp seen that C"'. and z h: C" -> R+ is upper semicontinuous, h(Xz) = IXIh(z), i, F E-:

h-OiffG=C". Proposition 2.2.1.

(a) c (0, ) < h in G; YG (0; ) < h in C". (b) For a E G the following statements are equivalent:

(t) c* (0, a) = h(a):

(ii)c*(0..)=h onGf1(Ca); (iii) YG(O;a) = h(a); (iv) yG(0, ) = h on Ca;

2.2 Balanced domains

19

(v) there exists a C-linear functional L : C" -+ C with I L I < h and I L (a) I = h (a). (c) The following conditions are equivalent: in G;

(ii) YG(0; ) = h in C"; (iii) h is a seminorm; (iv) G is convex. (d) yG (0; ) = h := sup{q : C" -> R+ : q is a seminorm, q < h); h is the Minkowski function of the convex hull G of G.

Proof. (a) Fix a E G (resp. a E C"). We have the following two possibilities: 1° h (a) = 0; then the mapping

C3A->AaEG is well-defined, and hence by Theorem 2. 1.1 we get a) (resp. YG(O;a)

c*.(0;1) = 0 = h(a). )c(0; 1) = 0 = h(a)).

2° h(a) > 0; take

E9A--* h(a)aEG. Then the holomorphic contractibility leads to

cc(0, a) < ce(0, h(a)) = h(a). (resp. YG(O;a) < YE(0;h(a)) = h(a)).

(b) Observe that if (v) is satisfied for a point a, then it is satisfied for each point from the line Ca. Moreover, if (v) is satisfied, then by the definitions

cc(0, a) ? IL(a)I = h(a), YG(O;a) ? IL'(O)aI = IL(a)I = h(a).

Thus, in view of (a) and the Hahn-Banach theorem, we have: (v) --> (iv) --> (iii)

-> (v) - (ii) -> (i). Suppose that (1) is fulfilled and let f E O(G, E), f (0) = 0, be an extremal function for cG(0, a), i.e. I f (a) I = h(a). Note that in the case

20

U. The Carath*5odory pseudodistance and the Carathdodory-Reiffen pseudometric

h(a) = 0 the implication (i) -+ (v) is trivial. So, assume that h(a) > 0. Define

E - .L f(-- a)EE. Then cp E O(E, E) and Icp(h(a))I = h(a). By the classical Schwarz Lemma we get cp(.l) = e'BA, I E E, for some 0 E R. Let L := f'(0). By (a) we have

IL(X)I = If'(0)XI < yc(0;X) < h(X),

X E C".

Moreover,

IL(a)I = If'(O)aI = IcP'(0)h(a)I = h(a).

(c) In view of (b) we have: (i) H (ii). Since yG(0; ) is a seminorm, we get (u) -> (iii). If (iii) is satisfied, then by the Hahn-Banach theorem condition (v) from (b) is fulfilled for any a E C". Hence (iii) -> (i). Obviously, (iii) and (iv) are equivalent.

(d) Since yc(0; ) is a seminorm. (a) implies that yc(0; ) <_ h. On the other 0 hand, by (c) we obtain h = yd (0; ) < yc(0; )-

Corollary 2.2.2. (a) Let Gj = Gqj

C" be balanced convex domains with Minkowski functions qt, j = 1, 2. Then the following conditions are equivalent: G2 with F(0) = 0; (i) there exists a biholomorphic mapping F: GI (ii) there exists a C-linear isomorphism L : C" -+ C" such that q2 o L = q,, i.e. G I and G2 are linearly equivalent. (b) (cf. [Pat]) Let G = Gk be a bounded balanced domain in C. Then the following conditions are equivalent: (i) Gh and B. are linearly equivalent (ill" is the unit Euclidean ball in C"); (ii) h2 E C2(C"). Note that h2 E C2(C") if h2 is C2 near 0 E C". Proof. (a) Obviously (ii) -+ (i). If F is as in (i), then by Proposition 2.2.1(c) we have

q2(F'(O)X)=yc2(0;F'(0)X)=yc,(O;X)=91(X),

X EC.

2.2 Balanced domains

21

(b) The only problem is to prove that (ii) + (i). Since AEC.ZEC", we get

h

a2

_,

(z) =

a

1:"

,_

h2(),z)

= Uh2(z),

a2h2

azjazk a.X h (A.z) = j.k=1

(Az)z;zk.

In particular, n

a2h2

0 < h2(z) =

aZj3tk

(0)zjzk,

z E (C")..

Consequently, h2 is a Hermitian form, and therefore we get (i).

Corollary 2.2.3. If G = G. is a balanced convex domain in C", a E G, and if ha E Aut(G) is such that h,,(a) = 0, then c* (a, z) = q(ha(z)), Z E G, yG (a; X) =q(ha(a)X), X E C".

Recall that in the case of the unit polydisc E" C C" we can take

ha(Z1,...,Zn)

_

z, - a, 1 - z,a,

zn - an ,

I - zndn

In the case of the unit Euclidean ball 13,, C C" we can use

IIal12(z - iia) - a + 4a ha(z)

-

1 - (z,a)

(2.2.1)

where (z, a) zjdj is the complex scalar product in C" and II II is the Euclidean norm; cf. Appendix AUT. Hence, in view of Corollary 2.2.3, we get

II. The Carathdodory pseudodistance and the Caratheodory-Reiffen pseudometric

22

Corollary 2.2.4. max(m(a1, z,): j = 1, ..

,

n}.

YE"(a;X) = max(y(ai)IX;I: j = 1,...,n), (I - Ila112)(1- IIzII2) I I - (z, a)I-IIXII`

I(a, X)I2

(2.2.2)

Z

1

X)-_ 11 -IlaII-+(I-ila112)2] Corollary 2.2.5 (Poincare Theorem) (cf. [Rein]). For n > 1 there is no biholomorphic mapping of E" onto

Proof. Use Corollary 2.2.2 and the fact that Aut(E") acts transitively on E". The Poincard Theorem may be generalized to the case of Caratheodory isometries.

A mapping F : G --+ D is said to be a c-isometry if ca ' (F(z'), F (z")) = c' ) (z', z"),

z', z" E G.

Recall that any biholomorphic mapping is a c-isometry.

Let k E N. a := (a1.... , al) E Nlk, and n := al + x

- + ak. Define

x lBaA

and observe that

Ba = (Z E C": qa(Z) < 1), where qa : C" -+ R_ is the norm given by the formula

qa(z)=max(IIziIIa.,....IIZkIIaj.

Z=(Z1,...,Zk) EC" =Cap X... XCap

II, denotes the Euclidean norm in C'). For a E L, let h.' denote the automorphism of ]l1 defined by (2.2.1). Then for any a = (a1, ... , ak) E Ba the mapping h7°') defined by the formula (II

haa)(z)

(h,,i)(zl , ... , hUak)(ZA)), )

Z = (z .... ZA) E C",

23

2.2 Balanced domains

0. Hence by Corollary 2.2.3 we

is an automorphism of Ba such that have

cBv1(a,z) =max(cB (aj,zj): j

a-(a,,....ak), z=(z,... .zx)EC".

(2.2.3)

We will study c-isometries F : Ba B1, where a = (a1, ... , at) E Ilk, _ Eli, n := a, + ... +.. ak, m :_ A + ... + A. ... , #I) E (/'1 = 1 has been already solved in Proposition Recall that the case k = 1 = 1, a = 1.17.

Remark 2.2.6. (a) Let F : Ba - BL be a c-isometry. By (2.2.2) and (2.2.3) the mapping F is injective, continuous, and proper. In particular, n < m; if n = m. then F is a homeomorphism and so F-1 is also a c-isometry (use the Brouwer theorem on the invariance of domain).

(b) Since the group Aut(Bf) acts transitively, to characterize all c-isometries F: Ba --> Bp it suffices to consider only those for which F(0) = 0. (c) Suppose that k =1 and let Fj : C", -p C'' be a unitary or antiunitary operator (in particular, we necessarily have a j < fl,), j = 1, ... , k. Set F = (F, , ... , FR ). Then, by (2.2.2) and (2.2.3), F is a c-isometry of Ba into Bp. (d) Let V: E -* E be given by the formula

(k if ImA>0 jl A

if IM), < 0

One can easily prove that m((P(A') V (X")) < m(A'. A"),

A'. A" E E,

with equality if (Im A')(Im A") ? 0. Let h 1, h2 E Aut(E) be such that hj (C j fl F,) _ j = 1, 2, where C, := B( 1 + e2e",j1, e), C2 := B( 1 + e2e3ir'4, s) are

"small" circles orthogonal to M. Put Fj = P o hj, j = 1, 2, F3 := co. Obviously,

m(Fj(A'), Fj(A")) < m(,'. A"),

A', A" E E, j = 1, 2, 3.

One can easily check that for any A. X" E E there exists j E 11. 2, 3. } such that ni(Fj(Ao), Fj(Ao)) = m(Ao. As).

II. The Caratheodory pseudodistance and the Carathdodory-Reiffen pseudometric

24

Hence the mapping F = (F1. F2, F3): E = Bii) -* E3 = B(t,I.,) is a c-isometry such that F1, F2, and F3 are neither holomorphic nor antiholomorphic. Moreover, F is not differentiable; cf. also Exercise 2.4.

Proposition 2.2.7. If k = I = I (put a = at and P = #t), then any c-isometry F: ]W -* Bp with F(O) = 0 is either unitary or antiunitary. Proof (see [Kuc-Rayj for a = fi). It suffices to prove that

either (F(z'), F(Z"))p = (Z', ")a, or (F(z'), F(z"))A = (z', z")a,

Z', Z" E Ba,

Z', Z" E 3a

((, ), denotes the complex scalar product in C5). By (2.2.2) we get IIF(z)IIp = IIZIia,

z E Ba,

and

Z', Z" E B..

I I - (F(z'), F(z"))fI = I I - (z', z"). 1,

(2.2.4)

Hence

F(-z) = -F(z),

z E

a.

(2.2.5)

Relations (2.2.4) and (2.2.5) imply that I (F (z'), F(Z")),I = I(z'. z"). 1,

Re(F(z'), F(z" ))R = Re(z', Z")a,

z% z" E lEa.

Thus, for any pair (z', z") E Ba X Ba \ {Im(z', z")a = 0) =: SZ there exists E(z'.z") E {-1, 1} such that Im(F(z'), F(Z"))p = E(Z', z") Im(z', z")a.

It remains to prove that the function s is constant. The set ) has two connected components S2_, Q+, say. Since E is continuous, it is constant on each of them: E = E_ on SZ_ and E = E+ on 52+. Note that (Z', Z") E SZ_

iff (-Z', z") E 52+.

Hence by (2.2.5) we get e_ _ E+.

In the general case the following result holds.

0

2.2 Balanced domains

25

Proposition 2.2.8. Let F: Ba -- Bp be a c-isometry with F(O) = 0. Then k < 1. Moreover, if k = 1, then the mapping F has the form described in Remark 2.2.6(c) up to permutations of (a, , ... , ak) and #t) (i.e. F is component-wise unitary

or antiunitary). In particular, we get Generalized Poincark Theorem. If k > 1, then there is no c-isometry of B, into Bp (cf. Corollary 2.2.5, where k = n > 2, a, = ... = an = 1, 1 = 1, ,8, = n).

ak = fil

Proof (cf. [Kuc-Ray] for k = 1, a, for arbitrary N E N we have

max{c4N, (a, b), c6N, (a, -b)) > IIbIIN,

,k). First recall that

a, b E BN, a 54 0.

(2.2.6)

Let es : Ba, -+ Ba be given by the formula es(xs)

= (0,

,

0, x 5 , 0, . . .

,

0 ),

xS E Ba,, s = 1, .. k.

T

s-th place

Set GS = Gs,!) := Foes : 1Ba, --3 Bp, s and xs E (Ba, ),. Since F is a c-isometry, we get cam, (-xs, xs)

k. Fix S E

k)

= c;.(es(-xs), es(xs)) = cB,(F(es(-x,)), F(es(xs))) = max(cHM (G5.r(-xs), G6.r(xs)): t = ,1}.

Hence there exists t = t (s, xs) E (1, ... , l } such that

c, (Gs.,(-xs) G..,(x.,)) = Since II Gs.$ (±xs) 11p,

cBo,(-xs.xs).

Ilxs IIa,, this implies that

G5.1(-xs) = -Gs.,(xs) IIG5.r(xs)IIp, = IIxsIIa,

(2.2.7)

(cf. the proof of Proposition 2.2.7). (2.2.8)

II. The Caratheodory pseudodistance and the Carathiodory-Reiffen ps udurnetnc

26

Take s' # s and x,, E B.,, with Ilx,'Ila,, < IIx,IIa, Suppose that

0.

Then (2.2.6). (2.2.7), and (2.2.8) give

c,,(e.'(Y,'). ex(±xs)) tx,

>max(c > IIGx.,(x,)IItS,

= llxslla,,

a contradiction. This shows that G,,,,

0, s'

S,

llxs' IIa,, _< 11x, Ila, (t = t (s, X.,)).

(2.2.9)

< ak. This permits Using a suitable permutation we may assume that a) < us to identify B., with ]3a, x (0) C 18a,. for 1 < s < s' < k. By virtue of the numbers (2.2.8) and (2.2.9) one can easily prove that for any x1 E t (l , xt )..... t (k, xt) are different. Thus k < 1. Now assume that k = 1. In particular, (t(1, xt ), .... t(k, xi )) = { 1.... , k). Fix

rt E

and let

t(s) := 1(s-ii),

s = 1,

..,k.

It is easily seen that

t(l) = t(l,xi)

for any xi E (18a,),

and

Gt. j(xt) = 0 for any x1 E Ia,

and

j 0 t(1).

Now, it suffices to verify the following property, which then applied induetivel) gives the required result.

If forsome sE(1,...,k)andtE{1,...,1} G,,j(x,)=0.

x,E3a,'J36 t,

then the mapping G,.1: 3, --> B/j, is a c-isometry and

.. xk) = G,., (x,), ......r&)

(XI, .... xk) E

In particular, by Proposition 2.2.7, G,., is unitary or antiunitary.

112._'.10)

2.3 Carathdodory hyperbolicity

27

For the proof of the above property observe that for xs, xs E Ba, we have

cB., (xs, xs) = c(es(xs), ef(xs )) = c;,(F(es(xs)), F(es(x: ))) = CB,, (Gs (x;), GS(xs )) = cBa (Gc.,(xs), G.,(xs ))

(use (2.2.10)),

and therefore Gs,, is a c-isometry.

Now suppose that F, (x) # Gs., (xs) for some x = (x i , ... , xk) E 13a. Let h E Aut(Ba,), g E Aut(B#1) be such that h(0) = xs and g(G5.,(xs)) = 0. Put cp := g o Gs,, o h : B , -+ Ba,. Obviously, cp is a c-isometry and V(0) = 0. Hence, by Proposition 2.2.7, (p is at least R-linear. Take Zs E Ba, such that IIzsIIa, = cB,(x,es(xs))

Then

IIzsIIa, = c4o(x, es(h(dzs))) >

max(cB, (g(F,(x)), (G(zs)), cBs, (g(F,(x)), -co(zs))} > Ilw(zs)IIp, = c;,, (G5.,(xs), G..,(h(zs))) = CBo (xs, h(zs)) = Ilzs Ila,

(use (2.2.6));

a contradiction. The proof is complete.

0

2.3 Caratheodory hyperbolicity In general, the pseudodistance c(*) need not be a distance, e.g. cC.i = 0. Note that

cG0 if YG=0 if where "H°°(G) C " means that all bounded holomorphic functions on G are constant, i.e. G is a Liouville domain. On the other hand,

CG is a distance iff H0O(G) separates points in G.

(2.3.1)

28

II. The Caratheodory pseudodistance and the Carathdodory-Reiffen pseudometric

If cs) is a distance, then we say that G is c-hyperbolic. If for any a E G the seminorm yG(a; -) is a norm, then we say that G is y-hyperbolic. Observe that

YG(a; ) is a norm iff d X Ear", X # 0,

3 f E HO0(G) : f'(a)X

(2.3.2)

0.

Proposition 2.3.1. If G C C', then the following conditions are equivalent: (i) G is c-hyperbolic; (ii) G is y-hyperbolic;

(iii) H°°(G) # C. In other words, if a domain G C C' is not a Liouville domain, then it is both c- and y-hyperbolic.

Proof (cf. [Sib I]). Obviously (i) -> (iii) and (ii) -+ (iii). Suppose that (iii) is satisfied and let fo E H°°(G), a,, a2 E G, be such that fo(al) 36 fo(al). Put fo(z)-fo(aj)

ZEG\{aj)

z-a,

fj (Z) =

z =aj

f0(aj),

Then fj E H°°(G), j = 1, 2. One can easily verify that the functions fo, f,, f2 separate points in G and that rank(fo, f1, f2) = 1 on G. 0 If n > 1, then there are domains such that cG # 0 (resp. yG # 0) but G is not c-hyperbolic (resp. y-hyperbolic). For example, take any balanced convex domain

G = Gq (; C" with q-'(0) 0 {0).

0 Relations between c- and y-hyperbolicity are still not understood*). 0 Proposition 2.3.2. If G C C" is a domain biholomorphic to a bounded domain, then G is both c- and y-hyperbolic. Proof. We may assume that G is bounded. Then Z ,, ... , Z. E HI(G) and so the result follows from (2.3.1) and (2.3.2). 0

Note that if G is a bounded domain, R := diam(G) (in the Euclidean sense), then by Theorem 2.1.1 and Proposition 2.2.1(c) we have CG (z , z

ca(z.,R) (z , Z ) =

YG(Z; X) > YB(z.R)(Z; X) *) Added in proof: see Exercise 2.10

Ilz - z11 R

= II X II , R

,

,

z , Z E G,

Z E G. X E C".

(2.3.3)

2.4 The Caratheodory topology

29

This is an alternative proof of Proposition 2.3.2.

2.4 The Caratheodory topology Let topcG} (resp. top G) denote the topology generated by cG (resp. the Euclidean topology of G). By virtue of (2.1.1) we have

a E G,r > 0.

B(G(a,r) = Hence top c* = top cG.

Proposition 2.4.1. (a) cG) is continuous. In particular, top c(') C top G. (b) If G is biholomorphic to a bounded domain, then top cG = top G. Proof. (a) Since c(;) is a pseudodistance. it suffices to show that limn-a c(,")(a, z) _ 0. In view of (2.1.1) it is enough to consider only the case of cG. Recall that

CG* (a, ) = sup{Ill: f E O(G, E). f(a) = 0}.

(2.4.1)

The family in (2.4.1) is equicontinuous, and therefore cG (a, ) is continuous.

0

(b) follows from (2.3.3).

Note that if B(a, 3r) C G, then by Proposition 2.2.1(c) C*G (i,, z")

YG(z: X)

< ce(z'.2,)(z, z") = ll: YB(z.r)(z: X)

=

ll

z'. z" E B(a, r),

(2.4.2)

z E B(a. r), X E C".

(2.4.3)

2r

IX, r

Relation (2.4.2) gives an alternative proof of Proposition 2.4.1(a). In C1 the situation is extremely simple, namely we have

Proposition 2.4.2. If G C C 1 is c-hyperbolic, then top cG = top G.

Proof. Let G

a, -> ao E G in top cc . Observe that

If - f(ao)I < 11 f - f(ao)11G .c (ao:').

f E H-(G)

H. The Caratheodory pseudodistance and the Caratht'odory-Reiffen pseudometric

30

Consequently, f f (ao) for any f E H00(G). Since H°°(G) C (cf. Proposition 2.3.1), there exists fo E H00(G), fo # 0, with fo(ao) = 0. Let

fo(z) = (z - ao)'kg(z), where g(ao) # 0. Clearly, g E H°O(G). Since condition (2.4.4) implies that a,, -> ao in top G.

z E G,

(2.4.4)

0 and g(a,) -> g(ao)

Unfortunately, for n > 3 there exist c-hyperbolic domains with top cG For n = 2 we do not know whether such a domain exists.

0

0,

top G.

0

Theorem 2.4.3 ([Jar-Pfl-Vig 1]). For any n > 3 there exists a c- and y-hyperbolic top G. domain G C C" such that top cG This theorem will be proved by a sequence of lemmas. The proof requires the use of manifolds. Observe that the notions introduced so far may be literally extended to the case of manifolds.

Lemma 2.4.4. Let V be a c- and y-hyperbolic connected complex subntanifold of top V. Then there exists a c- and yC", n > 2, dim V > 1, such that top c v hyperbolic domain of holomorphy G C C" with V C G such that top cG) # top G. Proof. It is known that there exist an open neighborhood Uo of V and a holomorphic retraction r : Uo -+ V ; cf. [Doc-Gra] (Appendix PSC 17). Put

U := {zEUo:z-r(z)EE"}. Note that U is an open neighborhood of V. Choose a domain of holomorphy G with V C G C U; cf. [Siul (Appendix PSC 18). Then

r(z")) < c )(z', z"), z', z" E G, CG (z', z") < C(v*) (z', Z"),

yv(r(z);r'(z)X) < yc(z; X), yG(z;X) < yv(z;X),

Z', Z" E V, z r= G, X E C", z E V, X E TTV,

where TZ V denotes the tangent space to V at z. In particular, =CV

ya(Z;X)=yv(Z;X),

(2.4.4)

ZEV,XETZV.

(2.4.5)

2.4 The Carathr odory topology

Put

31

fj(z):=Zj-rj(Z), zEG.Then fj:G-+E, j=1,...,n.

Take z', z" E G, z' j4 z". If r(z') = r(z"), then fj(z') fj(z") for at least one j E (1,...,m). If r(z') # r(z"), then we can use (2.4.4) to conclude that G is c-hyperbolic. For Z E G and X E C", X 36 0, we have the following two possibilities:

0:thenyG(z;X)>0by(2.4.5);

r'(z)X=0: then fj(z)X=Xj, j = 1, .

.

. , n, and yG (z; X) > 0 by (2.3.2).

Finally, (2.4.4) implies that top c( ) 36 top G.

Lemma 2.45. If M is a c- and y -hyperbolic connected Stein manifold such that top M, then there exists a c- and y-hyperbolic domain of holomorphy top cm C2 dim M+ 1 with top c(* ) top G. GC Proof. Use the embedding theorem (cf. Appendix PSC 16) and Lemma 2.4.4.

Lemma 2.4.6 (cf. [Hay]). There exists a c- and y-hyperbolic connected Riemann domain M spread over C with top c,y # top M. Proof. In the description of the desired Riemann surface M we omit the details of how to construct a Riemann surface by glueing together local pieces.

Let ak, bk E E (k E N) be sequences of real numbers in (0, 1) without accumulation points in (0, 1), such that 0 < ak < bk < ak+1 (k > 1), and put Ik := Eak, bkj

Moreover, to any k E N we assign nk pairwise disjoint subintervals Jk. j (l < j < nk) of Ik and we put oo

nk

Do:=E\ (UUJkj), k=1 j=1 n!

Dk:=E\(U11UUJ.j). k> 1; !96k

j=1

the value of nk will be found later. Then the Riemann surface M we need is given by the following glueing process:

Do and Dk are glued together along the cuts Jk.j, I < j < nk, by crosswise identification, k E N.

32

II. The Caratheodory pseudodistance and the Carathdodory-Reifl'en pseudometric

If we put Dk := E \ U,tk I,, then the subdomain Dk U Do of M, where Do denotes the part of Do corresponding to Dk, is a two-sheeted covering of Dk whose branch points are given by the ends of the subintervals Jk.1, j = 1, ... , nk-

For any holomorphic function f : M -> E we can define a new holomorphic E by

function fk : Dk

A (Z) = 4 [f(z) - f (Zk )]2 if z is not a branch point and where zk are the "two" points in Dk U Do over z. Here we have used the classical Riemann extension theorem for bounded holomorphic functions. Observe that fk vanishes at 2nk branch points.

Let z` := i/2; then by the Montel theorem we obtain I fk(z')I < (1/2)k for nk sufficiently large. Let zk denote the point `over" z' in M lying in the k-th "sheet" Dk. Then the above observation yields C ,(Z*, Zk)

koo + 0

whereas (zk )k does not converge in M. Hence top M 96 top c,',.

What remains to be verified is that M is c-hyperbolic, i.e. H°O(M) separates

points. We denote by r : M -> E the branched covering map of M, where ,r is given by "identification". It suffices to separate different points z' z2 with 7r (z I) = 7r (z2 ). Assume that z' belongs to the k1-th "sheet"of M. Without loss of generality, we may assume that k, > k2 > 0. Now, we construct a new Riemann surface R as follows: nj,

R is obtained from two copies of E \ U Jk,,i i=r by glueing them together along the cuts J&

..

crosswise.

E (as above). Moreover, Observe that there is a branched covering map n : R R such that r = n o *; here * maps the there is a holomorphic map rfi : M k, -th "sheet"of M into the "upper sheet" of R and the rest of M into the "lower sheet" of R again by "identification". Since R is obtained by glueing together a finite number of cut unit discs, HI (R) separates the points of R. Therefore, since *(z') 54 *(z2), we obtain a function f E HI (M) with f(zr) f (z2), i.e. M is c-hyperbolic. The y-hyperbolicity is obtained in an analogous way.

Observe that if the Riemann surface M from Lemma 2.4.6 could be embedded in C2, then Theorem 2.4.3 would be true also for n = 2.

2S Properties of c(*) and y. Length of curve. Inner Carath6odory pseudodistance

33

Remark 2.4.7. For a better understanding of top cG observe that if a c-hyperbolic domain G C C" satisfies one of the following conditions, then top cG = top G. (a) For any a E G there exists a basis B(a) of neighborhoods of a in topcG such that any U E 8(a) is connected in top G. (b) For any a E G there exists a neighborhood UQ of a in topcG such that UQ is relatively compact in top G.

For suppose that (a) is satisfied. Fix a E G and r > 0 with B(a, r) cc G. Put

e := min{c )(a, z) : Z E 8B(a, r)}. Then e > 0 (hyperbolicity). Let U E 8(a) be such that U C (a, e). Obviously C6 U (18B(a, r) = 0. Since U is connected in top G, we get U C B(a, r). If (b) is satisfied, then we put

e :=

z): z

B(a, r)}.

It suffices to prove that e > 0 (then B(.) (a, e) c B(a. r)). Suppose that e = 0 Cc

z

and let c*i(a,

B(a, r), v > 1. We may assume that z, E U0, v > 1,

where Ua is as in (b). Since Uo is relatively compact in top G, we may also assume

that zv - zo E G in top G. Clearly a 0 zo and by the continuity of cG*i we have c(*) (a, z,,) c(*)(a, zo) > 0, which gives a contradiction.

2.5 Properties of c(*) and ry. Length of curve. Inner Caratheodory pseudodistance First we complete the list of basic properties of c(*) and y.

Proposition 2.5.1. Let G C C" be a domain.

(a) If (Gv)_1 is a sequence of subdomains of G such that G, / G (i.e. Gv C Gv+,, v > 1, U0°_A, = G), then cG, \ cG and YG, N Yc (b) log CG E PSH(G x G), logc*G is separately plurisubharmonic, i.e. for each a E G the function log c* (a, ) is psh. (c) The function yG is locally Lipschitz and log-psh on G x C".

c

(d) lim Z

.z i cue-z"11

i #i X.

= YG (a; X), a E G, X E C", II X II = I.

II. The Caratheodory pseudodistance and the Carathdodory-Reiffen pseudometric

34

Note that in view of the continuity of YG condition (d) is equivalent to the following one.

(d') For any compact K C G and for any e > 0 there exists S > 0 such that

ICG)(z'.z")-yG(a,z'-z")l ! Ellz'-z"II, a E K. z', Z" E B(a, S) C G. Proof. (a) Use the Montel argument. (b) Observe that

logcG = sup{(logp) o 4 f: f E O(G, E)), where G x G a (z'. z")

-

(f (z'), f (z")) E E x E, and then apply Proposition

1.18 and Appendix PSH 14.

(c) Let B(a, 2r) C G. We will prove that for z', z" E B(a, r), X', X" E C" we have

r 11 Z' - z"IIIIX'll +

IYGW; X') - YG(z";

Ir II X' - X"ll.

(2.5.1)

First note that IYG (z'; X') - YG (z"; X") l

IYG (z'; X') - Ya

X') l + YG (z"; X' - X").

By Proposition 2.2.1(c) we get

YG(Z"; X' - X") <

IV, - X1,11 ,

r

z" E B(a, r), X', X" E C".

It remains to estimate IYG (z'; X) - yG (z"; X) 1. Fix z', z" E B (a, r) and X E C".

Suppose that yG(z'; X) > yG(z"; X). Let f E O(G, E) be an extremal function for yG(z'; X), i.e. f (z') = 0 and I f'(z')X I = YG(z'; X). Then IYG(z';X) -

YG(z";X)I

= If'(z')XI

-YG(z"; X)

lf'(z')Xl < l f'(z')XI - Y(f :5 lf'(z')X - f'(z")Xl :5 llf'(z') - f'(z")IIIIXII (z"))If'(z")Xl <

< max{ Il f"(z) II : z E [z', Z"]) 2 z

11 z' - z"IIIIXII

IIz' - z"IIIIXII

l f,(z )XI

2.5 Properties of c(*) and y. Length of curve. Inner Carathr odory pseudodistance

35

The last inequality follows from the Cauchy inequalities: 11f 11G

f EH°°(G),kEN,ZEG.

1 I1f(k)(Z)II < k! [dist(z, aG)]k'

Then, the plurisubharmonicity of log yG is a direct consequence of Property 1.3 and Appendix PSH 7,14. (d) In view of the relation cG = tanh-1 (ct) it suffices to consider only the case

of c,. Let B(a, 4r) C G. We will prove that CG* W, Z") - YG(zo;z' - z")I

(3I

(311z' - z"II +211z' - zoll) Ilz' - z"II,

Zo,Z',Z" E B(a,r). The proof of (2.5.1) gives

IYG(zo;z -z )-YG(z;z -Z") I 5

(3r)2

zo.z.z E B(a,r).

1Iz -zollllz -z il,

It remains to show that (33r)211

ICG(z', Z") - YG(Z',Z' - Z") 1 <

Z,

-

Z"112,

Z', Z" E B(a, r).

Fix z', z" E B(a, r). Put X := z" - z'. Then by the Cauchy inequalities for f E O(G, E) with f (z') = 0 we get

11f(k)(z')IIIIXIIk < (iixIi\k <

If(z'+X)-f'(z')XI:5 k__

k_2

r)IIX112. o

J

For a curve a : [0, 1] -+ G put N

LC(.)(a) G

:= sup(EcG't(a(tj_i), a(te)): N E N, 0 = to <

< tN = I }.

J=I

The number Lc (.) (a) is called the c(*)-length of a; if L,(.) (a) < +oo, then we say that a is cG -rectifiable. Observe that L,.- < LCG

36

II. The Caratheodory pseudodistance and the Carathdodory-Reiffen pseudometric

and for any holomorphic mapping F : G --> D we have

L,(.) (F o a) < L,(.) (a) V

D

with equality for biholomorphic mappings. If a: [0, 1] -+ G is a piecewise C'-curve, then we can also define its yc-length by the formula Lye, (a)

f

yG (a(t); a'(t)) dt.

0

Note that LyG(a) < +oo. Moreover, for F E O(G, D) we get

Ly"(F o a) < Ly,(a) with equality for biholomorphic mappings.

Lemma 2.5.2. (a) Lc; (a) = LcG(a) and, moreover, if a: [0, 1] -> G is a (III -rectifiable curve (i.e. if a is rectifiable in the Euclidean sense), then

L<(.) (a) < +oo. G

(b) If G is y-hyperbolic, then any cG rectifiable curve is II 11-rectifiable. (c) If a is piecewise C', then Lc(.) (a) = LyG(a) G

(d) If a : [0, 1] -* G is a II II -rectifiable curve, then for any e > 0 there exists a piecewise C' -curve a : [0, 1] -> G such that &(0) = a (O), &(1) = a (l), and L,(.) (a) - LC(.)(a)I < E. 4

6

Proof. (a) Since c = tanh cc, for any e > 0 there exists n > 0 such that cG(a(t'), a(t")) < cc(a(t'), a(t")) < (1 + e)cG(a(t'), a(t")),

0
Consequently,

Lc6(a) < (1 +e)Lc;(a). According to (2.4.2) there exist M > 0, n > 0 such that

a(t")) < MIla(t') - a(t") II ,

0 < t', t" < 1, It' - t"I < n.

2.5 Properties of c(*) and y. Length of curve. Inner Carathiodory pseudodistance

37

Hence

MLII 1(a) (< +oo if a is 1111 -rectifiable).

(b) Since yG is continuous and G is y-hyperbolic, there exists s > 0 such that yG(a(t ); X) > 2e 11X11,

0 < 1 < 1, X E C".

By Proposition 2.5.1(d') there exists q > 0 such that

eIIa(t') -all")II.

I ccil (a(t'), a(t")) - YG(a(t');a(t') - a(t"))I 0<

< 1,

(2.5.2)

Hence

cG l (a(t'), (r(t" )) > eIIa(t') - a(t") II ,

0 < t' t" < 1, It' - t"I < 17,

and finally

LC(.)(a) > eLll II(a)

(c) Without loss of generality we may assume that a is of class C'. Then, in view of the continuity of y, for any s > 0 there exists q > 0 such that

IYG(a(t');a(1') - a(t")) - YG(a(t'); (t' - t")a'(t'))I <- elt' -1I, 0 < t'. t" < 1 . It' - t"1 < q. We may assume that q is such that (2.5.2) is satisfied. Take 0 = to < with t! - ti-, < q, j = 1, .... N. Then N

N

I E CG'(a(tla(11)) 1=I

< tN = 1

YG(a(tJ-0; a'(tj-1))(tl

- ti-01 < c(LII 1(a) + 1).

.1=I

Thus

ILC(-)(a) - LyG(a)I < s(LII II (a) + 1).

(d) By Proposition 2.5.1(d') there exists q > 0 such that ICG

(a(t'), a(t")) - YG(z;a(t') - a(t"))I < eIIa(t') - a(t")II, 0 < t'. 1" < 1, It' - t"I < 1]. Z E (a(t'), a(l")] C G.

U. The Carath6odory pseudodistance and the Carath&odory-Reiffen pseudometric

38


Ftx0
N

LC c i=t

Define ai(r) := a(tj_1) + r(a(ti) - a(tj_1)). 0 < r < 1, j = 1, ... N, and put a := a1 U

U &N. Note that there exist zi E [a(ti-1), a(t;)] such that

Lre(ai)

=YG(zi;a(ti_1)-a(ti)),

j = 1,...N.

Hence by (c) and in view of the above estimates we get L,(.) (ii)

I

N

<s+

L,(.) (a) - Lye

Icc`)(a(ti-1),

a(ti)) - YG(zi;a(Ij-1) - a(ti))I

E(L0 0(a) + 1).

J=1

0 Define (cf. Corollary 1.15 and Remark 2.5.3(e)) c,',*)'(z', z") = inf{ L,G) (a) : a : [0, 1 J -- G,

a is a 11 11-rectifiable curve joining z' and z").

z'. z" E G.

Remark 2.5.3. (a) By virtue of Lemma 2.5.2(a) we have (CG*)

r -_ CG. r

Obviously, cc, < C'G.

(b) c'G is a pseudodistance and for any F E O(G, D) we have

c'Q(F(zr),F(z"))
G(zr,zn),

r

z,zn

E G,

with equality for biholomorphic mappings; the pseudodistance cG is called the inner Caratheodory pseudodistance for G. The pseudodistance CG is said to be inner if cc = ciG.

2.5 Properties of c(') and y. Length of curve. Inner Carathdodory pseudodistance

39

(c) For any curve a: [0, 1] -+ G we have N

N

E cG(a(ti-i),a(ti))

.r;J) =

[rr

i=I

l=I Consequently,

Lct = L,,(;.) G

(CG)' = CG.

(d) By Lemma 2.5.2(c)(d) we see that c`G(z', z")

_ (f YG)(z', z") := inf{Ly, (a): a:

[0, 1]

G,

a is a piecewise C'-curve joining z' and z"}.

Z'. Z" E G.

(e) By Lemma 2.5.2(b), if G is y-hyperbolic, then in the definition of c'G the II II-rectifiability of a may be omitted; in particular, the definitions of p' and m' given in Chapter I agree with the definition of 0 We do not know whether this is true in general.

0

Proposition 2.5,4. For any a E G, X E C", II X II = 1, we have lim #`., Z

c`

Ilz(-

Z', Z"

z"II =

yG(a; X); cf. Proposition 2.5.1(d).

Proof. Let B(a, r) C G. Z', z" E B(a, r). Then

CG W, Z") < CG (Z', Z") = (J YG)(Z' Z")

JI Y (Z'+t(Z"-Z);z'-z")dt=YG(;z'-z"), where

= i; (z', z") E [z', z"]. Thus CG(Z', z 11)

Ilz' - z"II

-

C'G(z',

zz'Yc(; - Zfr

Ilz' - z"II

Ilz' - z"II

40

11. The Caratheodory pseudodistance and the CarathEodory-Reiffen pseudometric

0

Now, we can use Proposition 2.5.1(d) and the continuity of yG.

Remark 2.5.5. By Proposition 2.5.4, c'G is continuous. Observe that the c`G-balls Be,,(a, r) are arcwise connected in top G. In particular, by Remark 2.4.7 we get the following implication: if G is c'-hyperbolic (e.g. if G is c-hyperbolic), then top c'G = top G. Consequently, if G is a domain as in Theorem 2.4.3, then CG

c', i.e. cG is not

inner.

To give more concrete examples of c-hyperbolic domains with CG 0 cG we will need the following

Lemma 2.5.6 (cf. [Vig I]). Let a, b E G, a 96 b. Suppose that there exists f E O(G, E) which is extrental for

f'(a)XI I

b) and such that

X E C", X

< YG(a; X),

0.

Then

cG (a, b) < cG (a, b).

Proof. By continuity there exist e, S > 0 such that YE(f (z); f'(z)X) + EII X II < YG(z; X),

Z E B(a, 3) cc G, X E C".

Recall that we always have

YE(f(Z);f'(Z)X) < YG(Z;X),

Z E G, X E C".

Let a : [0, 11 -+ G be a piecewise C ' -curve joining a and b. Denote by to the maximal t E [0, 1] such that a([0, t]) C B(a, S). Then I

LYG(a) =

J0

oYG(a(t);a'(t))dt+ J YG(a(t);a'(t))dt r0

f YE(f(a(t));f'(a(t))a'(t))dt+6 I0 Ila'(t)IIdt 0 Ly,(f oa)+sS> p(f(a), f(b))+eS=cG(a,b)+ES.

2.5 Properties of ci`i and y. Length of curve. Inner Caratheodory pseudodistance

41

Hence by Remark 2.5.3(d)

c'G(a. b) > cG(a, b) + eS.

O

Example 2.5.7 ([Jar-Pfl 41). Let

P:_{zEC: 1/R l), a:=1,b:=-1. If g E 0(P, E) is an extremal function for c;(l, -1), then

f(z) = 2(g(z) +g(z)),

z E P.

is also extremal and f'(1) = 0. Hence by Lemma 2.5.6 we get

cp(1, -1) < c' (1, -1). The full description of c`p will be given in Proposition 5.10. We conclude this section with a discussion of a surprising structure of Caratheodory balls. Example 2.5.8 ([Jar-Pfl-Vig 21). The aim of this example is to show that, in general,

B(( (a. r)

BAG (a, r),

where B,,; (a, r) denotes the open Caratheodory ball

B(.G(a.r):=(ZEG:cG(a,z) < r) and Bc, (a, r) is the closed Caratheodory ball

B,, (a, r) :_ {z E G: cG(a, z) < r) (the closure BCG (a, r) is taken in sense of the standard topology of G). Observe that, since CG is continuous, we always have

BcG(a,r) C B,G(a,r). One can easily prove that the condition Bc6 (a. r) = BcG (a. r)

for any

a E G, r > 0

(2.5.3)

42

Il. The Caratheodory pseudodistance and the Caratheodory-Reiffen pseudometric

is equivalent to the following "minimum principle":

For any a E G the function G 9 z -4 CG (a, z) has no local strictly positive minima.

0 We do not know of any example of a domain G C C' for which (2.5.3) is not satisfied. 0

On the other hand, we will prove that for any n >_ 2 there exists a bounded strongly pseudoconvex domain G C C" with real analytic boundary such that for some a E G and 0 < r2 < r, we have: (a) the ball B,G (a, r,) is relatively compact in G and disconnected, (b) B,., (a, r2) (; &. (a, r2) Moreover, we will construct a connected c- and y-hyperbolic Riemann surface

M such that for some a E M, 0 < r2 < r,, conditions (a), (b) are fulfilled with G = M; recall that the answer is not known for G C C1. Note that if (a) is satisfied, then CG is not inner; cf. Remark 2.5.5.

We now begin the construction. 1 ° For any c-hyperbolic domain G C C" condition (a) implies (b).

Proof of 1 °. Let S be a connected component of BCG (a, r,) such that a 0 S. Put

r2 := min(cG(a, z): Z E S)

and let b E S be such that cG(a, b) = r2. Observe that b E S, 0 < r2 < r,, b E B,G (a, r2), and s l Bc, (a, r2) = 0. Hence b E B,G (a, r2) \ BAG (a, r2).

O

Note that 1 ° remains true for c-hyperbolic manifolds.

2° For any domain G C C" we have

cGxE((a, 0), (z, l)) = max{cG"i(a, z), cE`i(O,,l)},

a, z E G,.1 E E.

Note that the above product-formula is a particular case of the product-property in Theorem 9.5.

Proof of 2°. Obviously, it suffices to consider only the case of c*. The inequality ">" follows from the holomorphic contractibility (w.r.t. the projections). Fix a, Zo E G and let C := cG(a, zo). First consider the case C = 0. Then for

every function f E O(G x E, E) with f (a, 0) = 0 we have f (zo, 0) = 0. Hence, If (zo. A)I : IXI for all A E E, and consequently cGXE((a, 0), (zo, A)) = IXI for

AEE.

2.5 Properties of c(*) and y. Length of curve. Inner Caratheodory pseudodistance

43

In the case C > 0 it suffices to prove the formula for all IXI = C. For, if 2` is true on the circle (IAI = C}, then by the maximum principle the formula follows for all I), I < C (the function E A - ccXE((a, 0), (zo, ),)) is subharmonic; cf. Proposition 2.5.1(b)). To prove the inequality "<" for C < IAl < 1 we use the maximum principle for the subharmonic function 1

// E, ? A -'i -CGxE((a. 0), (zo, A))

Now fix A0 with IXOI = C and let f be an extremal function for f (a) = 0, f (zo) = X o. Consider the holomorphic map

zo) with

GDz-+(z.f(:))EGxE. Then the holomorphic contractibility gives caxE((a, 0), (zo. ko)) = ccXE(F(a), F(zo)) < cG(a, zo) = C.

r) is relatively compact in G and disconnected, then 8,.,;,} ((a, 0). r) 3° If has the same properties w.r.t. G x E.

Proof of 3°. By virtue of 2° we get B,.(;F ((a, 0), r) = B,.,; (a, r) x Be(0, r). 4° If Go is a domain of holomorphy such that a ball B,.,;Q (a, r) is disconnected, then

there exists a bounded strongly pseudoconvex domain G C Go with real analytic boundary such that Bc,; (a, r) is relatively compact in G and disconnected (cf. (a)).

Proof of 4°. Let G / Go be an exhaustion of G by bounded strongly pseudoconvex domains with real analytic boundaries, cf. Appendix PSC 10. Then, by Proposition 2.5.1(a), B,,, (a, r) / B,.,;, (a, r), and therefore for sufficiently big v the ball B,,,, (a, r) is disconnected. On the other hand, if G is a bounded strongly pseudoconvex domain, then for

any zo E aG there exists a peak function f E O(G) with If I < I in G and if (zo)I = I (cf. [Kra 11, see also Appendix PSC 11). This shows that for any a E G, r > 0, the ball 8,,; (a, r) is relatively compact in G; cf. Example 7.4.2. 5° Properties I'' - 4° show that it is enough to construct a domain of holomorphy G C C2 such that at least one ball B,.,; (a. r) is disconnected. 6° (Sibony domains; see [Sib 11) Let (a,t )k'=t C E. be a sequence without accumulation points in E and such that any point E a E is a non-tangential limit of a

44

II. The Caratheodory pseudodistance and the Caratheodory-Reiffen pseudometric

subsequence of (ak )k 1. Choose (Ak )k i C R>o in such a way that 00

E Ak log 2 I > -Co k=1 and define

q (z) :_

2

YAk log

z E E,

,

k=1

expgp,

G:=((z,w)EExC:lwlexp*(z)<1}. Observe that G c E2 is a fat Hartogs domain of holomorphy. Then HO0(G) = H°0(E2)IG. In particular, cG((z,

w'), (z" w")) = cEXE((z w'), (z", w")) = max(m(z', z"), m(w', w")), (z'. w'), (z", w") E G. cf. Remark 2.1.4. (2.5.4)

7° There exists a Sibonv domain with disconnected Caratheodory balls; cf 5°. Proof of 7". Obviously, it suffices to consider c*-balls. We keep the notation of 6° and we assume that 3/4 < lax. l < 1, k > 1. Take 0 < e < 1 /2 and define

WIF

c2

01-) + log I

4

6 ,

z E E,

ff

F := exprpF. ((z, w) E E x C: lwl exp *e(z) < 1 }. Gr

Clearly, the domain G, has the same properties as G; cf. 6°. Put 1

A :_ (log -) 8

"` k=1

Ak,

l

B := 4 exp A, 4

and let 0 < C < B. Note that *f(z) >- Blz2 - e21 for I z 1 < 2,

(±e, w(e)) := (±e, exp(-Ce2)) E GF.

(2.5.5)

2.5 Properties of c(*) and y. Length of curve. Inner Carath6odory pseudodistance

45

Our aim is to prove that for sufficiently small E > 0 the ball

B(e) := B,.((-e, w(e)), 2s)

is disconnected. Recall that for - l < t < I and 0 < r < I we have

B(t,r) = B(t(1 -r-) r(1 -tz) 1 - t ''-r'''-r'-

cf. Exercise 1.1.

1 - 12r2

Consequently, if s is small, then by (2.5.4) we get

(s, w(s)) E B(s) c B(0, 2) x I E \ B(0,

1w(E)

- 2e ))

.

(2.5.6)

Suppose that B(s) is connected (s small). Then there is a continuous curve a: [0, 1] -> B(s) joining the points (±s, w(e)). Consequently, for some y E R and w* E C we get (iy, w*) E B(s). By (2.5.6) we have I

IYI < 2

and

w(8) - 2s Iw I >

1 - 2Ew(E)

On the other hand, by (2.5.5) we get

Iw*I < exp(-*e(iy))

exp(-BI(iy)2

- 821) <- exp(-B82).

Finally,

w(e) - 2E < exp(-Bs2),

1 - 2--w(E)

which is impossible for arbitrarily small s > 0; a contradiction.

0

The construction of a "regular" domain G C C", n > 2, with properties (a) and (b) is complete. We go to the construction of a connected c- and y-hyperbolic Riemann surface M with top cm = top M such that (a) and (b) are fulfilled. In view of V, the only problem is to realize (a). Fix 0 < qo < q1 < q2 < q3 < q4 < I with 2 gig2q3q4 < qo and then define 0 := E \ 071,172,173, q4). Let M denote the Riemann surface obtained by joining two copies of 0 along the two sides of each of the intervals (ql, q2) and (173,174) crosswise. Denote by n : M -+ 0 the standard covering projection. Obviously, M is connected, c- and y-hyperbolic, and such that top cm = top M. For Z E 0 let

46

11. The Carathe odory pseudodistance and the Carathbodory-Reiffen pseudometric

{z+ z_} :_ )T-(z). We will prove that 0_ E B. (0+, rlo).

(2.5.7)

Since

B,,(0+, 1o) C r '({z E E: I z I < no)) CC M, the ball B,.4(0+, 11o) is relatively compact and the points 0+, 0_ he in different connected components. In order to verify (2.5.7) take f E O(M, E) with f (0+) = 0 and set

f(Z)

4(f (Z+)

- f(Z_))2,

ZE

f(rlj)0, One can easily prove that f E Q(E, E). Hence by the Schwarz Lemma 1

Finally,

If (0J < 170. Remark. If G is a balanced domain, then c*(0,.ka) -<

a),

X E E, a E G.

For we observe that cG (0, e'ea) = c* (0, a) and we use the Schwarz Lemma for log-psh functions; cf. Appendix PSH 22 and Proposition 2.2.1(a). In particular, BCG (0. r) is connected and BC.(; (0, r) = &(; (0, r), r > 0.

The phenomenon described in Example 2.5.8 does not occur if we look at small Carathdodory balls. Moreover, such balls are even starlike with respect to sufficiently small neighborhoods of their centers.

Proposition 2.5.9 (cf. [Maz-Vig]). Let G be a bounded domain in C" and let be

a E G such that B(a, r) CC G C B(a, R) for fixed r, R > 0. For 0 < t < 1 we put s = s(t) := 1}4i Then for any z' E Z" E B,c(a't(R)2) the half-open segment (z', z") {z' + r(z" - z'): 0 < s < 11 is contained in B,G(a,t(L)2)

2.5 Properties of c(*) and y. Length of curve. Inner Caratheodory pseudodistance

47

Proof First observe (cf. (2.4.2)) that if z E k (a, a (R )2) with a E (0, 11, then )2) c B11 11(a, r). In particular, Uz - all < Rc*(a, z) -< a(R)r < r, i.e. or (-L)2) [z', z") C B11 11(a, r) C G.

Now, it suffices to prove that for any f E O(G, E), f (a) = 0, and for any E [z', z") the following inequality is true: If (i) I < t (R )'-.

Fix f and i and assume, without loss of generality, that f (i) = If (i)1. Set p := i=s I+srR We easily see that p > 1. Moreover, for X E pE we obtain that

llAz"+(I -,1)z'-all <- IAIIIz" - all+II - XlIIz'-a11 <

ptRr+(I +p)srr
This permits us to define the following function Ip E O(pE, E):

(I - X)z')),

SP(A) := 2 (f (AZ" + (I - A)z') + f

A E pE.

Directly from the definition we see that: (a)

cp

has real values on (-p, p),

(b)

cp(0) = Ref (z') < s(R )2,

(c)

go(1) = Re

f(z") < (R)2.

What remains to show is that cp(a) < l (L)2 )2 for all a E [0, 1). Otherwise, there would exist a' E (0, 1) such that W(a') = supo«, t(R)2. In particular, (p'(a') = 0. Now put

I - (P(a')c,(A)

X E pE.

Then g E O(pE, E) and, in addition, we have g(a') = g'(a') = 0. Let h denote the following automorphism of pE

P2 .l+a' p2+a'A'

lEpE.

It follows that g o h E O(pE, E) satisfies g o h(0) = (g o h)'(0) = 0. Hence from the Schwarz Lemma we find that Ig o h(A)l <

(I),I/p)2.

48

H. The Caratheodory pseudodistance and the Caratheodory-Reiffen pseudometric

In particular, for I = -a' we get the following inequalities:

I

((P(a')(P(0) < IS{0))

( p )Z <

p2

Because of cp(0) < s(R)2 < t(R)2 < cp(a') < 1 we conclude that

r 2 - t(R)2-s(R)2 (t -s)(R) 1 -st(R)4 < p2. from which we derive the following inequality

0 <s3+(1 -t)s2+(4t+ 1)s+t2-t = s3 - (1 + t)s2 ; a contradiction. 13

Corollary 2.5.10. Under the assumptions of Proposition 2.5.9 we have: BI 1(a p) C Be,; (a, (R)2) CC G, where p := r(R)2.

Moreover, foranyz" E 8B01(a, p) the function [0, 11 3 t -* cG(a,a+t(z"-a)) is strictly increasing.

Remark. (a) Proposition 2.5.9 shows that small Caratheodory balls in bounded domains are always starlike.

(b) The claim of Corollary 2.5.10 is also proved in [Cart 21 but by a slightly different method.

Finally, the following corollary shows that small Carathesodory balls behave nicely, i.e. differently from those in Example 2.5.8. Corollary 2.5.11. Under the assumption of Proposition 2.5.9 for any t r= (0, 11 we have B ,(a,t(R)2)=Bq(a,t(R)2).

2.6 Two applications In this section we present two interesting results whose proofs are based on the properties of the Carathdodory pseudodistance. Namely, we are going to describe the automorphisms of product domains and to discuss the existence of fixed points.

49

2.6 Two applications

First, we recall a few facts about the group Aut(G) of automorphisms of a bounded domain G C C". We always assume that Aut(G) is endowed with the compact open topology or, equivalently, with the topology of locally uniform convergence. Fix 4o E Aut(G). Then the following mappings

(b)

Aut(G) 3 ¢ -> 0-' E Aut(G), Aut(G) 3 0 - Oo o 0 E Aut(G),

(c)

Aut(G) 30 0

(a)

0 o Oo E Aut(G)

are homeomorphisms of Aut(G). In what follows we are interested in the connected

component of Aut(G) that contains the identity. This component is denoted by Aut;d(G).

Theorem 2.6.1 (cf. [Cart 2]). Let G3 C C111, j = 1, 2, be bounded domains. Then

for every 0 E Aut;d(GI x G2) there exist 4j E Aut(Gj), j = 1, 2, such that 4(z, w) = (01 (z), 02(w)), Z E G1. W E G. The proof of Theorem 2.6.1 will be given via several lemmas partially dealing with a slightly more general situation which may be of independent interest; see also Theorem 9.6. Lemma 2.6.2. Let Gj C C"" be bounded domains and let D C C"'+ be a domain satisfying G 1 x G2 C D C G 1 x C"'. Fix (at, a2) E G 1 x G2 and let ri, r2 > 0 be such that B(aj, 2rj) C Gj, j = 1, 2. Moreover, assume that a positive constant K is given with K diam G i < r2. Then for any points z', Z" E G 1, z' z", w' E B (a2, r2), w" E C"'-, satisfying W,-",', (Z", W") E D and 116-u""'111 < K, the following equality holds: II: -:II I

cn((z', w'), (z", w")) =

(z', z").

Proof. The inequality ">" is obvious from the contractibility property of c`. To verify the remaining inequality, fix f E O(D, E) with f (z', w') = 0 and f (z", w") = cp((z', w'), (z", w")). Then it is easily seen that the mapping G -3 Z -+ 1

w,+(w"-w')(Z-Z',z"-z')

liz' - z"1l2

E02

belongs to O(G1, B(w', r2)) C O(G1, G2). Therefore, the function g defined by g(z) := f (z, F(z)), Z E G1, has the following properties: g E O(G1, E), g(z') _ w') w")) g(z") = o 0,

cv((z',

(z"

11. The CarathEodory pseudodistance and the Caratheodory-Reiffen pseudometric

50

Lemma 2.6.3. Let G,, G2, a,, a2, r,, and r2 be as in Lenuna 2.6.2. Then there exist positive numbers a, fl with a < r, and iB < r2 such that the following statement (*,, *2) := 0-'. If the is true. Let 0 = (01. 02) E Aut(D) and denote by following inequalities:

110,(z,w)-z11
Il*1(z,w)-z11
II&(z, w) - wll < 0, 11*2(Z, w) - wll < 0,

(2.6.1)

hold for all (z, w) E C"'+"= with Ilz - a,11 < r,, 1Iw - a211 < r2, then O,(z. w) 01 (z, a2) for all (z, w) E B(a1, a) x B(a2, 6). There is an immediate consequence from Lemma 2.6.3, namely:

Corollary 2.6.4. Under the conditions of Lemma 2.6.3 it follows that there exists an F E O(G 1, G,) such that 01 (z, w) = F(z) for all (z, w) E D. Proof of Lemma 2.6.3. We start this proof by introducing some positive constants. First we choose R > 0 such that B(z, R/2) J G,; in particular 2r, < R/2.

l

iiz-a,II
Moreover, we fix p, a, 0, and K in such a way that the following inequalities hold:

p < r, (R )2 < 2 ,

a < 1.

K diam GI < r2,

6 < min(r2, K (2

- a)).

Recall from Corollary 2.5.10 that for any z E B (a, , r,) and X E C"', II X II = 1, the function [0, p) -D t -+ c*, (z, z + tX) is strictly increasing.

Now, we fix 0 = (01. 02) E Aut(D), * := (*,, *2) := 0-1, such that 0 and 41 satisfy conditions (2.6.1). Then we look at an arbitrary but fixed point (z', w') E C" x C"' with Ilz' - a, 11 < a, 11w' - a211 < f. Because of (2.6.1) it follows that

110, (z'.w')-4' (z', a2)11 < 110](z'.w')-z'11 +l1z'-01(z',a2)II < 2a
pin such a way that 01(z', w') _ -,01 (z'. a2)) for a suitable a > 0. Then

E C"' with 11 Z; - 01 (z', a2) II

Il01(z'.w')--III>p-2a>0. Moreover, observe that

III-a,11

III-Oi(z'.x2)11+ 110,(z',a2)-z'II+IIz'-a,11 < 2p
2.6 Two applications

51

In particular, the function

[0,11 a t -+ cG,

+ t (01(z', a2) - c) )

is strictly increasing.

a2) E D. Then, using (2.6.1) we get

Let (z", w") :=

a2)II < a and

III - z"II =11K -

11w" - a211 =

a2) - a211 < fl-

To be able to apply Lemma 2.6.2 we establish a sequence of simple inequalities; proofs here are mainly based on (2.6.1) and the choice of the constants: K fi K(p/2 - a) 110, (z', a2) -III - P < P <2

1102(z', a2) - a211

(i)

(ii)

11z"-z"II? 11w"-a211

0

<

K(p/2-a) = p-2a

<

11z"-z'11 - p-2a (iii)

Ilw - W'11 < 11z'-z"11

(iv)

1

- P-2a

II&2(z', w') - a211 <

K 2

(11 iv' - a211 + IIa2 - 02(x, a2)II) <

1

110,(z',w')-III

(1102

2f

p-2a

< K;

(z.w)-wll+11w-a211)<-
Then Lemma 2.6.2 implies the following chain of equalities:

c%J4 (z', a2) )

0)

`D(O(z', a2),

a2)) = cD(O(z', u,). O(z

w")) _

co((z., a2), («", w")) = CG, (Z, Z") = cD((z', w'), (z", III")) =

cD(d'(z', w'), 4(z", w")) = c* (0(-', w'),

a2)) = cG(01(:', w') )

cG t (¢i (z', a2) - )) leads to 01 (z', w') = 01 (z', a2). Since z', u,' were arbitrary, the lemma is established.

Finally, the strict monotonicity of i

After all these preparations we turn to the proof of Theorem 2.6.1.

52

11. The Carathe odory pseudodistance and the Caratheodory-Reiffen pseudometric

Proof of Theorem 2.6.1. First we show that the set M := (4) E Aut;d(G, x G2):

3Fj E O(G1, Gj): 0(z, w) = (F,(z), F2(w)),

(z, w) E G, x G2}

is a non-empty open and closed subset of Aut;d(G, x G2), and therefore: M = Autid(G, x G2). Obviously, idG, XG, E M. The closedness of M follows simply from the fact that if i C O(Gj), f,, f E O(G), then D°' reader. So it remains to show that M is open.

f. Details are left to the

Fix 46 E Aut;d(G) fl M. Because of the symmetry, Lemma 2.6.3 can be simultaneously applied for the two components of 0. Take ' E M such that *-1 o 0 satisfies the conditions of Lemma 2.6.3. Then we can find functions Fj E

O(G;, Gj) such that >r o 4)(z, w) = (F,(z), F2(w)), (z, w) E G, x G2. Hence 0 (z, w) = * (F, (z), F2 (w) ). On the other hand, k belongs to M, i.e. there are Hj E O(G;, Gj) describing *. So we get that 4)(z, w) = (H, o F, (z), H2 o F2(w)), (Z, W) E G, x G2. To conclude the proof it suffices to mention that the Fj's in the definition of M are obviously bijective.

Remark 2.6.5. (a) Observe that E x C 9 (z,, Z2) -* (z,, z, +z2) E E x C belongs to Aut;d(E x C) but it is not of the form stated in Theorem 2.6.1. (b) We mention that Theorem 2.6.1 remains true also in the manifold case (cf. [Pet]). Of course, the condition "bounded" has then to be substituted by another property, namely, by the hyperbolicity.

Theorem 2.6.6 (cf. [Rei 2]). Let G be an arbitrary domain in C". Then a holomorphic mapping f : G -+ G with f (G) CC G has a unique fixed point.

Proof. First we assume that G is bounded. Write R := diam(G) and let f satisfy the conditions of the theorem. Then, because of f (G) CC G, there exists a positive

number r with f (G) + B(0, r) C G. Now set it := r/(2R). If Z E G is fixed we define a holomorphic map gz E O(G, C") by setting g- (w) := f (w)+it (f (w)- f (z)), w E G. In view of the choice of µ it is clear that gz E O(G, G) and, moreover, we have that g,(z) = f (z), g' (z) = (1 + A) f'(z). Therefore we obtain

YG(f(Z):f'(z)X) = (I +µ)-'rG(g.(z):g'(z)x) _ (1 +u)-'YG(z;x), ZEG,XEC".

2.7 A class of n-circled domains

53

Then, applying Remark 2.5.3 it follows that

c'' (f (z'), f(z ))

(I + A)-'C' (Z'. z"),

Z', Z" E G.

From this inequality we conclude that CG(.f(k)(Z),

f(k+')(z))

< (I + u)-kcG(z, f(z)),

z E G, k E N,

where f (k) denotes the k-th iterate of f , i.e. f (k) := f o observing that G C B(z, R) for any z E G we have II f (k+r) (z) R

- f (k)(z)II =

- ca(I'"'(__).tr)(f(k)(Z)

(2.6.2)

o f (k times). Finally,

f(k+0(Z))


ECG(f(k+j-1)(Z), f(k+j)(Z))

l

j=I

G(<'..1(z)),

I

(I +µ)-(k+l-I)Ci

ZEG,k.1EN.

j=I

So we can see now that (f (k)(z))A 1 is a Cauchy sequence in C" for an arbitrary point Z E G. Fix Z' E G and set z" := f (k) (z'). By hypothesis, we have z" E f (G) C

G and f (Z") = limk-. f (f (k)(z')) = z"; i.e. z" is a fixed point of f. On the other hand, (2.6.2) shows that there is at most one fixed point of f . The unbounded case is an immediate consequence. For, let f be as in the theorem. Then choose a bounded domain D with f (G) CC D C G and apply to f ID what was said above.

2.7 A class of n-circled domains Up to now, the only domains for which we have explicit formulas are balanced convex domains (cf. Proposition 2.2.1(c)). We would like to find another class of domains G C C" for which cG may be effectively calculated (at least for some points). A natural candidate is the class of n-circled domains (cf. (Jar-Pfl 1,31, see also [Azu 5]). Throughout this section G denotes an n-circled domain in C" with 0 E G C n > 2. Recall that a set A C C" is said to be n-circled if f o r 0 = (01, ... , H") E 1[8"

54

II. The Carath6odory pseudodistance and the Caratheodory-Reiffen pseudometric

we have

&(A)=A, where

C"9(Zi,...,Zn)-*(ere'zi,...,ei0"z,,)EC"; n-circled domains are also called Reinhardt domains. It is well known that any function f E O(G) has the power series representation

f (z) = E aaza,

Z E G.

aeZ+

Observe that

IaaZal 5 max(If(RB(Z))I:9ER'},

ZEG,aEZ+.

Consequently, if f E H°°(G), then laal - IIzaIIG <_ IIf IIG,

a E Z+,

and so we get

Lemma 2.7.1. If f E HO°(G), then

f (z) = f (0) + E aaza,

z E G,

aeS(G)

where

S(G) :_ (a E (Z+).: Za E H°°(G)) is the set of admissible exponents for G. In particular, H°C(G)

C

i,,Q`'

S(G) = 0.

Elementary n-circled domains are defined as follows:

G(a, C) := ((ZI... , Z,,) E C": IZila' ... IZ"I0' < eC},

where a = (a 1, ... , a,,) E (lR )., C E R. There are two possibilities: either G(a, C) is of irrational type (i.e. a ¢ R R. Z")

55

2.7 A class of n-circled domains

or 0(a, C) i of rariana[ qpr (i&1 Y i - r); In the second case we may assume that a = (ai , ... ,

E (Z . ). and that

a l , ... , a are relatively prime. Put

H(a, C) ._ {x E R" : (x. a) < C}. ((, ) denotes the Euclidean scalar product in R"). For A C C" let log A be the logarithmic image of A, that is,

logA:={x=(x,,

,x,,)ER": ex=(ex'.

,ex")EA}.

Clearly, log G(a, C) = H(a, C). In general, we have

S(G)={aE(Z ).: 3C: GCG(a,C)) = {a E (Z+).: 3C: logG C H(a, C)). In particular,

S(G(a, C)) = 0 if G(a. C) is of irrational type. S(G(a, C)) = Na iff of E (Z'+)., &I--, a,, are relatively prime. (2.7.1) Using Lemma 2.7.1, one can easily conclude that in the latter case the mapping

H'(E)

V --> V o 4) E H"°(G(a, C)),

where 4)(z) = e-cz", is an isometry and an algebraic isomorphism. Thus we have proved the following

Proposition 2.7.2. If G = G(a, C), then (a) cG = 0 (yG = 0) iff G is of irrational type,

(b) i f a = (al,

a,,) E (Z!+). and a,, ..., a,, are relativel ' prime, then cC')(w, z)

= cE`l(4'(w), ID (Z)),

YG(Z; X) = YE(ID (Z); 4"(z)X),

w,

E G.

Z E G, X E C".

Now, we return to the general case. For a E S(G) put n" = n" (G) If T C S(G), then we define

M"(z):=sup{nIz"I: aET},

zEG,

z" II c

56

II. The Carathdodory pseudodistance and the Carathtodory-Iteiffen pseudometric

where MG o:= 0. Let MG := MS(G) and observe that

Our aim is to characterize some classes of n-circled domains G C Cn (with 0 E G) such that c* (0, ) = MG, or even more, cG (0, ) = MG for some T C S(G). We are interested in characterizing the minimal set T with this property and in situations

where the set T is finite (which is most important from the point of view of applications).

Note that if a is as in Proposition 2.7.2(b). then by (2.7.1) CG(a.C)(0") = MG(d.C).

Lemma 2.7.3. (a) The function MGT is continuous and invariant under the rotations

R©,9ER". (b) If T

0, then for any a E G there exists a E T such that MGT (a) = nalaal.

Proof. (a) The continuity of MGT holds since the family (nazi : a E Tj is equicontinuous.

(b) If MGT (a) = 0, then the result is trivial. Suppose that

0. Let

(aj) j , C T be such that naj l aa' I -. MG (a).

Put fj(z) := naiz°1', j > 1. In view of the Montel theorem, we may assume that

f j =4 f in G. Clearly, f (0) = 0, f (a)

0, and f E O(G, E). To prove the

required result it is enough to show that litn1. i o l aj l exists and is finite. Observe that

Ezkafj(z) = IajIfj(z), j ? 1. z k=1

k

Hence n

l llm laj l = 1: a., fl/a) k=1 8

0

(a).

Define

B* = B*(G) :_ (a E S(G): 3a E G: nalaal > nfiIa0I,

p E S(G) \ la)).

2.7 A class of n-circled domains

57

Lemma 2.7.4. For any T C S = S(G) we have MG = MG

iff

B*(G) C T,

that is, B' (G) is the minimal set determining the function MG.

Proof. The case S = 0 is trivial. So suppose that S # 0. For a, $ E S, a 74 P, put

Va,p := {Z E G: nalzal = nplz11 }.

It is clear that this set is closed and nowhere dense. Hence the set

Go:=G\ U V..{3 a.OES

aAfi

is dense in G. Take a E Go and let a E S be such that MG (a) = na I as I (Lemma 2.7.3(b)). By the definition of Go we see that na laa 1 n p l a,61 for all $ E S \ {a }.

Hence a E B(G) which shows that MG =

on Go.

Finally, the continuity (Lemma 2.7.3(a)) gives MG = MG = MG, then we get the inclusion B*(G) C T directly from the definition of B*(G) and from Lemma 2.7.3(b). 0

Let B = B(G) := S \ (S + S), where S = S(G); B(G) is called the set of all irreducible elements of S. Observe that for any a E S there exist k E N and t, , pk E B such that a = # + + /k (in general, k and $1, ... , $k are not 1

uniquely determined).

We say that G satisfies the cone condition if log G is a cone with vertex at a point x° E R", i.e. x° + t (x - x°) E log G for all t > 0, x E log G. In this case

na = e-(x°.a),

a E S(G),

and hence

B*(G) C B(G).

Moreover, we may assume that x° = 0 (use the mapping C" a (zi...... ") u (e-",z,

,

u e-"',z,,)

E C").

II. The Carathtodory pseudodistance and the (:u;nhrutlnr) Keillrn

58

Proposition 2.7.5. If G is an n-circled domain, 0 E G, +atisti ing the rune rendition, then

t'c(0

MG (=

M' '

=

Proof. We may assume that S = S(G) lp4 --f and that log G i, a cone ,%ith the vertex at 0 E R". Since both functions c* (0, ) and ate; are continuous and invariant under rotations, we only need to prove that c" (0. e'`) < Mc; (ex )

for .r in some dense subset of log G. Note that

MG(ex) =exp(sup{(x,a): a E S}).

x E logG.

Put

G' ._ { (

t' .

.

,

v

X""):

= ( v r ,....

E 1 - Iog G )

I

1

Then the set

log Gtr=(tv:t <0, vE(-logG)nZ") is dense in log G and

Mu(ett') = exp(tQ(v)) where Q(v) := inf((v,a): a E S}. Observe that Q(v) E N. On the other hand, if f E O(G, E), f (0) = 0, then the function

E.3A-+

fW''......

t")= &ES

extends to the whole E and the extension has a zero of order Q+ it at the origin. Hence

cc(0.

E E.

In applications, the most interesting case is when t2.7.2k

2.7 A class of n-circled domains

59

aN E (R'+)*, C1, ... , CN E R. We will always assume that the intersection is minimal, i.e.

k = l..... N.

G 54n G(a&, C;), j

(2.7.3)

k

The case N = I is covered by Proposition 2.7.2, so let N > 2. Put r := rank(a1, ..,aN). Lemma 2.7.6. (a) The domain G (as in (2.7.2), (2.7.3)) satisfies the cone condition

if

aI

C,

aN

CN

r = rank

(b)

U

S(G) = (z'+), n ( ,<

(R+ail + ... + R+a"

1<...
rank(& .....air )=r

(c) Let a" ... , aN E (Z n.)*, aj = (a...... an), and suppose that a...... ai are relatively prime, j = 1..... N. Then B(G) C {a1, ... , aN) U ((Z'+). n

U

([0. 1)a'' + ... + [0, l )a'r)) M.

,
rank(a' I ....,a'') =r

In particular, the set B(G) is finite. (d) If G (as in (2.7.2) and (2.7.3)) is an H''0-domain of holomorphy satisr'ing the cone condition and if the set B(G) is finite, then

In other words, if G satisfies the cone condition and if G is an H"-domain of holomorphy, then the situation described in (c) is essentially the only case where B(G) is finite. Proof. (a) Exercise. (b) Let S denote the right hand side of the formula in question. Clearly, S C S. Fix a E S. Then there exists C E 1R such that

H(al,C1)n...n H(a",CN) C H(a,C).

60

H. The Carath6odory pseudodistance and the Carath6odory-Reiffen pseudometric

< i, < N with rank(a" , ... a'") = r

One can prove*) that there exist 1 < i, < and such that

H(a'', C;,) n ... n H(a',, C;,) C H(a, C).

To simplify the notation put i, = µ, µ = 1, ... , r. By the above inclusion, a E Ra' + + Rar, i.e. a= tai i + + t,ar for some E R. It remains to show that ti, ... , t, > 0. Let

R" 9 x -+ ((x,

al), ..., (X,ar)) E Rr.

Note that L is sutjective. Put t := (t,, {( I.

..., t,). Then

r) E Rr: j < Cj, j = I__ , r} = L(n H(aj, Cj)) j=I C L(H(a, C))

E Rr: (l;, t) < C)

..., t, > 0. (c) Take aEB(G).By (b)we have a=t,a"+ +t,a'',where t,,...,tr>0 r. Suppose that i,, = µ, µ = 1, ..., r. Observe that and rank(a" , ..., which implies that t,,

of

r

=

r

(>[tjlaj) + (E(tj - [tjl )aj) =: f' + fi2, j=1

j=1

where [ ] denotes the integer part. It is easily seen that $1, #2 E S U (0). In view

of the definition of B(G) either a = Y2 or a = a`° for some io E 11, ..., r}. (d) We may assume that log G is a cone with the vertex at 0, or equivalently = CN = 0. In particular, na = 1 for any a E S. Put C, =

G := n G(fi, 0). BEB(G)

Since B(G) is finite, it suffices to prove that G = G. Since G C G is an H°°domain of holomorphy, we only need to prove that H°O(G) C O(G)IG; cf. Remark 2.1.3. Take f E HO0(G), f (z) = f (0)+EaEs aaza, z E G. To prove that the series is convergent on G it is enough to show that the sequence aaza, a E S, is pointwise + fik E S, bounded on G (G is a Reinhardt domain!). Fix b E G and a = # I + fk ... , E B. Then by the definition of G and by the Cauchy inequalities we *)

See the Appendix in M. Jamicki & P. Pflug, Existence domains of holomorphic functions of restricted growth, Trans. Amer. Math. Soc. 304(1987), 385-404.

2.7 A class of n-circled domains

61

have

Ia"b"I = Ia«Ilba'I...IbO'l <_ la"I

Corollary 2.7.7. If G = G(a', 0) n ... n

IIlz"IIG II11c

=

IIfIIG,

0

a E S.

G(arv. 0), ai = (ai , ... , a) E (Z ).,

j = 1.... , N, then the set B* (G) is finite and c* (0. z) = max(Iz"I: a E B`(G)),

Z E G.

While is given by monomials if G satisfies the cone condition, general situation is not yet completely understood

0 the

'.

Remark 2.7.& If G does not satisfy the cone condition, then in general c"(0, ) MG. For example, let

G := ((zi,z2): IziI < I, IZ21 < 1, IZIZ21 < 2}.

(2.7.4)

Then MG (1-7) = maxi 12,11, 1Z21.21z1Z21) (use Lemma 2.7.6(b))

but cG(0, z) > max{Izj 1, Iz21.

2

1Z21). 21ziz21) MG(z)

In fact, one can prove that

c*(0.;.) sup( IQ(z)1: Q is a homogeneous polynomial, IQI < I on G} (cf. Exercise 2.8, see also [Jar-Pfl 1)) and that C(*, (0, z) = MG(Z)

if z belongs to the shadowed part in Fig. 2.1 (cf. Exercise 5.7).

62

11. The Caratheodory pseudodistance and the Carathcodory-Reiffen pseudometric

1/2

Fig. 2.1

0 The complete formula for

is not known.

0

Using the same argument one can easily prove that for the domain G := ((z,, z2. z3): IZIZ3I < 1. 1Z2Z31 < I. IZIZ2Z21 < 2

(2.7.5)

we also have

On the other hand, there are domains without the cone condition for which MG. For instance, we have

Proposition 2.7.9. if G = G(a'. C,) fl

fl G(aN, CN), where a..... a'v E

Z+ ' x 111, then z) = max(e-c' Iz"' I : j = 1, ..., N) (= MG(z)),

ZEG.

Proof. The inequality ">" is obvious. Let a' = (g5', 1), j = I,-, N. Observe that C"-' x (0) C G and note that G may be written as the Hartogs domain: G = ((Z', Z") E C"-' X C: u(Z')Iz"I < 1).

63

2.7 A class of n-circled domains

max{e-c; I(z')" : j = I , ..., N}, Z' E C"-I where u(z') := If f E O(G, E) with f (0) = 0, then the Liouville theorem implies f (z', 0) = 0 for all z' E C". Hence, by the classical Schwarz Lemma (applied to the function I

f(z'. )), we get max{e-c, lz°'

I f (z', z") 15 u (z') Iz.l =

l : j = I-- NJ,

(Z'. Z,,) E G.

0

Example 2.7.10. If G := {(ZI, Z2, Z3) E C3: IZIZ31 < I. IZ2Z3I < I. IZIZ2Z3I < 2}, then

z) = max{IzIZ31, IZ2Z31, 2lzIZ2Z31} (= MG(Z)).

Note that G seems to be similar to the domain in (2.7.5), but the form of is totally different.

We conclude this section with some examples for n = 2. First observe that if G C C2 is a 2-circled domain satisfying the cone condition, then either G = G(a, C) or G = G(a1, CI) n G(a2, C2) with rank(a1, a2) = 2. In the latter case, we may assume without loss of generality that CI = C2 = 0. We will consider only the most elementary case where a' = (1.0). a2 = (p, q) E N2, and p, q are relatively prime.

Lemma 2.7.11. If G := {(Z1, Z2) E C2: Iz, I < 1, Izi z; l < I } and p, q E N are relatively prime, then

B*(G) C B(G) C ((1, 0), (p, q)} U {([p] + 1, v): v = I_- , q - 1). q

In particular, #B*(G) < q + I. Proof. Use Lemma 2.7.6(c).

O

We will show that the above lemma gives an effective method of finding B(G) for

G = Gp.9

pZq

l(Zl. Z2) E C2: IZII < 1. Iz1

< I)

64

II. The Caratheodory pseudodistance and the Carathdodory-Reiffen pseudometric

(p, q E N, p, q relatively prime). For let s(v).-I [9] +1, P.

v=0,...q-1 v=q

By the definition of B'(G) we have: (s(v), v) E B'(G)

iff

there exists (zI. Z2) E G such that Iz1

z21 > Iz1

z21 for all µ # v. (2.7.6)

In particular, (1, 0), (p, q) E B* (G ), and therefore we are done if q = 1. If q > 1, then by the definition of B(G) (recall that B*(G) C B(G)) we must exclude all multi-indices (s(vo), vo) of the sequence (s(v), v), v = 0, . . . , q, for which

s(vo) =s(µ)+s(vo-µ) for some µ E (0,...,q) (note that this is possible only for q > 3, 2 < vo < q - 1 and 1 < it < vo - 1). Let (ao(v), lo(v)), v = 0, ..., qo, denote the remaining multi-indices with

flo(g) < fio(v) for 0 < it < v < qo (if q = 2, then this sequence coincides with the initial one). Obviously

B(G) = I(ao(v), fo(v)): v = 0, ... , qo}. Now, we begin the main reduction procedure.

If qo = 1, then there is nothing to do and B'(G) = ((1, 0), (p, q)). If qo > 1, then by (2.7.6) we exclude f r o m the sequence (ao(v), io(v)), v = 0, ... , qo, the first multi-index (ao(vo), $o(vo)) such that

max(ao(vo) -ao(µ). µ =0,..., vo- 1) X(vo) - X(A) ao(vo) - ao(µ) > min(P,

q

fio(vo) - j6o(µ)

. µ = vo + l.... , qo)

(note that this is only possible for 1 < vo < qo - 1). If such a multi-index does not exist, then

B*(G) = ((ao(v) io(v)): v = 0, ..., qo)

and we are done. Otherwise, we get a new shorter sequence (cq(v),'6j(v)), v = 0, . . . , qi = qo - 1, and we can repeat our reduction procedure again.

2.7 A class of n-circled domains

65

Example 2.7.12. Let G := ((ZI, Z2) E C2: IziI < 1, Iz;°z25I < I). Then

z) = maxi 1z,1. IZIZ21, IzizzI, Iziz21,1z1 z2'I, Iz1

25I)1

see also Exercise 2.9.

Notes

In the twenties C. Carathr odory was the first who used the pseudodistance CG (now bearing his name) as a tool in complex analysis of several variables. The infinitesimal version of this pseudodistance (i.e. yG) was intensively studied no earlier than in the sixties by H. J. Reiffen [Rei 1,2]. Most of the results we presented in this chapter have become a part of standard knowledge and they can be found in many textbook (e.g. [Din], [Kob 4], [Pol-Sha], [Fra-Ves]).

The results on the Carathdodory isometries (the end of § 2.2 and Exercise 2.4) have the following general aspect. Assume that for any domain G C C" with arbitrary n we are given a pseudodistance dG : G x G -- R+ such that dE = p and suppose that the system (dG)G is holomorphically contractible, that is, do(F(z'), F(z")) _< dG(z', z") for any domains G C C", D C C' and for any holomorphic mapping F : G -> D; cf. Remark 2.1.2. Suppose that we want to use the spaces (G, dG) to characterize properties of holomorphic mappings. The question is how to distinguish between "good" and "bad" systems (dG)G. One possible criterion is that "good" systems have only few isometries F: (G, dG) -+ (D. dD) that are neither holomorphic nor antiholomorphic (at least in the case where G and D are subdomains of the same space C"). Observe that the problems related to the c- and the y-hyperbolicity and to the Carathdodory topology do not occur as long as we study bounded domains (cf. Propositions 2.3.2 and 2.4.1). The first counterexample was found by J.-P. Viguc [Vig 2] who constructed a complex c-hyperbolic space M with top M # top cm. Theorem 2.4.3 and Lemma 2.4.6 give counterexamples in the domain and the manifold case, respectively. Nevertheless, it seems worthwhile to look for more explicit examples. The first examples showing that CG is not inner in general are due to T. J. Barth [Bar 2]. Next, J.-P. Vigu6 [Vig 1] constructed a domain of holomorphy G C C2 such that (G, CG) is complete but CG is not inner. It is clear that the counterexample given in Example 2.5.7 is the simplest possible. Example 2.5.8 shows that there is a big difference between topological and geometrical structure of the space (G, CG Y

66

U. The Carathr odory pseudodistance and the Carath&odory-Reiffen pseudometric

Exercises

2.1 (cf. [Ear-Ham], [Lew]). Prove that for arbitrary z', z" E G C C" we have CG (Z', Z") = 2p(0, dc(Z', z (z,)

where dG(z', z") := sup(If

- f(z )I : f E O(G, E)}

Hint. If (p E O(E, E), then I(p(-A) - V(1)1:5 21X1,

A E E.

2.2.LetH+:=(AEC: Re), > 0). Prove that A'

cy+

A'

I

It + In

(AEI

'

I

E H+.

2.3 (Borel-Carathdodory Lemma). Let V : E -> C be a holomorphic mapping such

1, AEE.Then

that 9(0)=0 and Re (p(A) 1(

21A1

(A)1 t 1

- III

,

A E E.

B,,,, X B,,,, be a c-isometry, 2.4. Theorem (cf. [Zwo]). Let F = (F,, F-2): B,, n < m1, m2.Then Fi or F2 is a c-isometry; cf. Remark 2.2.6(d) and Proposition 2.2.7. In particular, up to a permutation of the components of the mapping F, we

have Fi = 4i o U. where U : C" -+ C" is unitary or antiunitary and 4? E Aut(3311, ).

Complete the following sketch of the proof. Consider the following two cases: (1)

min(cL.,, (F, (a)), F, (b)), cB,,, (F2(a), F2(b))) < c ,,(a. b),

(II) 3a c- 3,: db E B,: ca., (F1 (a)), F1(b)) = c6,, (F2 (a), F2 (b)) _ cB, (a, b).

67

Exercises

C a s e (/). For any a E lg" there exist j E { 1, 2) and open sets U1, U2 C B", a E U1, such that

(w, z) E U, X U2.

ce,, (Fj(w), Fi(z)) = CB. (W, Z),

(*)

First, we prove that

Va E B,,: F, or F, is holomorphic or antiholomorphic near a.

(**)

Fix a E B,, and let j, U,, U2 be as in (*). Suppose that j = 1. Taking 4'o F, o 41 instead of F,, where %P E Aut(B") and 4' E AUt(Bm,) are suitable automorphisms,

we may assume that a = 0, F, (0) = 0, and F, ((r, 0, ... , 0)) = (r, 0, ... , 0) for some 0 < r < 1 with (r, 0, ... , 0) E U. Moreover, we can also assume that U,, U2 are disjoint balls. Define

u, := re,, Uk := re,+

Eek,

k = 2, ... , n,

vk:=re,+isek, k= 1,...,n, where e is so small that Uk, vk E U2, k = 1, ... , n (and where e, , ... , e" denote the vectors of the canonical basis in C"). Then

F,(tuk)=tF,(uk)foralltERsuch that tukEU,UU2,

k=I,...,n.

Consequently, using induction over k, one can prove that there exists a unitary transformation T : C", -+ C" such that T o F,(uk) = Uk,

k = I,...,n;

as usual, we identify C" with C" x {0} C C for T o F,. Applying (*) to (w, z) = (tu4, ToF,(Vk) E (Uk, Vk),

Clearly, formula (*) remains true E U, x U2, we get

k = 1,...,n.

Now, taking (w, z) = (tuk, z) E U, x U2 and (w, z) = (tuk, z) E U, x U2, we prove that

ToF,(z)E(z,z},

zEU2.

Hence by the c-contractibility of T o F, we get

(***)

either ToF,(z)=z, zEU2,

or

ToF,(z)=i, zEU2,

68

II. The Carath8odory pseudodistance and the Carath8odory-Reiffen pseudometric

which completes the proof of (**). Now, we use the c-isometricity of F to prove that Fi or F2 is holomorphic or antiholomorphic on the whole 1B,,. Finally, by (* * *) we get the required result.

Case 11. We can assume that a = 0 and F(0) = 0. Consider the following two subcases:

min(cBm, (Fj(b), Ft(-b)). cB.,(F2(b), F2(-b))} < c ,(b, -b), (a) 3b E cB., (F, (b), F, (-b)) =cBR, (F2(b), F2(-b)) = cj (b, -b). (b) bb E B,: In the case (a) we may assume that CB., (F1 (w), F1 (z)) = cB.(w, z),

(w, z) E Ub X U_b,

where Ub c B,, is an open ball with the center at b, 0 It Ub, and U_b = -Ub. Next, we proceed similarly as in Case 1.

In the case (b) we put Uk := rek, k = 1, ... , n, for some r E (0, 1). Similarly as in Case I we have

TjoFj(uk)=uk,

j = 1,2, k= 1,...,n.

The c-contractibility of Fj gives Re(Tj o Fj(z))k = Rezk and Ilm(Tj o Fj(z))k1 < I IM zkl,

zER,,, j=1,2,k=1.....ra. Consequently, since the mapping (Ti o F1, T2 o F2) is a c-isometry, we have Tj o Fj(z) E lz, 0 for z in a certain nonempty open subset of the set

k=

1.....

rt).

and therefore either Tj o Fj(z) = z or Tj o Fj(z) = z for z in a nonempty open subset of D, j = 1, 2. Proceeding similarly as in Case I we complete the proof. 2.5. Prove that the Hayashi surface (cf. Lemma 2.4.6) gives another example of a connected c- and y-hyperbolic Riemann surface with disconnected Caratheodory balls.

Izii + Iz21 < I } be the 2.6 (cf. [Sch 1,2]). Let G := lz = (zi, z2) E C2: Ilzllt !t-unit ball in C2. Fix z' = (zi, z2) E G \ (0) and X', Z1, wt E C such that X'' c G, (zI, zZ) E G, and (wt, z;) E G.

Exercises

69

Prove that

cG(ZA'z') = m(IIz'III. A'IIz'III)

and co((zI, z2), (wi, z2)) = m(zl/(1

- Iz2I), wl/(1 - 1z21))

Hint. Use the mappings g E O(E, G), g(A) := 1Az'/Ilz'll I for A E E, and f E for Z E G, to obtain the first equality. O(G, E), f (z) := z,e''Arg:; + zee-' Proceed in a similar way with the second claim. A`8zz=

2.7 (cf. [Sch 2J). Let G be as in Exercise 2.6 and fix z° = (z,°, z_,-,°) E G. Assume

that there are positive numbers r and p with 0 < r < I and a point z' E G such that B, (z°, r) = BI1 III (Z' P) Prove that z° _ (0, 0) (i.e. the only Caratheodory balls in G that are II II I-balls are those with the centers at the origin). Hint. First, observe the following elementary geometrical fact:

if la + r,e'te'+'°i) + lb + r2e'(°=+v)l = constant for every cP E II8 (here a, b E C, rI, r2 > 0, and 01, 82 E R), then necessarily a = b = 0. Suppose that z° z°

0. Then, apply Exercise 2.6 to show that IIZ°III

r) = 8B11 11,(z', p),

z° E

where 22r2

A(w) := 11z°III

+r

l-

IIIef0 2

1-IIzIIII - Ilzllir1

°

(p E R.

Use the geometrical fact mentioned above to conclude that ° 1-r'1-llz°Ili z'=zIlz°II,r2 forj=1,2and P=rllz°IIir2

Finally, employ Exercise 2.6 again to show that (w, (IG), zi) E 8B,. (z°, r) = aB11 11. (z', p).

where

or)+(1-Izzl)r (1 - lz2I)` - Iz, I r-

(1-I_2I22-IoZ12"p

(1 - Iz21)

012r2

Iz,

70

II. The Carathe odory pseudodistance and the Carathr odory-Reiffen pseudometric

Then, by the geometrical fact again we have °

z;(1 - IzZI)(1 - r2)

(1 - Iz21) (I

- Iz21)2 - Iz,°IZr2

°

- z' 1

I - r2

- IIzopzr2

which gives a contradiction. The remaining case z° 0 0, but z° z° = 0, can be treated in a similar way.

2.8. Let G be as in (2.7.4). Prove that sup( IQ(z)I : Q is a homogeneous polynomial , IIQIIG

1)

= max(Izi I, Iz21, 3 (Izi I + Iz21)), 21ziz21) =: HG(Z),

Z E G.

Let P(zi, z2) := a(zi +z2) -b(z2 +zZ)+cziz2, where b, c > 0, 5b < 2c < 4(b+1) (4+5b-2c)/6. Verify that IIPIIG = 1 and P(2/3, 2/3) > HG(2/3, 2/3). and a 2.9. Using the procedure described after Lemma 2.7.11 write a computer program to calculate B* (G p.q) (p, q relatively prime). Next, verify Example 2.7.12 and the following list of examples: B*(G89.I44) = 10, 0). (1. 1). (2, 3), (5, 8), (13, 21). (34, 55), (89, 144)), B*(G123.199) = ((1,0), (1, 1), (2, 3), (5, 8), (13, 21), (34,55), (123, 199)), B`(G144.199) = ((1.0), (I, 1), (3, 4), (8, 11), (21, 29), (55, 76), (144, 199)).

Note that

1 < p, q < 100, p, q are relatively prime) = 6, max(#B* (G p,q) : 1 < p, q < 200, p, q are relatively prime) = 7, and #((p.q): I < p,q < 200, #B* (G p,q) = 7) = 3. max(#B* (G p,q) :

2.10. Let M be a y-hyperbolic Riemann surface such that H°°(M) does not separate points; cf. M. Hayashi, M. Nakai, & S. Segawa, Bounded analytic functions on two sheeted discs, Trans. Amer. Math. Soc. 333 (1992), 799-819. Prove via § 2.4 that there exists a y-hyperbolic domain of holomorphy G C C3 which is not c-hyperbolic.

Chapter III

The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

In the previous chapter we discussed the Carathdodory pseudodistance and we observed that if dG: G x G R+ is a function with p(f (z'), f (z")) <_ d0 (z', z"). z', z" E G, f E O(G, E), then CG < dG (cf. Remark 2.1.2). In this chapter we will study the opposite case, namely, a family of pseudodistances (kG)G such that whenever dG is a pseudodistance on G satisfying doff (A'), f ()l")) < p(A', h"), A', Al" E E, f E O(E, G), then dG < ko. Similar investigations will be presented on the level of pseudometrics. The main tool will be the space O(E, G) of "analytic discs" in G "antipodal" to the space O(G. E) which was the basis for our studies in Chapter II.

3.1 The Lempert function and the Kobayashi pseudodistance First we introduce a family (ko)6 of functions k6 from which kG will be derived as the largest pseudodistance below kG. Before presenting the formal definitions, we make the following observation.

Remark 3.1.1. Let G be any domain in C" and fix two points za, z)' in G. Then there exists a curve a : [0, 11 -+ G connecting the points za, Using the Weierstrass approximation theorem we find a polynomial map P : [0, 11 -+ G with P(0) = zd and P(l) = zo'. Then it is easy to choose a simply connected domain D C C, [0, 1] C D, such that P (X) E G for JA E D. By the Riemann mapping theorem we can conclude that z,,, zo lie on an analytic disc 4p: E --o. G with W(O) = z'

andV(a)=zo'(0
G. We put

kc(z', z") := inf(p(A', ),") :

E E : 3cp E O(E, G) : V(A) = z'.

(P(A")

= z") =inf(p(0, A"): A" E E: 3cp E O(E, G): y(0) = z', (p (.V) = z"),

k; :=tanhko,

72

111. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

and we call kG the Lempert function for G.

Remark 3.1.2. Observe that: (a) kG : G x G -* [0, oo) is a symmetric function; (b) (kG)G is a contractible family of functions with respect to holomorphic mappings, i.e. if F E O(G, D), then ko(F(z'), F(z")) _< kG(z', z"), z', z" E G; (c) in particular, we have ko(F(z'), F(z")) = kG(z', z") whenever F: G -+ D is a biholomorphic map; (d) kE = p (use the Schwarz-Pick Lemma); (e) CG < kG.

Let us summarize what we have found so far: if there is a family (dG)G of functions dG : G x G -+ [0, oo) contractible under holomorphic mappings with dE = p, then dG < kG.

But here, in contrast with the situation of the previous chapter, it turns out that, in general, the Lempert function is not a pseudodistance.

Example 3.1.3 (cf. [Lem 1]). For V E N we define domains G, C C2 by

G,, := {(ZI, Z2) E C2: IZjI < 1, j = 1.2, IZIZ21 < v].

Moreover, fix z' := (1/2,0) and z" := (0, 1/2) and observe that these points are contained in all of the we easily obtain

By the holomorphic contractibility of the system (kG)

kG, (z', 0) + kG, (0, z")

2p(0, 2 ) =: A.

If we assumed the triangle inequality kG, (z', z")

kG,. (z', 0) + kG, (0, z"),

then it is possible to find holomorphic maps cp E O(E,

0(a,)=z" with

such that

and

Applying Montel's theorem, we may assume that

a, E [0, 1). E O(E, C2). fP(0) = z'. So(a) = z" with o = lim V-00

3.1 The Lempert function and the Kobayashi pseudodistance

73

Write V = (g, h) and since gyp,. E O(E, G,.) it follows that 1

g(0)=2.

1

and

h(er)=2

which obviously contradicts the identity theorem. Hence for sufficiently large v the triangle inequality does not hold for the Lempert function kG,,.

Example 3.1.4 (cf. [Pol-Sha]). Let G := {z E C2: 1z1Z21 < I and 1:21 < R} with R > 1. Then the following inequality is true: kG((1,0). (0. 1)) > kG((1.0).0) +kG(0, (0, l)). The details are left to the reader (cf. Exercise 3.1).

Remark 3.1.5. Already here we draw the attention of the reader to the fact that in the case of a convex domain G c C" the triangle inequality for the Lempert function kG is always valid. This is the consequence of a very deep result of L. Lempert [Lem 1,2,31 and H. L. Royden & P. M. Wong [Roy-Won] (cf. Chapter VIII), namely, that kG = cc; for such domains. There also exists a direct proof due to L. Lempert [Lem 11; see Exercise 3.10. To overcome the difficulty connected with the triangle inequality we modify the function kG in such a way that the new function becomes a pseudodistance. For z', z" E G we put

kc(:

i ,

z)i := inf{

N

i

kc(zi-l. zj): N E N, 7-U = Z .:1,

, iN-I E G, ZN = <'"},

j=1

kG := tanhkG. The function kG is called the Kobayashi pseudodistatce for G.

Remark 3.1.6. Notice that the following properties hold for the system (kG)G: (a) kG is the largest minorant of kG that satisfies the triangle inequality: (b) kg is a pseudodistance on G; (c) if F E O(G, D), then ko(F(z'), F(z")) _< kG(:', z"), i.e. the system (kG)G is contractible with respect to holomorphic mappings;

(d)kF=kE=p. Even more, we have: (e) if (dG )G is a system of pseudodistances dG : G x G

stated in (c) and (d), then dG < kG;

[0, oo) with the properties

74

M. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

(f) in particular, cG < kG. To be able to continue the discussion on the Lempert function and the Kobayashi

pseudodistance we need at least a few examples for which these objects can be calculated.

Example 3.1.7. Let q be a seminorm on C". Denote by G :_ (z E C": q (z) < 1 } the associated open unit q-ball. For z E G, we claim that the following formulas are true

kG(0, z) = kG(0, z) = p(0, q(z)) To prove these equalities we remind the reader of the similar formula for the Caratht odory pseudodistance. Thus, it suffices to verify that kG(0, z) < p(0, q(z)). In the case q(z) 0 0 defining V(JA) Az/q(z) we obtain an analytic disc cp E

O(E, G) with ip(0) = 0 and cp(q(z)) = z, which implies k(0, z) <_ p(0, q(z)). In the other case q(z) = 0 we consider the family of analytic discs gyp, E O(E, G),

V,(X) :_ Xtz for t > 1 with rp,(0) = 0 and yo,(1/t) = z. Hence we find that

+

kG(0, z) -< P(0. 1/t)1

0.

In particular, we mention the following special cases.

Example 3.1.8. _ (a) kEM (0, z) = kEM (0, Z) = max(p(0, Iz, 1): 1 < j < n), (b) kBM(0, z) = kaM(0, z) = p(0, IIzII).

As a consequence of this example we obtain Proposition 3.1.9. The function kG : G X G -+ [0, oo) is continuous. Proof. In view of the triangle inequality it suffices to prove that kg (zo, -): G -+ R+ is continuous for fixed zo E G. To see this, we take wo E G and W E B(wo, R) C G and we estimate IkG(zo, w) - kG(zo, wo)I < kG(wo, w)

< k8(

R)(wo, w) = p(O,

Ilw

Rwoll)

< Cllw - wolf

if

11w - woI <

R

Here we have used the contractibility of the Kobayashi pseudodistance under the embedding B(wo, R) -+ G and Example 3.1.8.

Our next aim is to present an example which shows that the Lempert function is not continuous in general.

3.1 The Lempert function and the Kobayashi pseudodistance

75

1 } denote a balanced pseudocon vex PropV iUen al1191 k( 9 ;= f j E C': domain with Minkowski function it, i.e. h: C" - [0, oo) is a psh function with h(Az) = IAIh(z) (A E C, Z E O"). Then the Lempert function of G is given by kG(0, z) = p(0, h(z)), z E G.

Observe that in the case where it is a seminorm this formula was already derived in Example 3.1.7. Similarly, we also obtain here that kG(0, ) < p(0, h(.)) without the assumption that G is pseudoconvex.

Proof Assume tp: E --* G to be a holomorphic map with (p(0) = 0 and V(a) = z (0 < a < 1). Then tP can be written in the form tp(A) = with tp E O(E, O" ) and hence we have it o tp(A) = IAIh o cp(A) < 1. By the maximum principle for the subharmonic function it o we obtain it o W < 1, hence h (z) = h o tp (a) _ O ah o 0(or) < or, and therefore kG(0, z) > p(0, h(z)). Remark 3.1.11. Before we continue the discussion of the Lempert function, we

want to point out that a it does not seem to be known how to calculate the Kobayashi pseudodistance kG (0, ) in the situation of Proposition 3.1.10 N. The following example shows that for kG(0, ) there is no formula like in Proposition 3.1.10.

For 0 < e < 1/4 we put G := {z E C2: Izil < 1, Izzl < 1, Izizzl < E} so that ziz2l s}. By Proposition 3.1.10 we see that ifs < t < f,

h(z) = max{Izi 1, Iz2I. then

kG(0, (t, t)) = p

(o._).

On the other hand, the definition easily implies the following inequality:

kG(0, (t, t)) < k. (o, (t, 0)) + kG((t, 0), (t, t))

p(0,t)+p(0, = l 2

r2

l /

E+1Z)

log(1+t I -tE-:=

< 2 log

+ t = kG(0, (t, t))

if t is near e.

Observe that this is yet another example showing that the Lempert function does not satisfy the triangle inequality.

Because of the existence of many bounded balanced pseudoconvex domains whose Minkowski functions are not continuous, Proposition 3.1.10 shows that

76

111. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

the Lempert function is not continuous (in general), not even as a function of one variable. Nevertheless, kc is always upper semicontinuous (cf. Proposition 3.1.13).

To give the reader some feeling of how bad the behavior of kG may be, we present the following example of a balanced domain due to J. Siciak (cf. [Sic 1]).

C" -* Example 3.1.12. We claim that if n > 2, then there exists a psh function 0 but >/i = 0 on a [0, oo) with *(Az) = IAI* (z) (.k E C, z E C") such that dense subset of C". If for a while we assume the existence of such a then h(z) := *(z) + IIzII defines a bounded balanced pseudoconvex domain G = Gh := {z E C": h(z) < 1) whose Lempert function kc(0, ) = p(0, h(.)) behaves very irregularly. What remains is the construction of the function >1,. We write Q2n-2 fl Cn-1 = (rj: j E N) and we define the linear functionals l/ : C" --> C by lj(z) := (z, (1, r;)). Denote by Lj := kerb and set L := J,__1 L/. Then we construct a sequence of psh functions by

I/i I/I(Z)...1;(Z)I \sup(I1 (w) ...I (w)I: Ilwll <_ 1)1

,'(z) .-_ ( Observe that

*i > 0,

* ()z) = I AI *i(Z),

*ll u)_,L. =O'

Moreover, by the maximum principle, there are points z;, Ilz1II = 1, such that

*i(zi) = 1. By the Hartogs Lemma for psh functions (cf. Appendix PSH 19), it turns out that there is a point z*, 11z* 11 < 2, with lim supj.. 4rj(z*) >_ 2/3. So taking an appropriate subsequence (ty/,) C (*j )j with i/. (z*) >_ 1 /2 and defining

*(z) =1

we obtain a psh function on C" with

* IL = 0, *(Z*) ? 2,

and

41 ().Z) = I)IV1 (Z) (A E C, Z E C").

77

3.2 Tautness

Proposition 3.1.13. The Lempert function kG is upper semicontinuous. Proof. Fix two points zo, wo E G. Then for a positive number a we can choose an analytic disc V E O(E, G) with p(0) = zo, qp(a) = wo (0 < or < 1),

and

p(0, a) < kG(zo, wo) + e.

The compact subset V(E) has a positive boundary distance, say q. If we take two points z, w E G with Ilz - toll < qa/6 and Ilw - woll < )?or/2, C" by we are able to define a holomorphic map h : E

(A) + [(z - zo)(a - A) + A(w - wo)] .

I

Simple estimates yield h r= O(E, G) with h(0) = z and h(a) = w, which implies that

kG(z, w) < p(0, a) < kG(zo, wo) +e.

In the remaining case when or = 0, i.e. zo = wo, we choose R > 0, a > I such that

B(zo, R) CC G and p(0, 1/a) < E. Now, if llz - toll < R/4a and 11w - woll < R/4a, we get a holomorphic map * E O(E, G) defined by *(A) := z+Aa(w-z) with *(0) = z, *(1/a) =,w; thus we obtain kG(z, w) < p(0, 1/a) < e. Hence the upper semicontinuity of kG has been established.

0

3.2 Tautness

Although, in general, the Lempert function is not continuous, there is a sufficiently rich family of pseudoconvex domains whose Lempert functions are continuous. Let G be a domain in C". Then G is called taut if the space O(E, G) is normal,

i.e. whenever we start with a sequence (fj)jEN C O(E, G), then there exists a f E O(E, G) or there exists a subsequence (fj, ) subsequence (f j,) with fj, which diverges uniformly on compact sets, i.e. for any two compact sets K C E, L C G there is an index vo such that fj,(K) fl L = 0 if v > vo. The notion of taut domains was introduced by H. Wu (cf. [Wu I]). It can be reformulated in terms similar to the Kobayashi pseudodistance (cf. [Roy]).

78

III. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

For a natural number m we introduce m

k(m)W, z") := inf{ E ko(Zj_I, Zj): Z' = Zo, ZI E G, ... , Zm-I E G, z,, = Z"} j=1 M

= inf{ 1: p(O,aj): aj E E: 3wj E O(E,G) with rpl(0) = z', j=1

'Pj(aj) = Wj+1(0), 1 <- j < m, com(am) = Z"},

Z', Z" E G.

Observe that

kG=k()>k(M)>kG+l)>- l-oo limkg)=kG. With the above notation in mind, we have the following characterization of taut domains.

Proposition 3.2.1. The following statements are equivalent: (i) G is a taut domain;

(ii) for any to E N. R > 0, and zo E G the set {z E G: k(m)(zo, z) < R} is a relatively compact subset of G;

(iii) for any R > 0 and zo E G the set (z E G: kG2)(zo, z) < R} is a relatively compact subset of G.

Remark 3.2.2. (a) In particular, the proposition says that if all kG-balls (with finite radii) are relatively compact subsets of G, then G is a taut domain. (b) The statement converse to (a) is false as we will see in Chapter VII.

Proof of Proposition 3.2.1. (i) -a (ii). Suppose (ii) is not true, i.e. we can find

m E N, R > 0, zo E G, and a sequence (Z,)veN of points z E G with the following properties (m)

kG

(zo, zV) < R

and

[z,V-+- 00z E 8G

or zv --+ oo]. V-00

(*)

By definition, we can choose functions V,. j E O(E, G), I < j < m, and points a,,, j E E, 1 < j < in, that share the following properties (Vv.l(0) = Z0, w.j(OL.j) = cov.j+1 (0),

1 <j <M, m

p(0, a,,, j) < R.

Sov,m (av.m) = zv and j=1

79

3.2 Tautness K

C (V,.,) with cpI,, I ==*VI E Since 9,1(0) = Zo, there is a subsequence ai E E and we arrive at O(E, G) and V, (0) = zo. We may assume that alv.1 0

v-+00

(Pi (0) = Zo

and o (ai) = lim (Pi v. i (si v. i ) = lim 0,AO).

Repeating this argument several times we /end up with subsequences ((p

1) C

(av.l ), ... , (amv.m) C (av.m) with

(Vv.I ), ..., (gmv.m) C ((pv.m) and (amv.l) K

comv.j =* cpj E O(E, G),

amv.j

V-00

v-00

aj E E

and VI

= zo, coi(al') = cj+I (0), 1 -<j < m, com(am) = Vlim00Z"""

which contradicts (*). What remains is to prove that (iii) is sufficient for G to be taut. (iii) - (i). Now, we have to begin with a sequence (fj) j c O(E, G). We assume that this sequence is not compactly uniformly divergent, i.e. there exist compact sets K C E and L C G and a subsequence (ff j) c (f j) with f, j (K) fl L H for all

J.

Take points i; j E K with f3 ( j) =: Z j E L. Because of the compactness we may assume that and

Zj - Z`EL.

Now, for suitable r > 0 and jo E N, we have I, zj E B(z*,

B(z`, r) C G,

r) if j > jo.

Defining gj (A) := z` + 3A(z j - z*) gives a sequence of holomorphic mappings

gj E O(E, G) with gj(0) = z' and gj(1/3) = zj = f1j(0Therefore, for 9 E E with inequalities

kc)(zs,f.j(0))

9) < R (R arbitrary) this leads to the following

`= I4

(gj(0),gj(I))+k(

P(0,

3) + PW, 0 + PW09)


80

III. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

By our assumption, (**) implies that the sequence (f,) is locally uniformly bounded. Thus, Montel's theorem can be applied and we obtain a further subsequence (f2 j) C (f,) with f2;=4f E O(E, C"). Moreover, (**) shows that if 6) < R, then f (6) belongs to the C"-closure of (w E G: k(2)(z`, w) < R} which, by the assumption, is contained in G. Hence f E O(E, G), which completes the proof.

0

Before we continue our investigations of the Kobayashi pseudodistance, we like to make a digression about tautness. We will collect only the most important results on this subject. We first recall the Kontinuitatssatz of Complex Analysis (cf. [Kra 1]). Applying this theorem, it is easy to see that a taut domain in C" is necessarily a pseudoconvex domain. We will see that inside the class of pseudoconvex domains there is a rather large subfamily of taut domains.

Remark 3.2.3. (a) Any bounded hyperconvex domain G C C" (cf. Appendix PSC), i.e. any bounded domain G for which there exists a continuous negative plurisubharmonic exhaustion function w: G (-oo, 0), is a taut domain. (b) Even more is true (cf. [Ker-Ros]). If G is a bounded pseudoconvex domain fulfilling the following boundary regularity:

for any zo E 8G there exist a unit vector u E C", a neighborhood U of zo, and a positive number s such that: (U fl G) + (0, s) v c G, then G is a taut domain. Observe that any bounded domain with C' -boundary satisfies this boundary regularity. Hence any bounded pseudoconvex domain with Ct-boundary is taut. We point out that these domains are also hyperconvex (cf. [Ker-Ros]) so that (a) yields tautness, too. (c) In the case of a balanced pseudoconvex domain G := (z E C": h(z) < 1) with Minkowski function h, there is even a complete characterization of tautness (cf. [Bar 41), namely:

G is taut if h is continuous with h > C11 II for a suitable C > 0. In particular, any bounded complete Reinhardt domain of holomorphy is taut.

(d) A domain G C CI is taut if C' \ G contains at least two points; cf. the proof of Theorem 1.24. We hope that this series of remarks conveyed the idea how rich the family of taut domains is.

Now, we come back to our discussion on the Lempert function. Namely, we want to study this function on taut domains. Our first result is the following one.

81

3.2 Tautness

Proposition 3.2.4. Let z', z" be two points of a taut domain G C C". Then there exist a holomorphic map V E O(E, G) and a number Or E [0, 1) with (p(0) = z'. 9p(a) = z", and kG(z', z") = p(0. a). Proof. By definition we find a sequence ((pj)j C O(E, G) satisfying

cpi(0) = z', 9i (0j) = z' (0 < aj < 1),

and

p(0, ai) \ kG(z', z").

Since G is a taut domain and V (0) = z', we can choose subsequences ((pi j) C ((pj) co E O(E, G) and a, j -> Or E [0, 1). From this we and ( a t ) C (a1) with co j :

conclude that (p(a) = z". cp(0) = z', and p(0, a) = kG(z', z"). Hence, we have proved that there always exist extremal discs through two given points.

Remark 3.2.5. (a) We note that the claim of Proposition 3.2.4 is not longer true if G is not taut. For example, take Go := B2 \ {(1/2, 0)} and z' := (0, 0), z" := (1/4,0). Using the analytic maps (pR(A) := (RA, s(R) A(X - 1/(4R))), R < 1, s(R) << 1, one can easily deduce that kG,,(z', z") <_ p(0, 1/4). Now suppose that there exists V = (01, 92) E O(E, Go), 9(0) = z', 9(a) = z" such that kG (z', z") = p(0, a). Thus a < 1/4. On the other hand, the Schwarz Lemma implies that 1 /4 < or. Hence VI(114) = 1/4, i.e. ppl (A) - X. Since (P E O(E, p2), it turns out that cp2 - 0. In particular, 9p(1 /2) _ (1 /2, 0) which contradicts the definition of G.

(b) Let qp E O(E, G) be an analytic disc through the points z', z" E G such that cp(0) = z', 9(a) = z" (0 < or < 1), and kG(z'.z") = p(0,a). Then for all 0 < a' < a" < or we also have

kG(9(a'), 9(a")) = P(a', a"). For the proof observe that kG(z', z") < kG(co(0), V(a')) + kG(w(a') V(a")) +

92(a))

< P(0, a') + p(a, a") + p(a", a) = p(0, a).

Proposition 3.2.6. If G is a taut domain in C", then the Lempert function k(; is continuous on G x G. Proof. Since we already know that kG is upper semicontinuous, suppose that kG is not lower semicontinuous at (Zo, wo) E G x G, say. This means that kG(zo, wo) >

0 and, moreover, there are sequences

(w,,),, C G with

z = zo,

III. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

82

IimV w = wo, and kG(zV,

kG(zo, WO) - E E (0, oo) for a suitable e > 0. Proceeding as usual, we find holomorphic mappings gyp, E O(E, G) with

(p,(0) = zV, V,,(a,,) = wV (0 < a, < 1),

and

p(0, aV) < kG(zo, wo)

- 2*

On the other hand, since G is taut, there are subsequences (vi,) c (v,,) and K 3 E O(E, G), a,V V-00 -p ao E [0, 1), and therefore (a,,,) C (a,) with (piV= V-00 (p(0) = zo, V(ao) = wo. Now, we can conclude that e

kG (ZO, wo) < P(0, ao) =V lim00 P(0, U10 < kG(zo, wo)

- 2,

and this contradiction finishes the proof.

3.3 General properties of k We already know that the II-topology of a domain G C C" is stronger than the kG-topology on G. We remember that in the case of the Carathdodory distance the II II-topology can be different from the c(;-topology. To discuss the analogous question for the Kobayashi distance we need the following observation; see [Kob 51. II

Proposition 3.3.1. The Kobayashi pseudodistance is inner, i.e. if z', z" E G, then kG (z', z") = inf(LkG (a) : a : [0, 1) -* G is continuous and II 11-rectifiable with a(0) = z', a(1) = z"),

where N

N E N, 0 = to < t] < ... < IN = 1)

Lk,;(a) i=1

denotes the kG-length of a.

Proof. First observe that LkG(a) > kG(z', z") for any such a competing curve a. To verify the opposite inequality fix E > 0 and choose points at, ..., ak E [0, 1)

3.3 General properties of k

83

and maps pl,..., Vj. E O(E, G) with vi (0) = z', co1(6i) _ pj 1(0), 1 < I < k. gok (ak) = z". k

and E p(0, aj) < ko (z', z") + s. i= I

Of course, we may assume ai > 0. Then we are able to define a piecewise C' -curve in G connecting z' and z" by

a(t):=Vj((t-7k 1 f kaj)

if tE

Ijk l,kJ

Therefore, we obtain k

Li, (a) :

L.

Ltt ([0, Qi l) _

P(0, 6i) < kG (z', z") + E,

which concludes the proof.

Remark 3.3.2. (a) Recall that, in general. the formula analogous to that of Proposition 3.3.1 fails to hold for the Caratheodory pseudodistance. (b) Later (see §3.6) we will have another method to prove Proposition 3.3.1 (cf. Exercise 3.7). It is obvious that a necessary condition for the II II-topology and the kG-topology

to be equal is that kG is a distance. We say that a domain G c C" is k-hyperbolic if its Kobayashi pseudodistance is a distance.

Remark 3.3.3. By the well-known inequality CG < kG it is clear that any c-hyperbolic domain is also k-hyperbolic. Therefore, any bounded domain G is k-hyperbolic.

So far we have shown that the Kobayashi pseudodistance is inner and continuous. These conditions suffice to prove the following comparison property of the topologies (cf. [Bar 11).

Proposition 3.3.4. If G is a k-hyperbolic domain in C", then its equal to the kG-topology.

Pmof. Exercise; cf. Remark 2.5.5.

II

II -topology is

84

III. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

For the rest of this section we turn to the question of how the Kobayashi pseudodistance behaves under certain set-theoretic operations.

Proposition 3.3.5. (a) Let G be a domain in C", G = U' ' G, where

is an

increasing sequence of subdotnains. Then for z', z" E G we have kG(Z

, ,

,, Z) = lim V-00

,

, z) and kG(Z , , Z) = lim kG, W, Z ). V-00

(b) If a domain G C C" is taut and has a C'-boundary and if (Gj) j is a sequence of domains G j C C" with G j DD Gj+1 for all j E N. G = njo°_i Gj, then for every pair of points z', z" E G the following is true lim kG, (z', z") = kc(z', z"). j-.oc

Remark 3.3.6. (a) The assumptions in (b) can be reformulated by saying that G is a bounded pseudoconvex domain with C1 -boundary. (b) Exercises 3.3 will give examples showing that the assumptions in Proposition 3.3.5(b) cannot be considerably weakened. 01f we replace the Lempert function by the Kobayashi pseudodistance we do not know whether Proposition 3.3.5(b) remains true.

0

Proof of Proposition 3.3.5. (a) For given points z', z" E G there is a sufficiently large index vo such that z', z" E G if v > vo. For these v the following inequalities are obvious kG (Z', Z"),

kG, (z', z") ? kG, W-

z)

kG,.+, W, z) ? kG(z , Z") -

kG,(z', z") > A > kG(z', z"), we are able to select If we now assume that a function V E O(E, G) with

(p(0) = z', gp(a) = z" (0 < or < 1),

and

p(0, a) < A.

Then cp(E) c G if v >> vo, and so kG,,(z', z") <_ p(0, a) <

z"),

which leads to the expected contradiction.

Recall that kG is the largest pseudodistance below kG; hence kG(z', z") _ kG (z', z"). (b) Contrary to the situation just discussed, the sequence (kG, (z', Z")) j is in" < kc ( Z, , z "). creasing with lim j-ackc; (z, , z)

3.3 General properties of k

85

Now let us suppose that lim j.,0 kG1 (z', z") < A < kG (z', z"). By definition, we are able to select holomorphic functions W j E O(E, G j) with

Wj (0) = z'. (pi (aj) = z" (0 < aj < 1),

and p(0,aj) < A.

In view of Montel's theorem we may assume that cpj

l

1P E O(E,C"),

aj -* a E [0, 1),

'(0) = z',

and rp(a) = z".

Since the sequence (G1) j is strictly decreasing, it follows that 'p(E) c f l G j = 0. But, on the other hand, the assumptions of tautness and C'-boundary force the analytic disc (p(E) to be inside G. Thus 'p can be considered as a competitor in the definition of kG, which leads to kc, (z', z") <_ p(0, a) < A. a contradiction. In calculations of the Kobayashi pseudodistance holomorphic coverings often play an important role because of the following result of S. Kobayashi (cf. [Kob 4]).

Theorem 3.3.7. Let 7r: G G be a holomorphic covering. Then for x, y E G and i E G, n (x) = x, the Lempert function and the Kobayashi pseudodistance for G satisfy the following formulas:

y) = inf{kd(x, e): v E G. ir(v) = y},

(3.3.1)

kG(x,y)=inf{ke, {X,y):yEG.n(y)=y).

(3.3.2)

k(;

Proof By the contractibility of (kG)G we only have to verify the inequality ">".

Thus, for a given e > 0 we take a holomorphic map 'p E O(E, G) with the following properties:

'p(0) = x, 'p(a) = y (0
and

p(0, a) < &G(x, y) + E.

Remembering the properties of holomorphic coverings, we lift Sp to ip E O(E, G)

with 0(0) = i and r o0 ='p, in particular, n(O(a)) = y. Therefore, we conclude that

p(0,a) > kd(i,sp(a)) > inf(kd(i, y): y E G with;r(y) = y), which proves (3.3.1). The formula (3.3.2) may be proved similarly.

Remark 3.3.8. (a) From the previous theorem we conclude that if all ko-balls with finite radii are relatively compact subsets of G. then for x, y r= G, and x E G,

86

M. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

ir(i) = x, there exists a point y E G with 7r (y) = y and kG(x, y) = kd(x, y) (the same statement is also true for the Lempert functions provided that G is taut). (b) [] It seems to be unknown whether this sharp form of Theorem 3.3.7 remains true in general (see [Kob 41, p. 48). (c) In Chapter V, which is devoted to the annulus, we will see how Theorem 3.3.7 is applied in concrete situations. (d) Observe that if P = (A E C: 1/2 < JAl < 2} denotes an annulus, then there is the universal covering tr : E -> P, and therefore we can find two points A', A" E E with r (A') = 1, tr (A") _ -1, and kp(1, -1) = p(A', A"). But it is clear that tr is

0

not a k-isometry. (e) Since any domain G C C has E or C as its universal covering, Theorem 3.3.7 also implies that kG satisfies the triangle inequality; hence kG = kG. (f) We also mention that for G := C-{0, 1 } we have cG - 0 whereas kG(z', z") > 0 if z' z". The latter fact follows because of Theorem 3.3.7 and the well-known result that E is the universal covering of G. So far we know only few examples of domains G for which cG # kG (recall, for example, the domain constructed in Theorem 2.4.3). For plane domains we have the following complete characterization of such domains.

Proposition 3.3.9. (a) Let G be a c-hyperbolic domain in C and let us suppose that there is at least one pair of different points z'. Z" E G with kG(z', z") = CG(z', z"). cG. Then G is biholomorphically equivalent to E and so kG (b) If a plane domain G is not c-hyperbolic, then cG - 0 and either kG - 0 or G is k-hyperbolic.

Proof. In the case when G is c-hyperbolic there is the holomorphic covering n : E --> G. According to the above remark we choose points A', A" in the unit disc such that 7r (A') = z', n (A") = z", and kG(Z', z") = kE(A', A") = P(A'

On the other hand, CG W, z") can be written as CG W, z") = p(f (z'), f (z")) for a suitable f E O(G, E). For the function f o 7r E O(E, E) this implies P(f o 7r (k), f o 7r 01")) = CG W, z") = kG(z', z") = P(A', A").

Now, the Schwarz-Pick Lemma tells us that f o r is a biholomorphic map, and therefore n is biholomorphic. We turn to the proof of claim (b). Since G is not c-hyperbolic, we have cG =_ 0 (cf. Proposition 2.3.1). In the case where the universal covering of G is given by

C, it is clear that kG = 0. So we may assume that 7r: E -> G is the universal

3.4 An extension theorem

87

covering. Hence by 3.3.7 and Remark 3.3.8 we conclude that whenever z', z" E G, z' # z", then there are points X', A" E E, rr (A') = z', n (A") = z" with

kc(z', z") = p(x', A") > 0.

Corollary 3.3.10. Let P :_ (k: 1/R < IAI < R) (R > I). Then for z', z" E P, z' 0 z", we have

cp(Z',z")
3.4 An extension theorem The discussion in the previous chapter has shown that the Carathdodory pseudodis-

tance is preserved if a "small" set is removed from the domain considered. We will see that in the case of the Kobayashi pseudodistance a similar result is true. Nevertheless, the reader should note the following fact. Example 3.4.1. kE. # kE [E.XE. (E. := E \ 10)). To prove this non-identity we begin with two different points A'. A" in E.. It is clear that the equality would lead to the following equation CE. (A', V) = cF(A', A") = kE (A', A") = kE. W. ll"),

which would imply that E. is simply connected (use Proposition 3.3.9(a)).

Theorem 3.4.2. Let G be a domain in C" (n > 2) containing a relatively closed subset A with H'-"-2(A) = 0, where H'-" denotes the (2n - 2)-dimensional Hausdorff measure. Then the following statements hold.

(a) Any g E O(E. G) with g(0) ll A can be uniformly approximated on E by a sequence of analytic discs (g,.),,, g E O(E, G \ A); (b) kGI(G\A)x(G\A) = kc\A.

A weaker version of this theorem can be found in ICam-Oga); the formulation above is taken from [Pol-Shal. Proof. Taking (a) for granted we are going to deduce (b). First observe that for arbitrary z', z" E G \ A we have _

ka(z', z") =

zj): N E N, Zo = . . ' , Z 1 ,- . ,:.N-1 E G\A, ZN = Z"). j=1

III. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

88

This is a simple consequence of the upper semicontinuity of kG (cf. Proposition 3.1.13). (Note that the formula remains true for arbitrary nowhere dense subsets A C G.) In particular, if kG\A < kG on (G \ A) x (G \ A), then (b) is true.

Fix z', z" E G \ A and let cp E O(E, G) be such that z' = V(0), z" = cp(a) for some 0 < a < 1. By (a) there exists C O(E, G \ A) with cp -> cp uniformly on E. Hence Vv (a)) < p(0, a), v E N. Consequently, by the continuity of kG\A (cf. Proposition 3.1.9) it follows that kG\A(z', z") _< p(0, a). Since V is arbitrary, we get kG\A (z', z") < kG (z', z"). Hence (b) is verified under the assumption that (a) holds.

Before we proceed, we recall some results from geometric measure theory (cf. [Fed]).

1) Let D denote a disc in C and let F be a subset of D x C"-' with (2n - 2)Hausdorff measure zero. Then for any e > 0 there exists a vector v E C"-', 0 < II v II < E, such that the section D x { v) does not intersect F.

2) If F : G, -> G2 is a diffeomorphism, where Gj are open sets in R", then any subset M C G, of k-Hausdorff measure zero is mapped onto the set F(M) of k-Hausdorff measure zero. Now, we go to the proof of claim (a). We start with a holomorphic map g : U -+ G, where U denotes an open neighborhood of k with g(0) A. First we approximate g uniformly by regular holomorphic maps gc : U -+ C" (t > 0), where all these functions share the following properties:

(i) if 0 < t < to, then g,(U') C G for a suitable open neighborhood U' C U of E; (ii) 11 g, - gllu' t and g,(0) ¢ A if 0 < t < to;

(iii) g'(),) 00 if 0 < t
U X C"-' a (A, A2, ...,An)

9(A)-EAiv1 i=2

g(A,A2...... ,,).

3.4 An extension theorem

89

Observe that the Jacobian matrix of g at (Io, 0, ... , 0) is non-singular. Hence we find a small disc D around Ao and a positive number e(An) such that g maps biholomorphically D x (A E C"'' : IIXII < e(Ao)) into G. By a compactness argument, we cover the annulus {A E C: r < Ixl < 11 by a finite number of discs DI,.... Dk, such that for some vectors {v2.), ... , v;f) } and a suitable e > 0 the mappings

g;: D;x{AEC"':ll.lt<e}--' G, defined by

gf(JA, A2..... A,:) = 9(1) -

E

X" v(1,i)1

v=2

are biholomorphic into G. Since it is biholomorphic, we conclude that H2" ..({(A,,1) E D, X C"_' : 11 A11 < e, 91(A. A) E A}) = 0

Hence, according to our initial remarks, we find a small vector P) E C l t fulfilling the following conditions

g1(A, P)) ¢ A if I E rE U D,, and Ilg -

8i(. £('))IIE

is arbitrarily small.

Now, we proceed with the second disc. For D2 we only mention that

82(x, A2....., l") v=2

plays the role of it above. such that the image of the set rE U D, U D2 by

So we obtain small the function

X(2)) _ -k2

"

),(I)UM "

Is

L,

v=2

does not intersect A. After a finite number of steps we reach the desired function g E O(E, G \ A) which is near to g. Thus Theorem 3.4.2 is completely proven.

90

M. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

Corollary 3A.3. If A denotes an analytic subset of the domain G of codimension at least two, then the following formula is true kG\A = kG I (G\A) x (G\A)

Proof. Standard argument from measure theory leads to the conclusion that A has (2n - 3)-Hausdorff measure zero. Hence the corollary is a particular case of Theorem 3.4.2.

O

3.5 The Kobayashi-Royden pseudometric In Chapter II we have already learned that for the Carathdodory pseudodistance there is an infinitesimal version, the Caratheodory-Reiffen pseudometric, which measures the lengths of tangent vectors. A similar notion with respect to the Kobayashi pseudodistance will be introduced and investigated in this section.

Let G be a domain in C. The function xG : G x C' -* [0, oo) defined by xG(z; X) := inf(y(A)IaI : 3Sp E O(E, G), 3A E E: (p(A) = z, mp'(A) = X } is called the Kobayashi-Royden pseudometric for G. Observe that (a) xG(z; X) = inf{a > 0: 3(p E O(E, G): p(0) = z, acp'(0) = X}

=inf{a>0: 3rpEO(E,G): (p(0) =z, &V'(0) X); (b)XG(Z;AX)=IAIxG(Z;X), AEC,XEC",ZEGCC"; (c) xp(F(z); F'(z) X) < xG(z; X), F E O(G, D), z e G C C", X E C". Hence XG(z; ) assigns a length to any tangent vector at z, and moreover (c) shows that the system (xG)G is contractible with respect to holomorphic mappings. D is a biholomorphic map, then In particular, if F: G

(d) xn(F(z); F'(z)X) = xG(z;X), z E G, X E C"; (e) ,E(A;X) < y(A)IXI, A E E, X E C. Applying the Schwarz-Pick Lemma we obtain the following comparison result.

Lemma 3.5.1. For any domain G C C" we have yG < xG.

Proof. For a point Z E G C C" and a vector X E C" we choose a function f r= O(G, E) with f (z) = 0 and If'(z)XI = yG(z; X). Then for every E O(E, G)

3.5 The Kobayashi-Royden pseudometric

91

with (p(0) = z the composition f o qP E O(E, E) has the origin as a fixed point. Thus I(f o (p)'(0)I < I. In the case pp(0) = X (a > 0) we obtain

a ? I(f o w)'(0)I a = If'(z)XI = YG(z; X) -

0

Since (p is arbitrary, the claim follows.

Corollary 3.5.2. xE(A; X) = y(A)IXI, A E E. X E C.

Moreover, it turns out that whenever there is a system (SG )G of functions SG : G x C" -> (0, oo) (G C C") with the properties (b), (c), and SE = yE, then SG < xG for any G. We recall that we already know that also yG < SG is true. To see at least a few concrete examples, we calculate the Kobayashi-Royden pseudometric for balanced pseudoconvex domains.

Proposition 3.53. Let G be a balanced pseudoconvex domain in C" given by G := {z E C": h(z) < 1}, where h is its Minkowski function. Then

xG(0;X) = h(X),

X E C".

Proof First of all observe that if h(X) 0 0, then E A -`°. ,1X/h(X) is an analytic disc in G with (p (0) = 0, qp'(0) = X/h(X); hence we obtain xG(0;X) < h(X). The fact that the same inequality is also true if h(X) = 0 is left to the reader (cf. Example 3.1.7). On the other hand, let w E O(E. G) with V (O) = 0, a(p'(0) _ X (a > 0). As in Proposition 3.1.10 we observe that h(p(A)) = h(,kO(A)) < 1

and therefore h o 0 < 1. Thus we end up with h(X) = missing inequality.

ah o0(0) < a, which guarantees the

0

We emphasize that the proof above is based on the information that G is pseudoconvex, i.e. that the Minkowski function is plurisubharmonic. Indeed, the following example, partially due to N. Sibony (cf. [For-Ste]), shows that the above formula is false if G is not pseudoconvex.

Example 3.5.4. Let GE, 0 < s < 1, be the following complete Reinhardt domain: GE

{z = (zi, z2) E C2: Izi l < 1. IZ21 < e

or

Izi I < e. Iz2l < 11.

III. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

92

Note that GE is not pseudoconvex and its envelope of holomorphy H (GE) is given by

H(GE) :_ {Z E C2: IzI I < 1, IZ2I < 1, IZIZ21 < e). By hE (resp. hE) we denote the Minkowski function of GE (resp. H(GE)). Then the x-indicatrix of GE at 0,

I (xG,;0) := {Z E C2: xG,(O;Z) < 1}, is a balanced domain in C2 with GE C 1(XG,;O) C I (xHiG,i;O) = H(GE) 9 conv GE = conv H(GE).

First we show that 1(xG, ; 0) c H (GE) which implies that I (xG, ; 0) is not pseudoconvex, and therefore that xG, (0; ) is not plurisubharmonic. Fix to E (s, 1) and put Xo := (to, a/to). Now suppose that xG, (0; Xo) = 1. This implies that there exists a holomorphic map (P E O(E, GE) satisfying ip(0) = 0 and ip'(0) = Xo = (to, a/to). Observe that IgPIVp21 <_ e, where iP = (VI, ). Since V, (0)(p2' (0) = e, the maximum principle applied to QI92/A2 implies that (VI V2)(1) = eA2, A E E. Therefore, the non-tangential boundary values Spy of cps, j = 1, 2, satisfy Icpi Sp2 I = e almost

everywhere on a E. This can be read as V*(eie) E {Z E C2: IZI I = 1, IZ21 = e

or

IzI I = e, IZ21 = 1}

whenever co*(eie) = (qpi (eie), p (eie)) exists.

Now suppose that there are two different points ei0', eiez E 8E at which (p" exists and such that IV*,(eie')I = 1, livi(ere')I = e

and

I(vi (eie')I = e, Irp2(eie=)I = 1.

Choosing curves a in E U {ere' , eie= } with a (0) = e't', a,, (1) = erej, and supo<,
-

)

V --i 00

0 leads to curves q o a in GE connecting co (eie' )

with ip'(eret). Thus there exist points A E

1)),

1 with Iw1(Av)I =

192 (),,) I < E. So we have

s = lim e1Av12 = lim I(pi p2)(A1,)I < s2; p-00 a contradiction.

So we have that IVI = 1, IV2I = s a.e. (or conversely). Then the maximum principle implies that Icp2I < e (resp. I(p11 < e) and so the Schwarz Lemma gives

3.5 The Kobayashi-Royden pseudometric

93

I(pi(0)I < s (resp. I(p(0)I < s) contradicting (p2' (0) = s/to > s (resp. (pi (0) = to >

s). Hence xG,((0; (t, s/t)) > h, (t, s/t) whenever e < t < 1. For the final discussion of this example we will restrict ourselves to the case 0 < s < 2/(1 +e 2 ) : Let fi E [s, 1) with s < 2/(e2plE + 1). Under these assumptions we find positive numbers a and A such that

I < Act < min{as/2, a/(e°`p + 1)}. Then it is easy to see that the holomorphic map (p: E --> C2 defined by

c(X) := A(eforz -

l,e-0aa

- 1)

has its image inside GE. Since qp(O) = 0 and go'(0) = Aa(s, -f ), it follows that xG, A (s, 0))

Aa

<1

hE(s, $),

i.e., in general, the conclusion of Proposition 3.5.3 becomes false if the balanced domain is not pseudoconvex. We want to emphasize that ' we do not know the formula for xG, (0; )

0

Example 3.5.4 shows that, in general, the x-indicatrix is not a pseudoconvex domain. This phenomenon also occurs for very regular domains as the following example due to N. Sibony illustrates. Example 3.5.5 (cf. [For-Ste]). For e > 0 we set GE := {z = (Zt, Z2) E C2: e(Izi12 + IZ212) + Iz1 - zz12 < 1}.

Observe that GE is a strongly pseudoconvex domain with smooth C°'°-boundary; Appendix PSC. Moreover, GE is contractible. For example, take z -- (t3zi, 12Z2),

0
Since the map z --> (e31zi, elf z2) is a biholomorphic automorphism of GE, the (e3itz,, e2"z2) and z -> (e"zi, a" z2), t E R, and therefore I (xG,;0) is a Reinhardt domain containing the origin. Now, we claim that the point (2/3,4/3) E I (XG,; 0) for small E. To verify this we study the following holomorphic map op : E -* C2,

x-indicatrix of GE at 0 is invariant under the rotations z -

v(A) := (2A/3 + 2A2 +;13, 4.1/3 + A2),

94

111. The Kobayashi pseudodistance and the Kobayashi-Royden pseudoinetric

with cp(0) = 0 and ip'(0) = (2/3, 4/3). Because of rp1(A)22 - q(A)3 = (4/9)A2 + (8/27)A3, we obtain e(Io1(A)122+I

(A)I2)+Irp1(),)2-(p2().)312

< e(Iipl(),)I2+I

(k)i2)+(20/27)2 < 1

ifs is sufficiently small, i.e. V(E) C G. Therefore, xG, (0; (2/3, 4/3)) < 1. On the other hand, we will prove that the point (0, 4/3) does not belong to 1(xc,,; 0). Then this implies that I (.con; 0) is not a pseudoconvex (Reinhardt) domain, and therefore xG, (0; ) is not a psh function if s << 1. Now let us assume that (0, 4/3) a 1(xG,;O). Then there exist points (A;,, A;;) E I(xG,;O) with (A', A') V (0,4/3) and analytic discs gyp,, E O(E, Gf) with

0, a, (p'(0) = (A;,, A''), where 0 < a < I. Taking a suitable subsequence we obtain a map (P E O(E, Gfi) with V(0) = 0, o p'(O) = (0, 4/3), where 0 < a < 1. Therefore, (p can be represented as (p(A) = (A2h(A), 4Ag(A)/(3a)),

g, h E O(E) with g(0) = 1.

The maximum principle leads to IAh2(A) - (4/3a)3g3(A}I2 < 1, A E E. In particular. for A = 0 it follows that (4/3x)6 < 1; a contradiction. To summarize: there exist simply connected, strongly pseudoconvex domains G C C2, 0 E G, such that KG(0; ) is not a psh function.

Now, we come back to Proposition 3.5.3 and apply it to the cases of the unit ball and the unit polydisc. Example 3.5.6. (a) For z e B" and X e C", we claim that the following formula is true. 2

:m. (z;X) _

1/2

X)12

[I 11JIIZ 112 + (II _ 11z112)2 I

.

(3.5.3)

For the proof we recall that >% is invariant under A similar argument leads to the corresponding formula for the unit polycylinder.

xE»(z;X)=max{1IXz1I12,...,

1

I1

(3.5.4) ,,12L

From these examples we deduce the following consequences; cf. Corollary 2.2.2.

Corollary 3.5.7. (a) Let G1 = Ghj C" be pseudoconvex balanced domains with Minkowski functions h,, j = 1, 2. Then the following conditions are equivalent: (i) there exists a biholomorphic mapping F: G1 -+ G2 with F(0) = 0;

3.5 The Kobayashi-Royden pseudometric

95

(ii) there exists a C-linear isomorphism L : C" -+ C" such that h, o L = h,, i.e. G, and G2 are linearly equivalent. (b) (cf. [Pat]) Let G = G,, be a bounded balanced domain in C". Then the following conditions are equivalent: (i) there exists a biholomorphic mapping F: G,, - Is" with F(O) = 0: (ii) G,, and 18,, are linearly equivalent; (iii) h2 E C-(C").

Corollary 35.8. (a) For any compact subset K of any domain G C C" there is a suitable constant C > 0 such that the following inequality holds on K x C": XG(Z; X) 5 CIIXII,

Z E K, X E C".

(b) If, in addition, G is bounded, then there exists a constant C > 0 such that for

anyzEG,XEC"wehave xG(z;X)>C11XII Proof. Use (3.5.3) and the contractibility of the system (x(;)G.

O

Observe that the second part of Corollary 3.5.8 can be read as follows: XG is positive definite if G is bounded. In general, it turns out that xG(z;) need not be a norm on C". Example 3.5.9. Put G := {z E C'-: Izil < 1, 1221 < 1, 1Z1 Z21 < 1/2). Of course G is a bounded balanced pseudoconvex domain with Minkowski function h(z) = max(Iz,I, Iz21.

21ZIIIz2I)

Therefore we know that xG (0; X) = h (X). In particular, we have

XG10;12,2J I=

>2=xG(0;(1,21+XG(o. i)).

Moreover, Proposition 3.5.3 shows that, in general, >e(; (z-, ) is not continuous as a function of the second variable. For more details the reader should compare the analogous situation for the Lempert function in §3.1. The next example shows that the Kobayashi-Royden pseudometric is, in general, not continuous even as a function of the first variable. Example 3.5.10. (cf. [Die-Sib]). We are going to construct a pseudoconvex domain G C C2 satisfying the following two properties: (a) there is a dense subset M C C such that (M x C) U (C x (0)) C G; in particular,

xG(;.;(0,1))=0 forallzEA:=MxCandkG=_0;

96

III. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

(b) there exists a point z° E G \ A such that xG(z°; ) is positive definite, i.e. xG (z°; X) > C II X II (for some C > 0).

Construction of C. (i) As a first step we claim the existence of a subharmonic function u : C -> 1R_

with u $ -oo but such that the set M := (), E C: u(JL) = -oo} is dense in C. We construct u as an infinite sum. So, we start by choosing a dense sequence C B(0, k) \ (0). Then we put 00

IA - ajk) I

k)

Uk(A)

x1 109 2(j + k) i=1

where the numbers A

jki

> 0 are chosen in such a way that 00

E

Ia(k)I

A jk i log

'

E R.

2(j + k)

i=1

Observe that locally uniformly almost all of the summands become negative which shows that Uk is a subharmonic function on C; cf. Appendix PSH 12. Moreover, by definition we have

uk < 0 on B(0, k),

uk(ak)) _ -oo, J

and

uk(0) > -00-

Now, we repeat the same procedure. We choose new positive numbers Ak such that Ek°_I Akuk(0) is a real number. Then we obtain our function it by the formula 00

u(A) := E kkUk(A) k=1

The same argument as before leads to the conclusion that u is subharmonic on C, j E N}. u(0) -oo but u = -oo on the dense set M := (ii) We use the function u just constructed to define a new plurisubharmonic function * : C2 -+ R-,,, by *(z) := IZ21eXP(Iz112 + Iz212) exp(u(z1)).

3.5 The Kobayashi-Royden pseudometric

Then our domain C Is clklnaned

97

AS

G := {z E C2: iIi(z) < 1}.

We observe that the set A := (z E C2: zI E M} is a dense subset of G. Now, we are in a position to verify our claims (a) and (b). By construction it is clear that whenever z E A, then (zI } x C C G which implies that

xc(z;(0, I)) = XG(4(0);''(0)1) < X,',-(O; 1) = 0, where (D(A) := (zI, Z2 +A). Moreover, using the triangle inequality, it is clear that kG

0.

To prove (b) we select a point z° E C with u(z°) > -oo and t > 0 such that

°:=(z°. t)EG. Now, we observe that whenever (P E O(E, G) is an analytic disc in G with (p(0) = z°, then IIIP'(0)II < C for a suitable positive C which can be chosen independently of V. Let V be as above. Then the Cauchy integral leads to the following representation

of rpj(0) (W = (API,)): 0

1

27ri

J = I/2 2

Hence 2

n f"

II(P(2e'")I12d9.

Here we have used the Holder inequality. To continue the estimate we introduce the plurisubharmonic function

log *(z) - -IIzII'', for which the following inequality holds on G: i(z) < -112112/2. Then the mean value inequality for subharmonic functions shows that

-oo < I/r(z°)

o(P(0)

I

21r

c`p(2e'°)dO

I

0

2n

S - 4zr j0

II

(-,e"p)IIZd9.

98

III. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

Finally, combining the information obtained, we find that IIV'(0)1I2 <- -8V(z°). Hence we have also proved part (b) of our claim.

Remark 35.11. We emphasize that the above example can be modified to obtain a pseudoconvex domain G C C2 which is not k-hyperbolic but such that xG (z; ) is positive definite for every z E G. In fact, define

u(A) := 00E

1

j=2

max(log

1-k

21/11

I E E,

and

G :_ (z E E x C: *(z) := IZ21 (exp 11112) . expu(zi) < 11. Note that u is subharmonic on E, u (X) -00,1 E E. Hence G is a pseudoconvex domain containing larger and larger discs

((1/k, z2): '(1/k, z2) < 1}. Thus, the continuity of the Kobayashi pseudodistance implies kG((O, 0), (0, z2)) 0, whenever (0, z2) E G, i.e. G is not k-hyperbolic. On the other hand, following the argument of (b) we easily derive IIXII

xG Z;X > ( ) - -8(log>r(z) --2IIzII2)'

z E G.

To summarize, the last two examples show that, in general, the KobayashiRoyden pseudometric is continuous neither in the first nor in the second variable. Nevertheless, we will see that this function is always upper semicontinuous even in both variables simultaneously. Proposition 3.5.12. For any domain G C C" the Kobayashi-Royden pseudometric xG : G x C" -p [-oo, oo) is upper semicontinuous.

Proof. We start with a fixed point z° E G and a tangent vector X° E C" and we suppose that XG(z°; X°) < A. By definition we find an analytic disc (P E O(E, G) with cp(0) = z° and a a positive number, a < A. Without loss of generality we may assume that --p is holomorphic on E. Thus an e-neighborhood of V(t) remains inside G.

3.6 The Kobayashi-Buseman pseudometric

99

Now, we take Z E G with 11z - z° II < e/4 and X E C" with (Ilot) IIX - X°II < s/4. The following mapping 0: E C",

*W := (PO) + (z - z°) + (1l/a)(X - X°), has its image in G and it satisfies

00) = z, a*'(0) = acp'(0) + X - X° = X, which means that xG (z; X) < a < A, i.e. the upper semicontinuity of xG at (z°; X°) is verified.

Similarly as in the case of the Lempert function (cf. 3.2), we also obtain better results for the Kobayashi-Royden pseudometric on taut domains. Since the argument here is more or less the same, we only formulate the result. The proof is left to the reader as an exercise.

Proposition 3.5.13. Let G be a taut domain in C". Then

(a) for any z E G and for any X E C" there exists an extrenual analytic disc p E O(E, G), i.e. V(O) = Z and xG(z; X)rp'(0) = X. (b) the Kobayashi-Royden pseudometric is continuous on G x V.

Remark 3.5.14. K. T. Hahn [Hah 3] has introduced a family (hG)G of pseudometrics ho: G x C" -> [0. oc) (G c C") which is contractible only with respect to injective holomorphic mappings. This definition is as follows:

hG(Z; X) := inf(a ? 0: 3rp E O(E, G), (p is injective , cp(0) = z. acp'(0) = X). We only mention that the interesting properties of this pseudometric are based on the fact that hG is larger or equal than xG, but it may also be different from xG. For example, hc. is in fact a norm whereas xc. - 0. Along the same idea, K. T. Hahn also introduced a family of pseudodistances. For more information the reader is asked to consult [Hah 3]. [Jin], [Ves 3].

3.6 The Kobayashi-Buseman pseudometric We recall that, in general, the Kobayashi-Royden pseudometric xG (z°i ) does not give a seminorm on C" if G c C". Thus, for example, it may happen that the zero set of XG(z°: ) is not a linear subspace of the tangent space C". To overcome this

100

111. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

unpleasant fact S. Kobayashi [Kob 7] has recently introduced a new infinitesimal pseudometric following old ideas of Buseman (cf. [Bus-May]). Let G be a domain in C". Then the function XG : G x C" -). [0, oo) given by

is called the Kobayashi-Buseman pseudometric for G, where

rG(z) := (Y E c" : IY

ZI 51 for all Z EC" with xG(z; Z) < 11

denotes the polar of the unit Kobayashi-Royden ball.

Remark 3.6.1. At first we collect simple properties of this new function kG: (a) kG(z; ) <- xG(z; ), z E G; (b) kG(z; ) is a complex seminorm on C"; (c) XG(z; ) is continuous on C"; (d) IX E C": kG (z; X) < 1) is nothing else as the bipolar set of the unit xG (z; )-

ball 0 := IX E C": xG (z: X) < 1) and by a standard result from functional analysis it is the closed absolutely convex hull of 0; (e) with (d) in mind it is an easy exercise to see that { X E C" : XG (z; X) < 1) coincides with the convex hull of 0; (f) the system (CG)G of the Kobayashi-Buseman pseudometrics is contractible with respect to holomorphic mappings and ikE = XE.

We also have to mention that, in general, kG is not a continuous function of both variables as Example 3.5.10 can easily show. Nevertheless, the analogous result to Theorem 3.5.12 remains true (cf. [Kob 7]).

Proposition 3.6.2. For any domain G C C" the Kobayashi-Buseman pseudometric is an upper semicontinuous function on G x C". Proof. We begin with the following observation. Let z° E G and X° E C" be fixed. Then for Z E G, X E C" it turns out that XG(z; X) _< XG(z; X°) + kG(z; X - X°)

< xG(z,X°) +

I Xj

- XjIkG(z;ej),

j=1

where e j denotes the j-th standard unit vector. Hence it suffices to prove that for a fixed vector, for example for X°, the function X°) is upper semicontinuous at z°.

3.6 The Kobayashi-Buseman pseudometric

101

Assuming the contrary, there exists a sequence (z")" C G with z' --> z° and "H

xG(z"; X°) > A > iG(z°; X°). By definition of xG we find vectors Y" E C" such that IY" ZI < 1 for all Z with xG(z"; Z) < 1 and IY" Xol > A. Hence IY" Z1 < xG(z"; Z) < MIIZII for Z E C" and v > 1, where M is a suitable constant. In particular, the sequence (Y"), is bounded, and so, we may assume that Y' Y°. Hence we have I Y° X°I > A. Now fix an arbitrary vector Z with xG(z°; Z) < 1. The upper semicontinuity of xG ensures sure that xG(z"; Z) < I if v is sufficiently large. Thus we arrive Z I < 1, and therefore I Y° xG(z°; X°); a contradiction.

at I Y"

Z I < 1. Hence we have: xG (z°; X °) > A >

So far, we know that the Kobayashi-Royden pseudometric and KobayashiBuseman pseudometrics are upper semicontinuous functions. Therefore, they can be used to define the length of a piecewise C'-curve, and then the minimal length of all such curves connecting two fixed points will yield a new pseudodistance. For a piecewise C'-curve a : [0, 1] -> G we set:

L, (a)

f I xG(a(t):a'(t))dt. 0

Lx(;(a) :=

f

kG(a(t);a'(t))dt.

0

The numbers L,,,(a), L,«(a) are called the xG-length and the xG-length of the curve a, respectively. For points z', z' E G we define:

(f xG)(z, z") := inf{L,(a): a is a piecewise C'-curve in G from z' to z"}, (f xG)(z', z") := inf{Lx(,(a): a is a piecewise C'-curve in G from z' to z"}. f xG (resp. f xG) is called the integrated form of xG (resp. of XG). Remark 3.6.3. Observe that the systems (f XG)G and (f xG )G of pseudodistances are contractible with respect to holomorphic mappings and that f xE = f ;CE = p. Therefore, we obtain the following chain of inequalities: f xG < f xG < kG. The main result according to the integrated forms introduced above is that they coincide with the Kobayashi pseudodistance.

Theorem 3.6.4. If G is a domain in C", then kG = f xG = f xG.

Before we begin the proof, we recall the following result due to Harris (cf. [Hart).

102

M. The Kobayashi pseudodistanee and the Kobayashi-Royden pseudometric

Lemma 3.6.5. Let d : G x G --.%. (0, oo) be a pseudodistance which is locally uniformly majorized by the Euclidean distance. Moreover, suppose that S: G x C" -+ 10, oo) is an upper semicontinuous function with S(a; ),X) = IA1S(a; X) for

alla EG,AeC,andX EC".Set (fS)(z', z") := inff fo 3(a(t);a'(t))dt: a [0, 1] -+ G is a piecewise C1 -curve joining z', z"}.

Then d < f S if the following relation is true:

limsup

d(a,a+tX) t

r-+o+

<S(a;X), aEG,X EC".

Proof. We fix two points z', z" in G and we take a C'-curve a : 10, 1] -+ G connecting these two points. With f (t) := d(z', a(t)) we observe that if s, t are near t o c- [0, 1], then ! ! !

If (t) - .f (s)I = Id (z', a(t)) - d(z', a(s))I < d(a(s), a(t))

CIIa(s) - a(t)IF <

Is - ti.

Here we have used the fact that d is majorized by the Euclidean distance. Thus we know that the function f is locally Lipschitz. Hence f' exists almost everywhere

and d(z', z") = fo' f'(t) dt. It remains to estimate f':

If'(r)I

lim ho+

I.f(r+h)-f(r)I
hm sup

h

h-.o+

_< S (a

h-.o+

d(a(r) + ha'(r), a(z)) h

(z); a'(z)) + lim sup

h

+ lim sup

d (a (r) + ha'(r), a(r + h)) h

CIla(r) + ha'(r) - a(r + h) 11 h

h-.0+

= S(a(r);ai(r)) for almost all r E 10, 1]. Finally, we obtain

d(z', z") _<

/ S(a{z);a'(z))d-r.

o

r

3.6 The Kobayashi-Buseman pseudometric

103

By means of the triangle inequality for d the case of piecewise CI-curves is clear.

0 Proof of Theorem 3.6.4. First recall that xG is upper semicontinuous and that kG is locally uniformly majorized by the Euclidean distance, i.e. all the assumptions of Lemma 3.6.5 are fulfilled. So, by Remark 3.6.3, to obtain kG = f xG we have to verify the inequality of Lemma 3.6.5 for kG and xG. Let us fix z° E G and X° E C". Ifs > 0, then we are able to choose an analytic ,

disc (P E O(E, G) with 'p(0) = :°, acp'(0) = X°, and 0 < a < xG(z°; X°) + s. Observe that V can be written as cp(.k) = :° + (.k/a)X° + A2H(X), where H E O(E, C"). Then for positive t, to < 1, zo + tX° E G we obtain kG(z°, z° + tX°) < kG(:°, W(ta)) + kG((p(ta), z° + tX°) t t t kG(cp(0)-V(ta)) CIlt°a-H(ta)II < +

-

2t

log

+ta I - to 1

+Ca"tIIH(ta)II

Thus, in the limit case, we get

limsup r

kG(z° . z° + tX°)

°+


Since e was arbitrary, we get kG = f xG by Lemma 3.6.5. To prove the remaining inequality kG < f YG we will use Lemma 3.6.5 again. As above we begin with a fixed point z° E G and a vector X° E C. From what we have mentioned about (Y E C": irG(z°; Y) < 1), it is easy to conclude that for any e > 0 the vector X° can be represented as X° Xi with vectors Xi satisfying fit

1: >G (z°; X1) < srG(:°;X°) +s/2. i=I

Moreover, since xG is upper semicontinuous, we have xG (:; X j) < xG (Z°; X J) +

s/2m if z belongs to an appropriate neighborhood U = U(:°) of z°. Putting the

104

III. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

above together we get

kG(z°, z° + tX°)

lira sup 1-0+

< lim sup

kG(z°, z° + tX' ) t

t

rrr - I +limsupko(z°+t(X'

j=1

t

1-0+ m-I

<xG(zO;XI)+Elimsup 1=I

ro+ t

xG(z°+t(X'+

+Xi)+rXi+I;X.+I)dr

fo

xG(z°; XI)+...+xG(z°;X'")+ 2 <XG(z°;X°)+E,

0

which concludes the proof of Theorem 3.6.4.

Remark 3.6.6. We emphasize that the argument used to prove Theorem 3.6.4 is also valid in a more general context which will be discussed later on (in Chapter IV).

Before we proceed to apply the previous result we want to recall that the Caratheodory-Reiffen pseudometric has appeared as a kind of a strong derivative of the Carathe odory pseudodistance (cf. Proposition 2.5.1(d)). The following example (see [Ven 21) shows that there is no analogous relation between the Kobayashi pseudodistance and the Kobayashi-Royden pseudometric.

Example 3.6.7. Let us consider the following bounded balanced pseudoconvex domain in C2:

G := {z E C2: IzI I < 1, IZ21 < 1, IZIZ2I < a2)

(0
For z' = (z,, z2) and z" = (z',', z2) with Iz;l < a2 and Izi I < a2 we easily derive

kG(Z

,

Z") <_ kG(Z , (z',', z2)) + kG((zI

, Z2), Z)
3.7 Product-formula

105

Estimating the following general differential quotient we obtain lira sup

kc (a', a' + AX')

a'-0

ICI

X'-+(a,a)

1w0

< lim sup '_0

P(ai,a, +AXi) + lim sup P(a2,a2 +1X2)

= 2a.

a'-O

X', X;- (a.a)

X'-+(a,a)

1-0

A-0

On the other hand, we know that xc (0; (a, a)) = 1. Hence, the general differential quotient differs from the Kobayashi-Royden metric. The full discussion of differential quotients of pseudodistances will be given in Chapter IV. But already here we would like to see what happens for plane domains.

Remark 3.6.8. Let G be a domain in C. Then we claim that kc (z', z")

xc (a; 1) = km zz" -a Iz' Z'#

= km

kc (Z', z") IZ' - Z°I

a E G.

Z'#

Without loss of generality we may assume that G is a taut domain. Suppose that

(z')vEN, (Z;;),,EN C G with z;, # z', v E N, lim zv' = a = lim z' are such that lim kG(Z ' ; exists. Then there are holomorphic mappings cp,,: E - G, Iz,.-z;. I

with 0
v- 00

0.

Therefore, taking an appropriate subsequence we may assume that cp = cp with V-00 cp E O(E, G), cp(0) = a. Hence we obtain lim

= lim (P(0, av Iz" - z"I -00 av

av

kc (Z,', zv

)=

I(PV(0) - ((av)I

1

I(v'(0)1

On the other hand, take cp E O(E, G) with cp(0) = a, cp'(0) cp'(0) 0 we timV_+00 a, =

1).

0. Because of

E E, a' 0 a" , with

may assume that there are a' E E,

a = 0, such that cp(a;) = z',, cp(a,') = z' . Hence we get

)

kc(z, z) lim Iz' - Z"I

> xc (a;

lim sup

P(a', la;, - a 'I

which implies the remaining inequality.

I

"I

w(aJ')I

IcP'(0)I

106

III. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

3.7 Product-formula We conclude this chapter with a discussion of the way how to calculate the objects studied before on product domains. We begin with an information which can be derived directly from the original definitions.

Proposition 3.7.1. Suppose that two domains G C C" and D C C' are given. Then the following formulas hold: (z", w")) = max(kG(z', Z"), kD(w', w")), (a) kGxD((z', w'),

z', z" E G, w, w" E D; (b) MG.D((z, W); (X, Y)) = max(xG(z; X), xD(w;Y)),

ZEG,wED,XEC",YEC"'; (C) y=GxD((Z, w); (X, Y)) = max(5a(z; X), 3[D(w; Y)), z, w, X, Y as above.

Proof. By the contractibility property with respect to holomorphic mappings (here for projections), the inequalities ">" are obvious. Now, we proceed to prove the inverse inequalities. (a) Suppose that kGxD((z', w'), (z", w")) > A > max(kG(z', z"), kD(w', w")) Then, by definition, we find holomorphic traps (P E O(E, G) and ' E O(E, D) sharing the following properties: go(O)=z',

P(a)=z" (0
*(0)=w', *(r)=w" (0
F(A) := ((p(OA), *(A)) Z

with F(0) = (z', w') and F(r) = (z", w"). Hence kGxD((z', w'), (z", w")) < p(0, r) < A, contrary to our assumption. (b) The argument here follows exactly that used just before, so it is left for the reader.

(c) Assertion (c) is an immediate consequence of the formula (b) and the properties of )CG, that were formulated in Remark 3.6.1.

Now, we combine the results given in Theorem 3.6.4 and Proposition 3.7.1 to obtain the following product-formula.

3.7 Product-formula

107

Theorem 3.7.2. Suppose G C C" and D C C"' are given domains. Then the following formula is true on G x D

kGXD((z`, w'), (z2, w2)) = max(kG(z', z2), kD(w', w2)].

Proof. Obviously it suffices to prove the inequality "<". Let us suppose (without

loss of generality) that kG(z', z2) ? ko(w', w2). For any s > 0 Theorem 3.6.4 guarantees the existence of C' -curves

yi : [0, 11 -+ G, yi (0) = z', Y, (l) Y2: [0, 1] -> D,

z2, j xG (YI (t ); y'(1)) dt < kG (z' , z2) + E,

=

n(O) = w', n (j) = w2, f ' xn(y(1):Y'(t))dt < kp(w', w2) +E. 0

Then we choose continuous functions hj > 0 on (0, 11 with

xG(Y1(t);y1(t))

hi(t),

XD{Y2(1);Yi{0)

h2(t),

and

I

hi(t) dt < kG(z', z2) +E, 10

I

h2 (t) dt < kG(z', z2) + C.

It is easy to modify hj in such a way that we have fo h, (t) dt = fo h2 (t) dt. Thus we obtain two strictly increasing functions

F(t) := jhi(r)dr r

and

G(t) :=

ffh2(r)dr

with F(0) = G(0) = 0 and F(I) = G(l). Therefore, * := G-' o F gives a parameter transform of the unit interval with *'(t) = hi(t)/h2(>G(t)).

108

Ill. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

Then we observe that the curve y : [0, 1] -+ G x D. y(t) :_ (yi (t), y, o *(t)) connects (z`. w') with (z2, w2). Moreover, we obtain kGXo((:', WI (_2, u''-))

fo

xc o(y(t);y'(t))dt

max(xG(yi (t); y' (t)), 1

/o

xD(y o *(t).Y2 '(*(t))*'(t)))dt

_ J max{hi(t),h2oi/i(t)-*'(t))dt o

= J hi(t)dt
0

which implies the desired formula.

Remark 3.7.3. In Chapter IX we will discover that the analogous product-formula is also true for the Caratheodory distance.

Notes

About 1966 S. Kobayashi initiated studying his pseudodistance; cf. [Kob 3,41. H. L. Royden published the infinitesimal form in 1971 [Roy). Most of the material collected in this chapter is due to these two authors and also to T. J. Barth (cf. [Bar 1,4]). A fairly complete survey about the results up to 1976 can be found in the report of S. Kobayashi (see [Kob 6]). Recently, S. Kobayashi has introduced the pseudometric which is called here the Kobayashi-Buseman pseudometric (see [Kob 7]).

Exercises

109

Exercises 3.1. (a) Prove the inequality in Example 3.1.4. (b) For N E N construct N holomorphic functions fj E O(E, E) with the following properties:

EE: fj(A)=fj+i(A))=().j)CE for t<j 2. (i)

Define GF := {z E E x E: with Aa E E \ (A1,...,AN}.

fN1

1:2 - fj(zi)I < E) and set z(j) := (Ae, fj(Ao))

Prove that there is a sufficiently small a such that kG'-1)(z(1), z(j)) > k(j)(z('). z(j))

for

2 < j < N.

(*)

Use exhaustion to provide a strongly pseudoconvex domain G C GF with real analytic boundary such that (*) remains true for G. Compare Theorems 7.5.7 and 7.5.9 for another examples showing that, in general, k(N) # ko.

3.2. Let G be a plane domain and let f E O(E, G) be such that for two different points A', A" E E we have kG(f (A'), f (A")) = kF-(A', A"). Prove that f is a holomorphic covering.

3.3. (a) Note that G C1-boundary. Let G

(z E C2: Izi I < Iz21 < 1) is a taut domain without (z E (C2: IzI I < Iz21 + I/ V < 1 + 2/v). Prove that

lim kG,((0, 1/2), (0, -1/2)) < kc, ((0, 1/2), (0, -1/2)).

(b) Construct a domain G with C"-boundary, which is not taut, and a sequence (G,,),, as in Proposition 3.3.5(b) such that

lim kG,.((1/2, 1/2), (-1/2, 1/2)) < kc((1/2. 1/2), (-1/2. 1/2)). V

00

Hint. Start with G' := (z E C2: IziI < 1/2,1,,21 < I or 1/2 < Iz, I < 1, Iz21 < 1/21 and smooth it out. (c) To disprove the claim in Proposition 3.3.5(b) construct a sequence of domains C C3 with G. D D B(0, 1) but not with G,, » G,,+, and such that n G,, G = B3.

1 10

111. The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric

Hint. Build G, via thickening

G' :=B(0,

v

1)UU((A,A,jA(A-1/2)):AEE)

v and consider ka3 ((0, 0, 0), (1 /2, 1/2, 0) ).

(The ideas of the examples above are due to W. Zwonek.)

G:=(z=(zI,z2)EC2:IzII < I or1z21<1). 3.4.(a)Prove that (b) Let G := (z E C2: Izi I < 1, Iz2I < 1 and if Izi I = Iz21, then Izi I < 1/21. Prove that I < xG (0; (1, 1) ). F?] We don't know the exact value of xC (0; (1, 1) ).

0

3.5. How does the Kobayashi-Royden pseudometric behave under an increasing sequence of domains ?

3.6. Complete the argument to show that, in general, the Kobayashi-Buseman pseudometric is not continuous (cf. Example 3.5.10).

3.7. Use Theorem 3.6.4 to find an alternative proof of the fact that the Kobayashi pseudodistance is inner.

3.8. Prove the following analogue of Theorem 3.3.7 for the Kobayashi-Royden

pseudometric: let n : G -> G be a holomorphic covering; then xo (z; X) _

xd(z;Y) ifZEG with n(i)=zand ir'(Z)Y=X. 3.9. Prove the extension theorem for the Kobayashi-Royden pseudometric across closed sets with (2n - 2)-Hausdorff measure zero (cf. Theorem 3.4.2) and next, prove Theorem 3.4.2 using Theorem 3.6.4.

3.10 (cf. [Lem 11). Prove that if G is a convex domain in C", then the Lempert function kG satisfies the triangle inequality.

Chapter IV

Contractible systems

In Chapters II and III we proved that (CG)G. (kG)G (resP. (YG)G (XG)G) are "extremal" contractible systems of pseudodistances (resp. of pseudometrics). It is known that (CG)G and (kG)G are extremal contractible systems of functions. We also constructed some intermediate systems, e.g. (cG)G or (icG)G. Observe that there are many such intermediate contractible systems. e.g. dG`)

where

(I -t)cG+tka, S(`) := (1-t)yG+tXG (0 < t < 1). We are only interested

in the systems that appear in a natural way in complex analysis; most of them arise from certain extremal problems.

4.1 Abstract point of view Let 8 denote the family of all domains in all C"'s and let 00 be a subfamily of 13 with the following two properties: (a) E E O o,

(b) if G E 00 and D is a domain biholomorphic to G, then D E 00, i.e. i0 is invariant under biholomorphic mappings. Standard examples:

00 = 6, 80 = Oh

(G E 0: G is a domain of holomorphy),

Oo = 6,

(4.1.1) {G E 13: G is biholomorphic to a convex domain}. Oo = Oh:= {G E 13: G is biholomorphic to a bounded domain}. (4.1.2) Let d = (d(-,)GEmu be a system of functions dG : G x G -). R+ (G E 13o). We say that d is a (holomorphically) contractible famiy' of functions if

dE=p

(4.1.3)

and

d0(F(; ). F(; )) <- dd(:.: ).

r E G.

(4.1.4)

112

IV. Contractible systems

for every F E O(G, D) (G, D E so). If, moreover, for any G E Oo the function do is a pseudodistance, then we say that d is a contractible family of pseudodistances. R+ Now let 8 = (SG)GEeo be a system of pseudometrics SG: G x C"

(G E 0o, G a domain in C"), i.e. SG (a; AX) = IA ISa (a; X ), a E G, A E C, X E C". We say that 8 is a contractible family of pseudometrics if SE = YE,

and if for any F E O(G, D) (G, D E Oo) we have SD(F(z); F'(z)X) < 8G(z; X),

z E G, X E C".

(4.1.5)

(We use the following convention: when we write "SG (a; X ), a E G, X E C"", then it is automatically assumed that G is a domain in C".) Notice that we do not require that SG (a; ) be a seminorm nor that SG be upper semicontinuous. It is clear that if F is biholomorphic, then equalities hold in (4.1.4) and (4.1.5). If we substitute (4.1.3) by

dE = m,

(4.1.6)

then we say that d is an m-contractible family of functions (resp. m-contractible family of pseudodistances).

Remark 4.1.1. (a) c, c', k, k are contractible families of functions; c, c', k are contractible families of pseudodistances. (b) c*, k*, k* are m-contractible families of functions; c', k* are m-contractible families of pseudodistances. (c) y, x, x are contractible families of pseudometrics.

Remark 4.1.2. (a) If d = (dG)GEeo is a contractible family, then cc, < dG < kG, G E 60. If, moreover, dG is a pseudodistance. then CG : dG < kG ka. (b) If d is an m-contractible family, then c* < dG < k', G E O o. (C) If S = (SG)GE ft is a contractible family of pseudometrics, then yG < SG < xG, G E Oo. If, moreover, SG(a; ) is a seminorm, then yG(a; ) < SG(a; ) < XG(a; -) <_ xc(a; ). Remark 4.13. In view of Remark 4.1.2(b), if d is an rn-contractible family, then dG < 1. If d = (dG)GEeo is a contractible family, then we put dG := tanhdG, d* := (dc)GEe0. Observe that the operator

4.1 Abstract point of view

113

is a bijection between the class of contractible families of functions and the class of m-contractible families of functions. From now on, objects will

be marked by "*", e.g. d (this agrees with c', k*, and k*). Observe that if d is a contractible family of pseudodistances, then so is d`. We have introduced the notion of m-contractible families for technical reasons only (they are much simpler in calculations; cf. Chapter II).

Remark 4.1.4. In Chapter VIII we will prove that for any G E 0, (cf. (4.1.1)) we have CG = kG = kG, c* = k* = kG, and yG = xG.

Thus, in the class t the theory of contractible objects reduces to the Caratheodory case; the reader should always remember this fact. To complete these (tedious) general preliminaries observe that, unfortunately, our notions of contractible objects are too restrictive. They do not cover, for instance, the Bergman distance and metric (cf. Chapter VI) nor the Hahn pseudodistance and pseudometric (cf. Remark 3.5.14). Problems appear because of the requirement that d (or S) be contractible with respect to all holomorphic mappings. To avoid these difficulties one could proceed as follows. First, for any G, D E O o we fix a family .F(G, D) C O(G. D) such that (a) if F : G --). D is biholomorphic, then F E.F(G, D), (b) if F, E .F(G,. G2), F2 E .F(G2, G3), then F2 o F1 E .F(G,, G3). Next, in (4.1.4) (or (4.1.5)) we only take F E .F(G. D). For example:

.F(G, D) = .Fi(G, D) := IF E O(G, D): F is injective}; this family is good for the Hahn pseudodistance and pseudometric;

.F(G, D) =.Fb(G, D) := IF E O(G, D): F is biholomorphic)}; this family is good for the Bergman pseudodistance and pseudometric. We will not go into this generalization, but we advise the reader to verify which of the results may be extended to the above general case. Let us come back to the standard situation.

Remark 4.1.5. The functions dG and 3G in the definitions of contractible families of functions and pseudometrics look very arbitrarily. But this is not so. They must satisfy a lot of regularity properties. For instance, since kG is continuous, we get the following properties.

(a) If d = (dG)GEeo is a contractible family of pseudodistances, then for each G E 00 the pseudodistance dG is continuous (use Remark 4.1.2(a)). (b) Similarly, if d' is an m-contractible family and B(a, 3r) C G E 00, then

d'c(Z', Z") <

G(

)

k'e(z .2r)(Z . Z ) =

IIZ

2r

II

,

Z.Z

E B(a, r).

IV. Contractible systems

114

If d is a contractible family, then, of course, we have analogous inequalities with tank-' (iz.zi) at the end. This last term is not handy in calculations but since

tanh-I (t) '- t as t - 0+, we can reformulate the estimate as follows. (c) If d is a contractible family, then for any a E G E 00 there exist M, r > 0 such that

dG(z', z") C MIIZ'- z"II,

z', z" E B(a, r) C G;

we briefly say that dG is locally bounded by the Euclidean distance. (d) If 8 = (SG)GEeo is a contractible family of pseudometrics and if, moreover,

B(a,2r)CGE(so,then SG(z;X) < xG(z;X) <x$cz.r)(z;X) = IIXII,

r

z E B(a,r),X E

Let d be a contractible family of pseudodistances. We say that G is d-hyperbolic if dc is a distance. Let S be a contractible family of pseudometrics. We say that (a) G is S-hyperbolic if

VaEG 3M,r>0:6G(z;X)>MIIXII,

ZEB(a,r)CG;

(b) G is pointwise S-hyperbolic (briefly, p. 1-hyperbolic) if Va E G: SG (a; X) > 0,

XE(C")..

Notice that [G is c-hyperbolic]

[G is d-hyperbolic]

(G is k-hyperbolic]

t (cf. Th. 7.2.2)

[G is y-hyperbolic] - [G is 8-hyperbolic]

t [G is p. y-hyperbolic]

--> [G is >r-hyperbolic]

4 (cf. Remark 3.5.11) [G is p.8-hyperbolic]-+ [G is p. x-hyperbolic]. 4

In particular, if G E ob n oo (cf. (4.1.2.)), then G is d- and ¢-hyperbolic for any d and S; moreover, top dG = top G. In general, [top cG = top G] -- [top dG = top G] (cf. Proposition 2.4.1(b)). To justify the above general definitions of contractible families of functions and pseudometrics we need examples of such objects (different from the Caratheodory and Kobayashi cases). This will be done in the following sections.

4.2 Extremal problems for plurisubharmonic functions

115

4.2 Extremal problems for plurisubharmonic functions Let us begin with the following trivial remarks:

c* (a, z) = sup{u(z): u E M(')(a)},

(4.2.1)

yG(a;X) = sup{lim -u(a +AX): U E M( ')(a)},

(4.2.2)

;L-0 JAI

{If l: f E O(G, E), f (a) = 0}. Observe that

where

[0, 1): u is log-psh , M(l)(a) C ICG(a) := {u: G 3 M, r > 0: u(z) < Mllz - all, z E B(a, r) C G),

a E G E 0.

This suggests to consider systems (GG(a)) of subclasses GG(a) C KG(a), where

G runs through t5o and a through G, and to define corresponding contractible objects via certain extremal problems for GG(a). More precisely, we assume that such a system (GG (a)) satisfies the following conditions:

the function E

A -> IXl belongs to GE(0),

(4.2.3)

if F E O(G, D) (G, D E !5o)and iEGo(F(a))(aEG), then u o F E GG(a).

(4.2.4)

Observe that (4.2.3) and (4.2.4) imply that M( ')(a) C GG(a); in other words, (M(')(a))aEGEI is the minimal "admissible" family. Remark 4.2.1. One can easily prove the following families L = (GG(a))aEGEe are "admissible": (a) CG(a) = KG(a) (and so (ICG(a))aEGEC is the maximal "admissible" family);

(b) £G(a) = SG(a)

J

fu-: u: G

(c) GG(a) = M( )(a)

[0. 1). u is log-psh, u(a) = 0. and u is C2 near a);

{If IR : f E O(G, E), order f > k) (k E N), where

order f denotes the order of the zero of f at a. Clearly,

MG (')(a) C M(')(a) C ICG(a),

M(')(a) C SG(a) C KG (a),

a E G E l3.

IV. Contractible systems

116

Moreover, if G C C', then

f

(If I

E G.

If L = (GG(a)).EGEO satisfies (4.2.3) and (4.2.4), then we define

d( )(a. z) := sup(u(z): u E GG(a)}, SG )(a;X) := sup(limsup y--0

I

a. z E G E l'So,

u(a +AX): u E GG(a)},

a E G, X E C".

IXI

In the special cases we set (R)

,gG := dG

(K)

,

(S) SG := CIG ,(,NUB),

nt(A)

AG := aG

(S)

SG

= SG

(,N n' )

(k)

YG

SG

GE0.

By (4.2.1) and (4.2.2) we have In0) = GG,

YGI)

= YG,

G E 4.

Moreover, (k) MG < 8G,

(R)

YG< yd < AG, GGG :5- SG <8G YG $So
and in the case where G C C', we get

YG
(dG ))GEOc, is an m-contractible

Proof. In view of (4.2.4), the only difficulty is to show that dE) = m. It suffices to prove that dE (0. ,X) = IAI, A E E. Since .M(')(0) C CE(O), we get The inequality "<" follows from the Schwarz Lemma for subharmonic functions (cf. Appendix PSH 22).

0

4.2 Extremal problems for plurisubharmonic functions

117

Lemma 4.2.3 (cf. [Kli 2]). We have 9G (a, ) E KG (a), a E G E 0. Consequently, AG(a; X) = lim sup x-.o

1 gG(a, a + AX),

a E G, X E C".

ICI

(In particular, SG 1 < AG < +oo.) Proof. Fix a E G E 6 and let u* denote the upper semicontinuous regularization of the function u := gG(a, ). Obviously u* is log-psh on G. According to Proposition 4.2.2 (and Remark 4.1.5(b)) we have

u(z) = gG(a, z) <

liz -all r

z E B(a. r) C G.

Hence

u*(z) < IIZ -all ,

r

z E B(a, r),

and therefore u' E KG(a). Since u is extremal, we get u' = u. Proposition 4.2.4. For any C the system 6tc1 : _ (SGc} )G 0 is a contractible family of pseudometrics.

Proof. It is clear that (a;) is a pseudometric. Using the same methods as in the proof of Proposition 4.2.2, we get SF('C) (0; ) = ye (0; ). It remains to verify the holomorphic contractibility. More precisely, it suffices to prove that for any F E O(G, D) and for any u E Go(F(a)) we have lim sup x-»o

1

u(F(a) +X F'(a) X) < lim sup x.-o

IXI

1 (u o F)(a +,kX). IXI

This is a direct consequence of the following

Lemma 4.2.5 (cf. [Azu 3,4]). Let (pi, (p2: r1E --)- G (rl > 0) be holomorphic mappings with SPi (0) = (G2 (0) =: a and (p', (0) = (p2' (0) =: X. Then for any u E ICG (a) we have

o

limsuP II

lu(w2(A))

U(VI(A)) = limsuP ;L-0

I

IV. Contractible systems

1 18

Proof. Define a mapping

by

(RE) x C Note that 4'(., 0) = Bpi,

01) + N2 (X) E C".

1) _

.

Put 11 := 4)-'(G), S2o := 1 \ ({0) x C), and

let E S?,o.

Then v is psh in Slo. Let M, r > 0 be such that

u(z):<MIIZ-all,

ZEB(a,r)CG.

Write rpj(A) = a +AX +A2* (A), A E ?It (j = 1, 2) and set Mo := max(IIfI IInE, 111L211,1E}. Then for (,l, ) E 4)-' (B(a, r)),.k:34 0, we have

v(a. 0:5 M[IIXII + IAIMo(I1 - I + I VA. This shows that v is locally bounded in 1, and therefore by putting v(0, 4) := lim sup v(A, 4'),

4 E C.

X- O S/~S

we extend v to a psh function on the whole S2 (cf. Appendix PSH 21). By the above inequality we get

v(0,) < MIIXII,

t E C.

Consequently, v(0, ) = const; cf. Appendix PSH 25. Finally, lim sup

;.-0

1 u ((pi (A)) = lim'k-0sup v(A, 0) = v(0, 0) = v(0, 1) ICI

= limsoupv(A, 1) = Iimsup

IXlu((PZ(A))

D D The function gG is called the complex Green function for G, AG is the Azukawa pseudometric, SG is the Sibony pseudometric, meld is the k-th Mobius function, and M Yd is the k-th Reiffen pseudometric.

4.2 Extremal problems for plurisubharmonic functions

119

Remark 4.2.6. (a) If G C C', then - log ga(a, ) coincides with the classical Green function for G with pole at a; in particular, gG(a,) is of class C'- on G; cf. Appendix OR.

(b) If g2 (a, ) is of class C2 near a (e.g. G C C' ), then and

Now, we are going to present the basic properties of the above contractible families of functions and pseudometrics.

Properties of 9G and AG: Proposition 4.2.7. (a) If GL, / G, then gG, \ 9G(b) If G = Gh C C" is a balanced pseudoconvex domain with Minkowski function h, then 9G(0AGO;

h on G, h on C".

In particular, gG (0, ) = k* (0. ) and AG (0; ) = r.G (0; . ), cf. Propositions 3.1.10 and 3.5.3. (c) (cf. [K]i 1 ]) If P C G is a closed pluripolar set, then gG\P = gG1(G\P)x(G\P),

AG\P = AGI(G\P)x.C..

(d) (cf. [Dem], [Kli 3]) If G is bounded, then for any ball B(a, r) C G and for any s > 0 there exists S E (0, r) such that [9G (X,

9G (y,

z),

r, y E B(a, S), E G \ B(a. r).

(4.2.5)

(e) If G is bounded, then for any zo E G the function G 3P z -> gG(z. Zo) is continuous.

Note that for unbounded domains G the function G 9 z - gG(z, zo) need not be continuous; cf. Remarks after Proposition 4.4.2.

(f) For any G and for any zo E G the function G -3 z -+ 9G (Z, Zn) is upper semicontinuous.

(g) (cf. [Kli 1]) If G is hyperconvex, then limn-dG ga(a, z) = 1, a E G. (h) (cf. [K]i l]) For each a E G the function log gG(a, ) is a maximal plurisubharmonic function on G \ (a). In particular, if 1099G (a, ) E L°G(G \ (a), loc) (e.g.

IV. Contractible systems

120

if the set G is bounded), then (dd`loggG(a, ))" = 0 in G \ ]a); cf. Appendix MA.

(i) (cf. [Dem], [Kli 3]) If G is hyperconvex, then the function gG is continuous on G x G, where gG I G. aG

I: cf (g)

Note that for an arbitrary (even bounded) domain of holomorphy and for a E G the function G D z - gG (a, z) need not be continuous; cf. (b). (j) (cf. [Kli 2]) If G is a domain of holomorphy, then the function gG is upper semicontinuous. (k) (cf. [Azu 2], [Kli 2]) For any a E G the function C" a X - AG(a; X) is upper semicontinuous.

Note that this function need not be continuous; cf. W. (I) (cf. [Kli 2]) If the function gG is upper semicontinuous (e.g. if G is a domain of holomorphy, cf. (j)), then the Azukawa pseudometric is upper semicontinuous. (m) The function gG : G x G -+ [0, 1) is symmetric if for any zo E G the function G 3) z -+ gG(z, zo) is log-psh.

Note that, in general, for n > 2 the function gG need not be symmetric; cf. [Bed-Dem]; see also Proposition 4.4.2 and Exercise 4.4.

Proof of Proposition 4.2.7(a). Use standard properties of plurisubharmonic functions, in particular, Appendix PSH 12. Proof of Proposition 4.2.7(b). The inequalities "<" are obvious. It remains to observe that hIG E /CG(0): cf. Appendix PSC 6. Proof of Proposition 4.2.7(c). Since pluripolar sets are removable for upper bounded psh functions (cf. Appendix PSH 21), we obtain

aEG\P.

KG\P(a) = KG(a)IG\P,

Proof of Proposition 4.2.7(d). Fix B (a, r) C G and e > 0. We may assume that B(a, 3r) C G. Put R := diam G. Recall (cf. Remark 4.1.5(b)) that

gG(z' w)

<

Ilz

2rw11,

z, w E B(a, r).

(4.2.6)

Take 3 E (0, r/3) such that (2S) i +f <

.

R

(4.2.7)

4.2 Extremal problems for plurisubharmonic functions

121

Fix xp, Vo E B (a, d) and detinc

z-II u(z) :=

max([gG(XO

if z E B(a, 28) Ilz R+0II }

z)J'+t.

if z E G \ 8(a. 2S)

Observe that if IIz -- all = 28, then by (4.2.6) and (4.2.7) we get [SG(ro,..)]I+F

<

11-1, - yoll

R

Thus u is log-psh on G, and therefore u E /CG(yo). Hence gG(yo, z) ? u(z) ? [SG(XO,

Z E G\B(a,28) DG\B(a,r).

z)11+E,

Proof of Proposition 4.2.7(e). First, observe that by (4.2.6) the function gG is continuous on the diagonal of G x G (for arbitrary G). Next use (4.2.5). Proof of Proposition 4.2.7(f). Use (a) and (e).

Proof of Proposition 4.2.7(g). Let' : G - (-oo. 0) be a continuous psh function such that {>[i < -t} CC G whenever t > 0. If G C B(0, R). then for z E G we define

gp(z) := sup(v*(z) +

I (IIz112 V

- R2): V E N).

Observe that go is locally the maximum of a finite number of negative continuous strictly psh functions (cf. Appendix PSH), therefore (p itself is a negative continuous

strictly psh function. Moreover, (V < -t) cc G for all t > 0. Fix a E G and 0 < r < 1/2 such that B(a, 2r) c c G. Let X E Co (G. [0. 1j) be such that X = 1 on B(a, r) and suppX C B(a. 2r). For C > 0 let u(z) := exp(Ccp(z) + X(z) log 11z - all),

z E G.

Obviously, 0 < u < I and u(z) _< 11z - all, z E B(a, r). Moreover, since go is strictly psh, we easily conclude that for C >> 0 the function log u is psh on G. Let us fix such a C. Then U E 1CG (a), and consequently

gG(a, z) >- exp(C((z)), which directly implies the required result.

z E G \ B(a, 2r),

IV. Contractible systems

122

Proof of Proposition 4.2.7(h). Let Go be a relatively compact open subset of G \ (a)

and let v be a function upper semicontinuous on Go and psh in Go such that v < 1099G (a, -) on aGo. Define

u(z)

_ .

max(gG(a, z), exp(v(z))),

z E Go

gc (a z)

zEG \ Go

Then U E ICG (a) (cf. Appendix PSH 16), and therefore by the definition of gG we

have u < 9G (a, .). In particular, v < loggG(a, -) in Go, which finishes the proof of maximality. It remains to observe that if G is bounded, then Ilz - WV

diamG

g G (z , w) ,

z , w E G.

(4.2.8)

In particular, if G is bounded, then the function log gG (a, -) is locally bounded in G \ (a), and therefore, using Appendix MA 2, (dd` loggG(a, -))" = 0 in G \ (a}.

0 Proof of Proposition 4.2.7(i). Fix (a, b) E G x G and let G x G 9 (z,,, gG(a, b). Fix r > 0 with B(a, 3r) C G (a, b). We want to prove that 9G (Z,, and put R := diam G. We may assume that a # b (cf the proof of (e)) and that b 0 B(a, 2r). Let us assume for a while that (4.2.9)

the function gG (a, -) is continuous on G.

Take arbitrary e > 0 and let S = S(e) be as in (d). Note that by (g) condition (4.2.5) holds for x, y E B(a, S), Z E G \ B(a, r). In particular, [9G(a,

w.)J'+E <

gG(zv,

[gG(a,

v» 1.

w,)]±,

Then according to (4.2.9) we get [9G (a, b)]'+F < liminfgG(z,,, -00

-

,

limsuPgG(zv, wv) < [9G (a, b)]

vy0

Hence letting a -+ 0 we are done. It remains to prove (4.2.9). By virtue of (g) and Lemma 4.2.3 it suffices to prove that gG(a, -) is lower semicontinuous on G. of subdomains Since gG(a, -) is log-psh on G, we can find a sequence of CO0 log-psh functions g on G with G / G of G and a sequence and g,. \ gG (a, . ); cf. Appendix PSH 17.

4.2 Extremal problems for plurisubharmonic functions

123

Let V: G -+ (-oo, 0) be a continuous psh function such that {V < -t} CC G whenever t > 0. We may assume that

V(z) < -r for z E &(a, r).

(4.2.10)

Fix so E (0, 1), so < r, such that R >

52(I-e) > ee-11F,

0 < s < so.

(4.2.11)

Since so < r, condition (4.2.10) implies that

p(z) < -e for z E B(a, s),

0 < s < so.

(4.2.12)

For an arbitrary s E (0, So) let r) = r)(s) E (0, s) be such that

n 2r

(4.2.13)

<

Then using Dini's theorem we choose v(s) with v(s) -> oc as s -+ 0 such that for ve := g,(,) the following properties hold {rp < -s3} C G,,(F).

ve < (Eq)1-e on B(a, i) (use (4.2.6) and (4.2.13)) ,

(4.2.14)

ve(z) < I if (p(z) < -s3.

(4.2.15)

Now, we define ue : G - R by

us(z) :=

e-e(sllz - all)'-' max{e-e(sllz -all)`, e-evf(z)}

if z E B(a, rt) if z E &(a, s) \ B(a, rj)

e-eve(z) max{e-ev((z). ew(:)/e'}

if 92(z) <--s and z0 B(a,s). if - s < W(Z) < -s3

e'P(z)/e=

if V(z) > -s3

Note that by (4.2.12) the function ue is well-defined. Then (4.2.6), (4.2.8), (4.2.11),

(4.2.14), and (4.2.15) show that uF is a continuous log-psh function on G with values in [0, 1). In particular, ue/(I -C) E ICG(a). Moreover, ue -+ 9G (a, ) as s -+ 0, i.e. gG(a, ) =

sup(ut.l(1-e): 0

< e < so).

Hence gG(a, ) is a lower semicontinuous function.

0

124

IV. Contractible systems

Proof of Proposition 4.2.7(j). Any domain of holomorphy may be exhausted by an increasing sequence of hyperconvex domains (cf. Appendix PSC 11), and therefore (j) is a direct consequence of (a) and (i).

Proof of Proposition 4.2.7(k). Suppose that AG(a; Xo) < M. By Lemma 4.2.3 there exists R > 0 such that 0 < ICI < R.

9G (a, a + AXo) < MIAI,

Since the function 9G (a, ) is upper semicontinuous (Lemma 4.2.3), there exists a neighborhood U of Xo in C" such that

gc(a, a +AX) < MR,

IAI = R, X E U.

Fix X E U. Observe that the function u defined by u(),)

ICI 9G (a,

u(0)

AG(a; X )

a + AX),

0 < IAI < R,

is subharmonic (Lemma 4.2.3). Since u(A) < M for IAl = R, the maximum principle gives

u (O) = Ac (a; X) < M. Proof of Proposition 4.2.7(l). The proof is analogous to the proof at (k). If

gc(a, a + AX0) < Mlal,

0 < 111 < R,

then the upper semicontinuity of gc implies the existence of a neighborhood V of (a, Xo) in G x C" such that

gc(z, z + AX) < MR,

IAI = R, (z, X) E V.

Then, as in (k), the maximum principle leads to the required result.

Proof of Proposition 4.2.7(m). If gc is symmetric, then by Lemma 4.2.3 for any Zo E G the function u,(z) := gc(z, zo), z E G, belongs to iCc(zo) In particular, u4, is log-psh. Now suppose that for each Zo E G the function uZD is log-psh.

Fix zo and let B(zo, 2r) C G. Then (cf. Remark 4.1.5(b)) we have uz,,(z) _< Ilz - zoll/r, z E B(zo, r). Consequently, uzD E 1Cc(zo), and finally gc(z, zo) = u;. (z) -< gc(zo, z), z, Zo E G.

4.2 Extremal problems for plurisubharmonic functions

125

0 We do not know whether the function gG is upper semicontinuous for arbitrary domains G. The following example shows that, in general, there are no relations between gG and gcIGXG, where d is the envelope of holomorphy of G; see also Exercise 5.8.

Example 4.2.8 (cf. [Jar-Pfl 6]). Given 0 < a, i6 < 1, we define

G+=G+(a,fl):=E2\((zI,z2)EE2: Izil>a, 1221?fl), G- = G_(a, fi) := E2 \ ((ZI, z2) E E2: Izi I < a. Iz21 ? fl}. Observe that G+ and G_ are Reinhardt domains, G+ is balanced, but G_ is not balanced. They are not domains of holomorphy. The envelope of holomorphy of G+ is given by the formula

G+ = G+(a, fi) := ((:.i, z2) E E2: Izi

(-log(ilz,l-Toga < e-10ga10"};

the envelope of holomorphy G_ of G_ is just the unit bidisc

E2. Our aim is to

prove the following: (a)

gG+(0, z) = gG+(0, z) = max(1zil. Iz21,

h+(z) = the Minkowski function of G+,

z = (zI. z2) E G+;

(b) if rg > a, then gG_ (0. z) = gc_ (0. z) = max(Izl I, Iz21},

z = (zI, z2) E G-;

(c) if , < a, then

gc_(0,z)

-

max(IzI 1,

Iz2I}

if Iz21 < f

0gc_(U.<), z=(.i,z2)EG_.

goo

max(1zi I, 1 zzl'" }

if Izr I > a

IV. Contractible systems

126

Proof of (a) (cf. Fig. 4.1).

Let

h(z) := max{ IziI,

Iz2l,min(lal,

1011}.

By Proposition 4.2.7(b) and the remark after Proposition 3.1.10 we have

h+ = gc+(0, )

gc+(0, ) _<

h.

In particular,

h+(z) = gc+(0, z) = gG+(0, z) = h(z) = max(Izil, Iz21}

(4.2.16)

for all z = (zi, z2) E G+ such that IZ21 : fiIzi I or Izi 1 < aIz2I It remains to prove that

gc+(0, z)

h+(z)

for all z = (zi, z2) E G+ with

(0< zj
4.2 Extremal problems for plurisuhharmonic functions

127

Fix 0 t I:011 < a and dchnc the function it by u

I

U := (A E C: PIz°I < IxI < a Iz011) 3 A

gc, (0. (.A)).

Then u is subharmonic and by (4.2.16)

u(x)<

ifI;I=flI:,°I

Iz 10 11

1I`0

ifIAI_a1701

Hence, by the Hadamard Three Circles Theorem for subharmonic functions (cf. Appendix PSH 23). we get

u(a) = gc, (0, (z°. A)) < h+(z°. h).

AEU.

0

The case 0 < Iz2I < B is analogous. Proof of (b) (cf. Fig. 4.2).

a Fig. 4.2

Obviously,

h-(z) := max(Iz11,1z21) = S6(0. z) _< gc (0. z), Take

E 3G- with

z = (z,. z2) E G_.

I = I. Then by Appendix PSH 22 we have

9G-0,4) < IAI = h-(4).

A E E.

IV. Contractible systems

128

It remains to prove that SG_ (0, z) -< h- (z)

(4.2.17)

for all z = (ZI, Z2) E G_ with Iz21 ? Iz1I. Fix 0 < Izil < f and consider the function u given by

U:_{,tEC:1k1<_Iz2I+E}3A-gG_(0,(A,z2)) (E > 0 small). Then, by the maximum principle, we have

u().)-<1Zz1+E,

AEU.

Letting E -i 0 we get (4.2.17). Now let a < Iz°I < 1. We prove (4.2.17) by using the Three Circles Theorem for the function (X

E - 0.

Proof of (c) (cf. Fig. 4.3). A

1

0

a Fig. 4.3

4.2 Extremal problems for plurisubharmonic functions

129

Put if IZ21 < P

I maxilzi I, V Z211 U

k]La

max{Izi1, Iz2I

I

if Izil > a

One can see that u is well defined and that u EKG (0). Hence

gG. (0, ) > u. By the Schwarz Lemma (cf. the proof of (b)) we have gG_ (0, z) _< u(z) = Izi I

for all z = (zi, z2) E G_ with 1221 < @Ia I. So it remains to prove that gG._(0.z)

u(z)

(4.2.18)

for all z = (zi, z2) E G_ with

tIziI)

or la
In the former case one can apply the maximum principle (cf. the proof of (b)) to the function E C: iX { <

Iz°I + e)

A -> gG_ (0, (A, zz)).

The latter case is more complicated. It is sufficient to prove that (4.2.18) holds for a < Izi I < I and Iz21 < Izi Imo. Then one can apply the Three Circles Theorem to the function

(AEC:Iz°I1". -e
gG (0.(z,°,A))

(a0,ssmall). Take r, s E N with r/s > log O/ log a and consider the function

EDA -+ gG_(0,(A',A')) Then by the Schwarz Lemma for subharmonic functions

9G- (0, (A', A')) < IAI' = u(A', A'),

A E E.

IV. Contractible systems

130

Hence by the maximum principle we get gc_ (0, z) _< Izt I = u(z)

for all a < Izi I < I and Iz21 _< Izt I'Js We conclude the proof by takings O

Properties of SG: Proposition 4.2.9 (cf. [Kli 2]). Sc, (a; X)

= sup([(Cu)(a; X)){ : u: G -+ [0, 1), u log psh, u(a) = 0, and u C2 near a),

In particular, So (a; ) is a seminorm for any a E G E 6. Proof. It suffices to prove that if u is as above, then Urn sup

I

IZu(a+AX) = (Cu)(a;X).

Fix u and put v(A) := u(a + AX). Since v'(0) = 0, we obtain z

I

z

z

v(A) = v(x, Y) = 2 axe (0)x2 + axay (0)xY + 2 aye (0)y2 + o(IAIZ) P(x, Y) + o(IAI2) in a neighborhood of 0 E C.

We already know that u (z) < M 11z - a II2 for z near a. Hence the function A 2+ 71 v(A) (defined near 0) extends to a subharmonic function at 0 with 0 v`(0) := lim supa.,,,o v(A). In particular, by the Oka Theorem (cf. Appendix PSH 24) we conclude that lim sup, o+ v(tx, ty) is independent of (x, y). Consequently, a" (0) _ (0) and - may (0) = 0. the function Tpr(+-,YYT) is constant, and therefore Finally, lim soup

I

Iz

v(X) = 4(i v)(0) = (Cu) (a; X).

O

Example 4.2.10 (cf. [Jar-Pfl 6]). Let S` denote the upper semicontinuous regularization of SC. We show that the system (SG)GIO, is not holamorphically con-

4.2 Extremal problems for plurisubharmonic functions

131

tractible. In particular, there are domains of holomorphy G for which SG is not upper semicontinuous: see also Exercise 4.1. Let 00

kklogl

(PR' 11) k

_

I

-ak2+1n1),

rl)EC2.

I

where (ak)k_l C E. is a dense subset of E and Ak > 0, k > 1, are such that (p(0, 0) > -00, V E C2(C x C.) fl PSH(C x Q. Define G := {(ZI. z2) E C2: Izi lew(:2.o) < I), 1), D :_ {(ZI, Z2, Z3) E C3: Izi

G9(ZI,z2)-F*(ZI .Z2.0)ED. Note that G and D are Hartogs domains of holomorphy. Take (z°, z2) E Gfl(C x E)

and let u : G - [0, 1) be a log-psh function such that u (z', z2) = 0 and u is C2 near (z°, z2). Since C x {ak} C G, k > 1, we get u(zi, ak) = const(k), Zi E C, k > 1. Hence, since {0) xC C G, we have u(zi, ak) = const, Zi E C, k > 1. Recall that (ak)k1 is dense in E and u is C2 near (z°, z2). Therefore, u - const = 0 near (z°, zZ). This shows that SG = 0 in (G fl (C x E)) X C2, and consequently SS =0on (G fl (C x E)) X C2. is a log-psh On the other hand, the function D 3o (zi, z2, Z3) - Izi function on D which is C2 near any point (0, 0, 1) E D, t > 0. Hence

SD((0,0,t);(1,0,0))

=ew(o.r) > ew(o.o)

> 0.

Finally, we obtain that

SD(F(0,0);F'(0,0)(1,0)) = So((0,0.0);(1,0,0)) > limsupSD((0,0,1);(l,0,0)) > e"(o.o) > 0 =

The above example shows that there are contractible families of pseudometrics which are not upper semicontinuous. Moreover, if one forces these pseudometrics to be upper semicontinuous, then the new system is not longer holomorphically contractible. F?] It is an open question what are good sufficient conditions on domains of holomorphy G for SG to be upper semicontinuous.

IV. Contractible systems

132

Properties of mG ) and ryG ) Proposition 4.2.11. (a) For any a, z E G, and k E N there exists f E O(G, E) with orda f > k such that I f (z) I F = mG (a, z) (f is called extremal for mGk (a, z)). is continuous and it belongs to ICG(a). (b) The function (c) The function mG is upper semicontinuous. (d) If G E 6b, then mGk) is continuous.

(e) If G / G, then mck) \ me l Note that for k > 2 the function m(k) is in general neither symmetric nor continuous; cf. Proposition 4.4.1. Proof. (a) Use Montel's argument. (b) The family (If 1: f E O(G, E), ordp f > k) is equicontinuous; cf the proof of Proposition 2.4.1(a). (c) Let G 31 z,, -- Zo E G, G 3 w -> wo E G, and let f be an extremal v > 1. We can find a subsequence f,,,,=:f*fo E O(G, E) function for mkl (z, , (use the Montel argument). Clearly, ord;,, fo > k (hence fo: G -+ E), and consequently m()(zo, WO)

> Ifo(wo)Ir = 14-,00 lim

lim m(k)(zV,,, wIj. G

This proves that limsupiz.Wh(Z,WO) mGki(z, w) = mGl(zo, wo).

(d) We may assume that G is bounded. Fix zo, wo E G and let fo be extremal for

M(k)

(z(" wo). Define

Da fo(z)(w - Z)a, f (w) = fo(w) - V 1! a!

Z, W E G.

lal
Then f E HI(G) and ordz f > k. Let M(z) := max{1, IIf IIG}.

Since ord; fo > k, M(z) -+ I as z

zo. Put f := f /M(z). Then

m(k,(z. w) 2: IfZ(w)I{.

4.2 Extremal problems for plurisubharmonic functions

133

and therefore

liminf (z.w)-. (;o.wo)

mGk)(z, w) ? Ifo(wo)Ir =

mGk)(Zo. WO).

(e) Use the Montel argument; cf. Proposition 2.5.1(a).

Proposition 4.2.12. (a) yc)(a; X) = SUP( If(k)(a)XI1: f E O(G, E), order f k), where f (k) (a)X =

f(k)(a)X

E

J

Ia II=k

Da f(a)X' a!

(note that f(k)(a): C" -- C is a homogeneous polynomial of degree k). (b) Y (k) (a; X) = 1imx...o Iai mGk) (a. a + X X ).

(c) If G E 6b, then

yd (a; X) =

lim

Ilz' - z"II

z T,

aEG.IIXII=1.

(d) For any a E G, X E C" there exists f E O(G, E) with order f > k such that If(k)(a)XIF = y(k)(a;X). (e) The function yy (a; ) is continuous and log-psh on C". k

(f) The function yy is upper semicontinuous. (g) If G E 6b, then y(k6) is continuous.

(h) If G / G, then yc.(k )

\ Yc ) (k

Note that for k > 2 the function

Y6(A)

need not be continuous in general; cf.

Proposition 4.4.1.

Proof. (a) If f E O(G, E) and order f > k, then f (a+AX) = Ak f(k)(a)X+o(IAlk)

asI -+ 0. (b) By (a) we have I

Yc) (a; X) < lim inf -MG (a, a + kX ). A -0

IX I

IV. Contractible systems

134

Take 0 0 k,, -+ 0 and let f,, be extremal for mGkl (a, a + X, X), v > 1. By the Montel argument we get f , = fo. Hence ,,,

k)(

Y

a.X) > Ifo(k)(a)XIt = lim If,,,(a+x,,,,X)Ir µ-.+00 Ix", I 1

= limo Ill ImGi(a,a +A,,,,X). (c) As in (b) one can prove that lim sup --

N--,d

a

Il

W, ZII) < yck) (a. X).

X

By the methods analogous to those of the proof of Proposition 4.2.11(d), we can easily show that lim inf > y(6k) (a; X). (d), (h) Use the Montel argument. (e) The family of functions C" D X f(k)(a)X, f E O(G, E), orda f > k, is

equicontinuous on V. (f), (g) Cf. the proofs of Proposition 4.2.11 (c),(d).

O (k)

We end this section with a list of relations between m(k) G . Sc, kc and yc

,

AG,

xG for G C C'. Recall that if G C C', then mC1) YG

= c. 5m (ck) <sc = gc
Moreover, if G is biholomorphic to E, then cc

=MG(k)

= Sc =

Yc=Yc) =Ac=xc Proposition 4.2.13. (a) Let G C C' be a taut domain (i.e. a hyperbolic domain). Then the following conditions are equivalent. (i) There exist k E N and za, zo E G, zo 36 zo, such that

mck)(4, zo) = ka(zo, za).

(11W Lxtrcmal problems for plurl5uhharmonic functions

I i1

(ii) There exist k E N and zo E G such that YG

(iii) There exist z, za E G, -o

-0;1)=XG(-Lo,0.

z

,

such that

=ka(4zo) (iv) There exists zn E G such that AG(zo; l) = xG(zo; 1).

(v) G is biholomorphic with E. (b) Let G C C I be a domain biholomorphic to a bounded domain that is regular with respect to the Dirichlet problem; cf. Appendix GR. Then the following conditions are equivalent. (i) There exist z . 14 E G. 1-0 z. such that

cc; (zo, zO) = 9G(4, z'). (ii) There exists zq E G such that

Yc(o; 1) = AG(..o;l). (iii) G is biholomorphic with E.

Remark 4.2.14. (a) If G C C' is not taut, then assertion (a) in Proposition 4.2.13

is not true, e.g. G = C (c = k* = 0). (b) If G C C' is not regular with respect to the Dirichlet problem, then Proposition 4.2.13(b) is false, e.g. G = E. (c*. = 9E. = CE j E, x E, ).

(c) If G = P = {z E C: 11R < 1z1 < R) (note that P is taut and regular with E P. respect to the Dirichlet problem), then for k > 2 there are points zo', such that mr z) = gp(z,, z) and 1) = AG(z0; 1); cf. zo Proposition 5.9(c). (d) Observe that

logk;(a, .) is psh

if

gG(a, ) = k*(a, ).

Hence by Proposition 4.2.13 if G C C' is a multiconnected taut domain, then for any a E G the function logku(a, ) is not subharmonic; cf. Proposition 2.5.1(b).

IV. Contractible systems

136

Proof of Proposition 4.2.13. (a) The only difficulty is to prove that (iii) -> (v) and (iv) --* (v). Suppose that (iii) (resp. (iv)) is fulfilled and let V E O(E, G) be such that p(0) = zo and sp(a) = zo', where a := k* (z', z") (resp. (p(0) = zo and xG(4;1)rp'(0) = 1) ; cf. Propositions 3.2.4 and 3.5.13. We will prove that V : E -* G is biholomorphic. Let A E E,.

u(A)

Then u is subharmonic and u < 1 (by the holomorphic contractibility of 9G)- Put

u(0) := limsupu(A) (= AG(za;(p (0)) ).

Clearly, u is subharmonic on the whole E. Since u (a) = I (resp. u (O) = 1), we get u - 1. Let Xo E E,. Define w(-k))

,

X E E \ (,Lo}.

M (Ao, A)

Then, as above, v is subharmonic and v < 1. Since gG is symmetric (n = 1!), we get v(0) = u(Ao) = 1. Thus v - 1, and therefore gG(go(A'), sv(A")) = m(A',

A"),

A', 1X" E E.

This implies that gp is injective and proper. Consequently, (P is biholomorphic.

(b) Without loss of generality we may assume that G itself is bounded and regular with respect to the Dirichlet problem. In particular, 9G (Z', Z") > CG W, Z") > 0,

Z', Z" E G, z' # Z",

AG(z';1) YG(z';1)>0, z'EG, `lim gG(Z', Z) = 1, z' E G.

Nontrivial implications are (i) -+ (iii) and (ii) -+ (iii). Assume that (i) (resp. (ii)) is satisfied and let f E O(G, E) denote an extremal function for c*G(zo, zo') (resp. yG(zo;1)), i.e. f (zo) = 0, 1 f (4)I = c*(za, zo) (resp f (4) = 0, If'(zo)I =

4.2 Extremal problems for plurisubharmonic functions

137

YG(zo; 1)). We will prove that f : G -+ E is biholomorphic. Put

If(z)I

u(z)

z E G \ {zo},

gG(za, z)'

If'(zo)I u(zo) := limsupu(z) = AG(zo;1) Z,..4

Then u is subharmonic (since loggG(zo, ) is harmonic in G \ {zo}) and u < 1. Since u(za') = 1 (resp. u(zo) = 1), we have u - 1. Take Zo E G \ (zo) and put v(z) .= m(f(zo), f(z)),

z E G \ {zp}.

gG(zo, z)

Then v is subharmonic, v < 1, and v(za) = u(zo) = 1. Hence v - 1, and finally

m(f (z'),

f(z )) = gG(z', z"),

z', Z" E G.

In particular, f is injective and proper, and therefore f is biholomorphic. Remark 4.2.15 (cf. [Pol 2], [Pol-Sha]). For any domain G C C" define

gG(a, z) :=

inf ,pEO(E.G)

J

IAIaa,(w-a)

a, z E G.

Ixiordx(jp-a)

a, z E G.

AE -I a

aElp(E)

Then the following properties are true:

(a)

gG(a, z) =

inf yoEO(E.G)

W(0)=z

11

{ X-ap-'(a)

aErp(E)

(b) gG < gG for any domain G; (c) the system (gG)GEe is an m-contractible family of functions; (d) for any G and for any a E G the function gG(a, ) is upper semicontinuous. Ad.(a) Fix G, a E G, and zo E G, a 36 zo. Suppose that gG (a, zo) < C' < C. Let rp E O(E, G) be such that tp(O) = zo, a E V (E), and rl I,1j Iki < C', where (A, )j denote all zeros of (p-a and k j := orda, (9p-a). We may assume that IJ1; I < IA;+1 I Take N such that jj7 , Ix; Ik, < C'. Choose R E ( IAN I, 1) sufficiently near to I.

IV. Contractible systems

138

Define coR(A) := cp(RA). Observe that 'PR E O(E, G), SoR(0) = zo, and a E 9pR(E).

Because VR(Aj/R) = a for IAjI < R, we get

17 (ICJ II )kI < CSRjAjj
R

< C.

Since C is arbitrary, we are done.

Ad.(b) Fix G and a, zo E G, zo 0 a. Let u := ga(a, ). Recall (cf. Lemma 4.2.3) that u E 1Ca(a). In particular, u is log-psh on G and

u(z) <- MHz - aM for z near a,

(4.2.19)

where M is a positive constant. Take arbitrary tP E O(E, G) with cp(0) = zo and a E Sp(E). Let X j, j = 1, ... , N, denote the solutions in E of the equation 9,(A) = a counted with multiplicities. Define f := hA...... hA,r, where hAO(A) :=

(A-Ao)/(1-ioA). Put v := uo9/lf I. It is clear that v is sh on E\111_., AN) and v is locally bounded on E (use (4.2.19)). Consequently, v extends subharmonically

to E. Since lim sup,A,, t v(A) < 1, the maximum principle shows that v < 1. In particular, u(zo) = u((p(0)) < 1 f (0) 1 = fNt 11j 1. Now, by (a), 9G (a, zo) 5 gG(a, zo) Ad.(c) Directly from the definition it follows that the system (gG is holomorphically contractible. Observe that if G = E, a, zo E E, then the function rp = h-Z0 is a competitor in the infimum which defines gE(a, zo). Hence, by (b), m(a, zo) = 9E(a, zo) < 8E (a, zo) <_ Ih-1(a)I = m(a, zo). Ad.(d) Let zo E G be fixed. Because of (c), the function SG is continuous on the diagonal, and therefore we may assume that zo a. Take C > ga(a, zo) and let (p E O(E, G) be such that 9(0) = zo, a r= Q(E), and of , jAj I < C, where A1, ... , AN denote the zeros of 9p - a in E with multiplicities. Define N

(PAA) :=Sp(A)+(z-zo)fl(1 -A/Aj), A E E. j=1

Then 92(E) c G for z near zo, and therefore (pz is a competitor in (a). Thus 8a (a, z) <- C for z near zo.

Notice that g`a(a, ) E kc;(a) under the assumption that ga(a, ) is log-psh on G (use (c) and (d)). Then gG(a, ) < g6(a, ), and finally by (a) we get gG(a, ) gG(a, ). We mention that for arbitrary G and a E G the plurisubharmonicity of logic;

has been claimed in [Pol-Sha] and [Pol 21.

4.3 Inner pseudodistances. Integrated forms. Derivatives. Buseman pseudometrics

139

U Inner pseudodis!anees. Integrated forms. Derivatives. Buseman pseudometrics. C I -pseudodistances Roughly speaking, the aim of this section is to define abstract versions of the following operations: cG->c'G

(cf. § 2.5),

YG -> f yG (cf.§2.5), YG

(cf. § 2.5),

xG -: xG

(cf. § 3.6).

CG

Fix a domain G C CO. Denote by D(G) the family of all pseudodistances d : G x G -+ R+ such that Va E G 3M, r > 0: d(z'. z") < Mllz' - z" 11,

z', z" E B(a, r) C G.

(4.3.1)

Note that if d = (dG)GEOo is a contractible family of pseudodistances, then for each

G E 00 the pseudodistance dG belongs to D(G) (cf. Remark 4.1.5(c)). Observe that if d E D(G), then d is continuous on G x G. Moreover, if d E D(G), then tanhd E D(G). Let 9Jt(G) denote the family of all upper semicontinuous pseudometrics S: G x C" -+ 1[t+, i.e. S(z; ),X) = IAIS(z; X), Z E C. .l E C, X E CO. Observe that y, 1, MG, xG E 93(G); cf. Propositions 4.2.12(f), 3.6.2, and 3.5.12, respectively. Note that if S E M(G), then

VK CC G 3M > 0: S(z;X) < MIIXII. We will define four operators:

D(G)

D(G)

f

E K,X E CO.

(4.3.2)

9Dt(G)

9R(G)

Operator d - d*. Let d E D(G) and let a: [0, IJ -+ G be a continuous curve. Put N

d(a(tj_i),a(tj)): N E N,0 = to < --- < tN = 11.

Ld(a) := sup( i=1

IV. Contractible systems

140

The number Le(a) E [0, +oo] is called the d-length of a. If Ld(a) < +oo, then we say that a is d-rectifiable. Define

d` (z', z") := inf(Ld(a) : a is a II II-rectifiable curve in G joining z' and z"), z ', Z " E G.

Clearly, d < d'. Proposition 4.3.1. Let d E D(G). (a) If a is a 11 11 -rectifiable curve, then it is d-rectiflable.

(b) d' : G x G -+ R+ is a pseudodistance satisfying (4.3.1), i.e. d' E 0(G). (c) Ld, = Ld; in particular, (d')' = d'. (d) Any d'-ball Bdi (a, r) is arcwise connected in top G; in particular, if d' is a distance, then top d' = top G. (e) Ld = Lthd; in particular, d' = (tanhd)'. We say that d' is the inner pseudodistance for d. A pseudodistance d is said to

be inner if d = di. Proof (a) Cf. the proof of Lemma 2.5.2(a). (b) By (a), d' is finite. It is clear that d' is a pseudodistance. It remains to verify (4.3. 1) for d'. Fix a E G and let M, r be as in (4.3.1). Then for any z', z" E B(a, r) we have

d'(z', z"):5 Ld([Z', Z"]):5 MLt 1([z', z"]) = Mlli - z"ll. (c) Cf. Remark 2.5.3(c). (d) Cf. Remark 2.5.5. (e) Cf. the proof of Lemma 2.5.2(a).

0

Operator 6 -- f 6. Let 8 E JDR(G) and let a : [0,1 [ - * G be a piecewise C1-curve. Put I

La(a) := f 8(a(t);a'(t))dt. 0

The number La(a) E R+ is called the 8-length of a. Define

(f S)(z', z") := inf(La(a): a: [0, 11 --> G is a piecewise C'-curve joining z'and z"),

z', z" E G.

4.3 Inner pseudodistances. Integrated forms. Derivatives. Buseman pseudometrics

141

Proposition 4.3.2. Let a e wt(G). (a) f d : G x G -- R+ is a pseudodistance with (4.3.1), i.e. f S e 4) (G).

(b) L fd(a) < L6 (a) for any piecewise C1-curve a; in particular, (f S)' = f S. We say that f S is the integrated form of S. Proof. (a) It is clear that f S is a pseudodistance. We only need to verify (4.3.1 ).

Fix a E G and let B(a, r) C G. Let M be as in (4.3.2) with K := B(a, r). Then for any z'. Z" E B (a, r) we have (fS)(z', z") < L8 ((z'. z",1) =

Jo '

-

S(z' + t(z"

z'); z" - z') dt < Mllz' - z'"II

(b) Let a : [0, 1] -> G be a piecewise C' -curve and let 0 = to < ... < In _ Then N

1.

N

E(fS)(a(ti-i),a(ti)) < FLo(ali= La(a).

0

Operator d -> Dd. Let d E 1)(G). Define (Dd)(a; X) := lim sup x-o Z_U

1 d(z, z + AX ),

a E G. X E C".

ICI

Note that according to (4.3.1) the function Dd is finite. Lemma 4.3.3. Let d E 3J (G). (a) For each a E G the function (Dd)(a. ) is a C-seminorm. (b)

(Dd)(a; X) = lim sup A-o :-a

I

d (z, z + AX'),

a E G, X E C",

P'I

d (z' z")

(Dd)(a;X) = limsup "-a Ilz' - z"II

aEG.XEaB,,.

-X R+ is an upper semicontinuous pseudometric, i.e. Dd E JR(G). (c) Dd : G x C" (d) d < f (Dd ). (e) D(f S) < S for any S E 9)t(G).

We say that Dd is the derivative of d.

IV. Contractible systems

142

Proof. (a) Directly from the definition we get

(Dd)(a;).X) = IAI(Dd)(a;X),

A E C,X E C",

(Dd) (a; X' + X") < lim sup ).-0

d(z, z + )X) + lim sup x_»o

IAI

1d (z + )X, z + AX' + AX") IAI

(Dd)(a; X') + (Dd)(a; X"),

a E G, X', X" E C".

(b) Fix a E G and X E C. Let M, r be as in (4.3.1). Then for Z E B(a, r12), IAI < (r/2)/(11XII + (r/2)), and X' E B(X, r/2) we have

d(z. z+AX') < d(z, z+AX)+d(z+AX, z+AX') < d(z, z+AX)+MIAIIIX'-Xll, which implies the first equality in (b). Now, the second equality directly follows from the relation

d z,z

_

Ilz' - z"II

z-z 1d(z,z+AX'), where z=z',A:=-Ilz'-z"II,X':= Ilz'-z"II' 11

IAI

(c) Fix a E G, X E C" and suppose that (MG) (a, X) < A. By (b) there exists rl > 0 such that -

do(z, z + AX') < A,

IAI < rl, z E B(a,, ), X' E B(X, t)).

1XI

Hence (using again (b)) we get

(Ddo)(z, X') < A,

Z E B(a, 17), X' E B(X, rI).

(d) We only need to verify that for any z', z" E G and for any piecewise C1-curve a : [0. 11 -3- G joining z' with z" we have d(z', z") -< 10

(Dd)(a(t);a'(t))dt.

0

Fix z', z" E G. Clearly, it is enough to consider only C'-curves a joining z' and

z". Fix such an a and put cp(t) := d(a(O),a(t)), *(t) := (Dd)(a(t};a'(t)},

4.3 Inner pseudodistances. Integrated forms. Derivatives. Buseman pseudometrics

0

r

143

1. our goal i to ProYC Mat i(1) t fo r(t) dt. It suhces to On that P(tr) - (!ri) I <

lim sup r',r"-r r, #r

*(I),

t E [0, 1)

(then rp is a Lipschitz function and tp' < fr a.e. on [0, 11). Since a is of class C', we can write

a(t") = a(t') + (t" - t')X (t', t"),

t', t" E 10, 1),

where

lim X(t', t") = a'(t),

'X1 r, #r..

t E [0, 1].

Fix to E [0, 11. Then by (b) we get

limsup I cP(t') r'.r"-.rr,

-

I

r' - t"

< lim sup dG(a(t'), a(t")) r'.111,10 It' t"I

r'or"

-

r'Ar`

= limsup

dc(a(t'), a(t') + (t" - t')X(t', t")) <

to

t'.r"pro

(e) Fix aEGandXEC".Then (D(fS))(a:X)=limsup A-o


I

(fS)(z,z+XX)
ICI

L IxI

1

La([z,z+AX))

ICI

S(z+tAX;AX)dt o

lim sup f S(z + tAX ; X) dt (use the Fatou Lemma)

za j [lim sup S(z +

X )[ dt (use the semicontinuity of

)

S(a; X).

0

IV. Contractible systems

144

Operator 6

Suppose that h : C" -> R+ is a function such that h(AX) = IAIh(X), h(X) < MII X II,

A E C, X E C", X E C",

where M > 0 is a constant. Put

1'(h) :_ (Y E C": IY ZI < h(Z), Z E C"}, where "

Y.Z=EY1Zj =(Y,Z), Y=(Yt,...,Y"),Z=(Zi,...,Z") i=1

Note that r(h) c B(0, M). Define

Y Er(h)),

X EC".

Remark 43A. (a) h < h; (b) h is a C-seminorm; (c) I'(h) = l: (h); in particular, (h) = h;

(d) his a C-seminorm if h = h; (e) (h < 1) is the closed absolutely convex hull of (h < 1); cf. Remark 3.6.1(d). Now let S E W1(G). We define S(Z; X) :_

Z E G, X E C".

Note that the definition of S extends to all pseudometrics S : G x C" --> R+ such that for any a E G there exists a constant M > 0 such that S(a; X) < M11 X II, X E C".

Proposition 4.3.5. Let S E WI(G). (a) S is an upper semicontinuous pseudometric, i.e. S E TT(G).

(b) f S = f S. We say that h is the Buseman pseudometric for S; cf. [Bus-May].

4.3 Inner pseudodistances. Integrated forms. Derivatives. Buseman pseudometrics

145

Proof. (a) Assume that G 3 a -> ao E G, C" 3 X - Xo E C" are such that S(a,,,

A. Fix Y, E r(S(a,; .)) with

IX,,.

V> 1. V

By virtue of (4.3.2) we may assume that (for a constant M > 0) we have

S(a,,;X) <MIIXII,

X EC",v> I.

Hence II Y,, II < M for v > 1, and therefore we may assume Y -+ Yo E C". Then for any Z E C" we have I Yo

ZI = lim

I Y,,

ZI < lim sup S(a,,; Z) < S(ao; Z).

Thus Yo E r(S(ao; )), and therefore A = IXo

Yo1 < S(ao; Xo).

(b) The inequality "<" is obvious. The inequality ">" follows from Proposition 4.3.3. For by Proposition 4.3.3(e) we have D(f S) < S. Hence using Proposition 4.3.3(a) we obtain D f S < S. Finally, by Proposition 4.3.3(d) we get

f S > fD(f3)> fS. Remark 4.3.6. Let us summarize the relations between the operators and "' acting on the families V(G) and TZ(G).

', f

,

D,

o f = f (Prop. 4.3.2(b)), D (Prop. 4.3.3(d)(e)), (6) D o f <" (Prop. 4.3.3(a)(e)), (2) V o (3) f o D > ' (Prop. 4.3.3(d)), (7) f o = f (Prop. 4.3.5(b)), (4) "' o D = D (Prop. 4.3.3(a)), (8) - o =" (Remark 4.3.4(c)). (1) ' o

' (Prop. 4.3.1(c)),

(5) '

Remark 43.7. (a) In general, the inequality in Remark 4.3.6(6) may be strict. For by Example 3.5.10 there exists a pseudoconvex Hartogs domain Go C C2 such that kG,, = 0 and xao # 0; recall that kG = f xa (cf. Theorem 3.6.4). In particular, D(f ma.) = Dkaa =_ 0. Thus D(f XGo) icao.

(b) The same is true for the inequality in Remark 4.3.6(3). For example take G = C, d(z', z") := I Re z'- Re z" I, z', z" E C (it is clear that (4.3.1) is satisfied). Then D(a; X) = IXI, a, X E C. Hence (f Dd)(z'. z") = Iz' - z"I, z'. Z" E C. In particular, (f Pd) (0, i) = 1. On the other hand, d' (0, i) = Ld ([0, i J) = 0.

IV. Contractible systems

146

We say that d is a C' -pseudodistance if

VKCCG, Ve>03q>0: Id(z

z") .

- (Dd)(z; z -

I

e II

z,

-

z

z E B(z, q)',

cf. [Rei 1,2].

Remark 43.8. One can easily prove (cf the proof of Proposition 4.3.3(b)) that for d E I)(G) the following conditions are equivalent: (i) d is a C'-pseudodistance; (ii) Dd is continuous on G x C" and (Dd)(a; X) = lim thd(z. z + AX'), ),_0 1XI

a E G, X E C'.

Zya

X'- X

(iii) Dd is continuous and

(a; X) =

d(z', z") Ilz' - z"II'

lim

aEG,XEdP,,.

X

Proposition 4.3.9. Let d be a C' -pseudodistance. Then (a) Ld (a) = Lad (a) for any piecewise C' -curve a : [0, 11 -+ G; (b) d' = f (Dd) (cf. Remark 4.3.7(b)); (c) d' is a C'-pseudodistance (recall that Dd' = Dd, cf. Remark 4.3.6(2)). Proof. Use the same methods as in the proofs of Lemma 2.5.2(c)(d) and Proposition 2.5.4.

Having finished the general constructions we come back to the case of contractible families.

Theorem 4.3.10. (a) If d = (dG )GEao is a contractible family of pseudodistances, then the family d' := (dG)GEmo is also a contractible family of pseudodistances. (b) If S = (SG)GE0o is a contractible family of pseudometrics such that for each (f SG)GEmo is G E t the pseudometric SG is upper semicontinuous, then f a contractible family of pseudodistances. (c) If S = (8G)GE0o is a contractible family of pseudometrics, then the family 8 (SG )GEtlo is also a contractible family of pseudometrics.

4.3 Inner pseudodistances. Integrated forms. Derivatives. Buseman pseudometrics

1 47

(d) If 4 = (d(;)GEOo is a contractible family of pseudodistances such that for each G E e5o the pseudodistance dG is of class C', then the family Dd := (Ddri,)GEI" is a contractible family of pseudometrics. (e) CG and c`G are C I -pseudodistances; DcG = DcG = yG. (f) DkG <

Observe that (f) with Remark 4.3.6(3)(7) give an alternative proof of Theorem 3.6.4. Note that, in general, we do not have equality in (f); cf. Remark 4.3.7(a).

0 We do not know whether the family Dd is contractible for an arbitrary contractible family d of pseudodistances: even the case d = k is open. 0 Proof. (a),(b), and (c) directly follow from the definitions. (d) Let F : G -+ D be a holomorphic mapping. By Remark 4.3.8 for any a E G and X E C" we have

(DdD)(F(a); F'(a)X) _ slim

I dD(F(a), F(a) + tF'(a)X)

0+ 1

lim

lim Idc;(a,a+IX)=(DdG)(a;X). IdD(F(a),F(a+tX))< - 1-0+ t t

(e) See Propositions 2.5.1(d) and 2.5.4. (f) (Cf. the proof of Theorem 3.6.4.) According to Proposition 4.3.3(a) we only

need to prove that DkG < xG. Fix a E G and X E V. Let (p: E -* G be a holomorphic mapping with (p(0) = a, (rp'(0) = X for some a E (0, 1). It suffices to show that (DkG)(a, X) < a. Let M. r be as in (4.3.1) with d = kG. For any z E G define

Wz().) := v(A) + (1 - X) (z - a),

A E E.

Clearly V, (O) = z. Since II Vz - w II e < 2111- -all, we may assume that Vz : E -+

G, Z E B(a, r) C G. Let 0 < e < (r/2)/11X11 be chosen in such a way that

*) Added in proof: see also M.-Y. Pang, On infinitesimal behavior of the Kobayashi distance, preprint (1993).

IV. Contractible systems

148

92(A) E B(a, r/2) for any IAI < e. Then for Z E B(a, r/4) and IXI < t we get kG (Z, z + AX) < kG(z, (Pz(0A)) +kG((Pz(QA), z + AX)

< kG(z,tpz(a),)) + MIIrpz(ax) -z - xxll


(DkG)(a;X)
I 1XI

0 As an application of the above results we present the following proposition (clarifying Remark 3.1.11 and Example 3.6.7).

Proposition 4,3.11 (cf. [Ven 3]). Let G = Gh C C be a balanced domain with Minkowski function h. Then (a) h = yG(O: ) = (DkG)(0;') = kG(0; ), cf. Proposition 2.2.1(4): (b) if G is pseudoconvex, then for any a E G the f ollowing conditions are

(i) h(a) = h(a), (ii) YG(O;a) = h(a),

(iii) c*(0;a) = h(a), (iv) .G(O:a) _)CG(O;a), (v) kG(0;a) = kG(0, a). Proof. (a) By Proposition 2.2.1(d), Theorem 4.3.10(1), and the proof of Proposition 3.5.3 we get

h = YG(O; ) < (DkG)(0; ) < kG(O;') < xG(0; ) = h.

Since kG(0, ) is a seminorm, we obtain xG(0; ) < h, which completes the proof of (a).

(b) By Propositions 2.2.1(b)(d), 3.1.10, 3.5.3, and (a), we only need to prove

that h(a) = h(a) if k*(0,a) = h(a). By Remark 3.2.5(b), if k*(0.a) = h(a), then k* (0, ta) = th(a), 0 < t < 1. Hence (DkG)(0;a) > h(a), which by (a) gives

h(a) = h(a).

0

4.4 Example - elementary n-circled domains

149

4.4 Example - elementary n-circled domains Let

G := (z E C": Iz" I < I),

where a = (a1, ... , a,,) E N", a1, ... , a are relatively prime (n > 2); cf. Propo-

sition 2.7.2. Put 4)(z) := z' and r(a) := ord Go := ((zi, ..

z,,) E G: zi

,

E+(t) :=

.. z 36 0). Fort > 0 let

4)(a)]. Moreover, we define

ft

if tEN

I + the integer part oft

if t 0 N

Proposition 4.4.1. Let a E G and let r := r(a). Then for k E N we have ni

(t') a. Yd (a;

X) = j [yE(4'(a); 4'(r)(a)X)]'

l0

where 4'(r)(a)X

z E G,

(a, z) = [m(4 (a), (D(z))]f E+0),

if rlk otherwise '

X E 0,

(4.4.1)

(4.4.2)

I3I=r

Remarks. If k = 1, then the assertion reduces to Proposition 2.7.2(b). If r(a) = I (e.g. a c- Go), then mG)(a, z) = m(I(a). c(z)).

Z E G, k > 1,

y(a; G ) X) = yE(ID (a);'V (a)X ),

X E Cr. k > 1.

Fix k > 2 and b E Go. By (4.4.1) we have ID(b)IIE-(A

m(rk)(0, b)

=

)

>

m(')(b, 0).

This shows that the function mC) (k > 2) is not symmetric. If D is a subdomain

of G (with 0, b E D), then by Proposition 4.2.11(e) for any k > 2 we have m((k)(0, b) > mn)(b, 0) provided that D is sufficiently close to G. Thus there exist

"very regular" domains in C" (e.g. bounded Reinhardt domains with real analytic boundary) such that in p) is not symmetric (k > 2). (for k>2) Let Go m )(a,,, b) = m(4,(aL), `1(b)) - I4;(b)I <

M(k)

(0, b).

150

IV. Contractible systems

Consequently, mg is not continuous in the first variable; cf. Proposition 4.2.11. Moreover, for X E C" we get lim yGiai)(a;X) Goaa-0

=

(a)X) = IV(0)XI.

lira

Go3a-+0

On the other hand, yylal)(O;X) = IX'I'1lal Comparing the zero sets of the functions X -+ 4'(0)X and X -> X", we see that there exists Xo such that I4'(0)XoI 0 IXo Ian"I. Therefore, the function y cf. Proposition 4.2.12.

( . ; Xo) is not continuous at 0;

Proof of Proposition 4.4.1. Let

f (z)

-t (a) 1 - I(z) (_4(z)

E+(;)

ZEG.

I(a)

Then f E O(G, E) and orda f = rE+(k/r) > k. Hence mGk)(a, z) > If z E G, which gives the inequality ">" (in (4.4.1)). Similarly, Y6 )(a; X) >

(Z)I1/k,

I f(k) (a) X I Ilk

which is the right hand side of (4.4.2). Now let f E O(G, E) with orda f > k be arbitrary. Then f = (p o I (cf. §2.7), where cp E O(E, E) and orda f = rord,.(a)(p > k. Hence ord4,(a)W > E+(k), and therefore If (z)I ilk < [m(4) (a), t(z))JFE+(%) (m = m(E), l > 1). The proof of (4.4.1) is finished.

To prove (4.4.2) observe that if r t k, then orda f > k and so f(k)(a)X = 0. If k = !r, then f(k)(a)X = rp(!)(4 which directly implies the required result.

Proposition 4.4.2. Let a E G, r := r(a). Then 9G (a, z) = [m(t(a), 4 (z))I AG(a; X) = [YE(1D(a);

,

z E G,

X E C".

Remarks. 9G (a, ) = mG)(a, ) iff rIk iff AG(a; ) = y( ) (a; ). In particular, the equalities hold for a E Go.

If b E Go, then 9G(0, b) = I4 (b)I"/I'I > gG(b, 0). Thus gG is not symmetric and, by the same methods as in the case of the Mtibius functions, we can find "very regular" subdomains D of G for which go is not symmetric (cf. Proposition 4.2.7). An argument as in the remarks after Proposition 4.4.1 shows that the functions gG and AG are not continuous in the first variable.

4.4 Example - elementary n-circled domains

151

Proof of Proposition 4.4.2. Proposition 4.4.1 gives the inequalities ">". The converse inequalities are immediate consequences of the following lemma.

Lemma 4.43. For any u E ICc (a) there exists a function v : E - (0, 1) such that u =VO4)andV' E ICE(4)(a)).

Proof. ForA.EEletVA:=(zEC": z°

A). Fix A E and let J2 be a fixed

an-th root of X. Then the mapping -n, ... °,. ,...,wn-1,µw1

iy

1

is a holomorphic surjection of (C.)r-1 onto Va (an exercise). The psh function u o %P extends to a bounded psh function on C"-I which implies (cf. Appendix

PSH 25) that u o' = const and hence ulva = const =: v(A). It is clear that u = const =: v(0) on Vo. Thus we have constructed a function v : E --> (0, I) such that u = v o 4). Now, we prove that V' E ICE(4)(a)). For we take ko = 4)(b) E E. and we observe that

ub

v (k)

'k

b

b",,-' n-I

b(" 1

for ), near A

where the branch of the a,,-th root is chosen in such a way that (bn' )1/° = b,,. This shows that v is log-subharmonic near Ao, and consequently in E.. Moreover,

if 4)(a) $ 0, then taking b = a we get A

V0.)

MI(

a,, 1 < MIIJ. - 4) (a) I for A near 4) (a).

a1

1

-

It remains to prove that v is log-subharmonic at 0 E E (since v is bounded, it suffices to show that v(0) = lim supA_o v(A), cf. Appendix PSH 21) and that V' E ICE(O) in the case where 4)(a) = 0. We have

limsup v(A) > limsupv(4)(z)) = limsupu(z) = u(0) = v(0). A-0

Coat-0

Z-0

On the other hand,

limsupv(A) = limsup u(Ar,...'X ) < limsupu(z) = u(0). A-0

A-.0

z-»0

IV. Contractible systems

152

Now suppose that 4 ?(a) = 0. We may assume that at, ... , a., # 0, as+i = = a = 0, 0 <s
v(A)=u(a''

...,a,,(

ac"

a«,) s

I

(a°'

...

a,)')<M,III

"

s

Proposition 4.4.4. sG(a, z) =

SG(a;X) =

( gG(a, z)

if g'(a, ) is C2 near a

I0

otherwise

AG (a; X) if g2 (a. -) is C2 near a 0

otherwise

Remarks. The function gG (a,.) is C'- near a iff #(j: aj = 0) < 1 (in particular, 92 (a, ) is of class C2 near a if a E Go). Proof of Proposition 4.4.4. In view of Proposition 4.4.2, we only need to show that

ifa,,...,a,

0O,a.,+,=...=a"=0,0<s
q

Recall that q is a C-seminorm and (by Proposition 4.4.2)

q(X) < AG(a, X) = la" ... ar,Xa+1'

where r = r(a) = as+I +

Xp^Il =: MIXs+i'

SG(a;0. Xc

+ a,,. In particular,


Notes

Most of the material presented in the chapter may be found in [Din], [Fra-Ves], and [Har]. The complex Green function has been introduced by M. Klimek in [Kli 1], the Azukawa pseudometric by K. Azukawa in [Azu 3], and the Sibony pseudometric by N. Sibony in [Sib 4]. Another definition of a (symmetric) complex

Green function has been proposed by U. Cegrell in [Ceg]. Higher order Mdbius functions and Reiffen pseudometrics were studied in [Jar-Pfl 6,7]. The examples in §4.4 are based on [Jar-Pfl 6,7].

Exercises

153

Exercises 4.1 (cf. Example 4.2.10). Let (4)4-logl

-1) l1kI

,

SEC".

k=2

where kk > 0 are such that V(O) > -oo. Define G = {(Zl, Z2) E

Prove that limsup,.

0

C2:

IZI

Iew(Z2)

< 1 }.

SG((-t.0); (O. 1)) > SG((O, 0); (0, 1)).

4.2. Let F : G --* D be holomorphic (G C C', D C CI). Let a E G and let r := ord"(F - F(a)) = min{ord"(Fj - Fj(a)) : j = 1, ... , m}. Prove that m(k)(F(a), F(z)) < [mk')(a, z)]',

z E G.

4.3. Let G = Gh be a balanced domain in C" (h is the Minkowski function of G). Fix a E G and k E N. Prove that the following conditions are equivalent (cf. Proposition 2.2.1(b)): (i) m'k)(0, a) = h(a);

(ii) MG (0, ) = h on G fl (Ca); (iii) y(k)(O;a) = h(a); on Ca; (v) there exists a homogeneous polynomial Q of degree k such that h(a) and IQI11k < h. 4.4 (cf. [Pol 21). Let rp E O(E, C2) be defined by (p(A) := (A2 -1/4, A(A2

-1/4)).

Fix a := (0,0) = (p(±l/2) and b := (-1/4.0) = cp(0). Let D C C2 be the

connected component of the set {z E C2: I - 16(z, + 1 /4)'-(z, - 1/4) + 16221 < 11

that contains sp(E). Prove that go(a, b) < 1 /4 < 1/2 <

a).

Hint. Use the log-plurisubharmonic functions u, (z) := 116z,/(15 - 4z, )I and 16(z, + 1/4)2(z, - 1/4) + l6z2I"2. u2(z) 4.5. Prove that f (DkG) = kG.

Chapter V

Contractible functions and metrics for the annulus

For R >

I

let P = P(R) := (z E C: 1/R < Izl < R). Note that for any

0 < r, < r2 < +oo the annulus Q := {z E C: r, < Iz) < r2} is biholomorphic to r2/rI )The aims of this chapter are - to present effective formulas for gp, Ap (cf. Proposition 5.2), mp , y,, (cf. Proposition 5.5), and kP, xp (cf. Proposition 5.7) (recall that kP = tanhkp = tanhkp, cf. Remark 3.3.8(e)), - to study the relations between the above objects (cf. Proposition 5.9), - to characterize c`, (cf. Proposition 5.10), - to prove an analogue of the Schwarz-Pick Lemma for P (cf. Proposition 5.14), - to characterize c- and y- isometrics of P (cf. Proposition 5.15). P(

First observe that P is invariant under rotations, and therefore to calculate dp (resp. Sp) it is enough to describe dp(a, ) (resp. Sp(a;1)) for any a with

1/R
Put q := 1 / R2. For 1 / R < a < R define

f (a, z) = (I - Z)n(a, z). a where

II(a z):=

(li(lazq ?')(1 - zq-)-1)

Remark 5.1 (cf. [Cou-Hil]). (a) The function f (a, ) is meromorphic on C. It has simple poles at z = R4k-2/a and simple zeros at z = aR4k, k E Z. In particular, f (a, ) is holomorphic on P and the only zero of f (a, ) in P is the simple zero

atz=a.

(b) f (a. z) f (a, I/(R2z)) = 1, f (a, z)f (a, R2/z) = R2/a2, and f (a, z) _ f (a, z).

V. Contractible functions and me[ncs for [he. annulus

15:

In particular,

If(a, z)I =

1

if IzI = l/R

R/a

if IzI = R

Lets = s(a) E (0, 1) be such that a = R1-'. Proposition 5.2. For any a E (1/R. R) ve have: If (I

(a)

gp(a,z) =

(b)

Sp(a; l) = Ap(a; l) =

Z E P;; 1

H(a. a)

(Ra)'(°)

a

Proof. (b) is a direct consequence of (a); cf. Lemma 4.2.3. To prove (a) we only need to observe that the function P 9 z - - log(I f (a. z) I / I Rz I' (°)) is the classical Green function with pole at a (cf. Remark 4.2.6). Put

Iks(a) (a) bk(a) := 1&.

ifks(a)EN

1 + the integer part of ks(a) R,-20k(a)-ks(a))

if ks(a) 0 N

k E N.

(5.2a)

(5.2b)

Remark 5.3. (a) 1i(a) = 1, b1(a) = 1/a.

(b) bk(a) E 8P if bk(a) = R iff ks(a) E N. For 0 E R define ek(z) = ek(a.0. z) :=

f (hk(R)

- a)e-":) e

.

[f (a. z)Ik.

z e P.

where f (R, .) := 1.

Remark 5.4. We have ek E 0(P), orda ek ? k (Remark 5.1(a)), and Iekl = l on 8 P (Remark 5.1(b)). In particular. mpk)(a, z) ? lek(z)I'lk.

Z E P.

V. Contractible functions and metrics for the annulus

156

Proposition S.S. For any a E (I/ R, R) we have: (a)

rf(bk(a). -IZI)l (k) mP (a, z) = - IL

J

IRzI''(a)

(b)

Y(k)la+ 1) =

Z E P.

If (a. z)I.

I/k

[f(bk(a), -a) (Ra)IA(a)

Ilk

,

H(a,a) , a

Note that the right hand side of (a) is equal to

k E N.

I ek (a, arg z, z) I i 1k . Moreover,

condition (b) follows from (a); cf. Proposition 4.2.12(b). The proof of Proposition 5.5 will be based on the following

Lemma 5.6 (Robinson's Fundamental Lemma, cf. [Rob]). Let I/R < b < R, 0 E R, c := -eieb, and let cp E O(P \ (c)). Assume that V has at most a simple pole at c and that lim sup.yap IQ(z)I < 1. Then (a) Igp(erex)I < 1, 1/R < x < R; (b) if Igp(erexo)I = I for some 1/R < xo < R, then rp - consL

Proof. Without loss of generality, we may assume that 0 = 0 (c = -b). (a) Fix xo E (1/R, R). We may assume that rp(xo) > 0. Put

4'(z) :=

I [rp(z) + (p(z)],

Z E P \ {C}.

Then 4' E O(P \ (c)) and 4' has at most a simple pole at c. Moreover, 4' satisfies 1 and 4' = Re(p on (P \ {c)) n R. If 4' has no singularity at limsupzyap I4'(z)I c, then obviously I4'(xo)I = V(xo) < 1. Assume now that 41, has a simple pole at

c. Fore>0put

PE:={zEC: 1/R+e
(note that limsupME < 1), E-0

-t,:= -t/ME. Choose e so small that c, xo E PE. Because 4E has a simple pole at c, I4EI < 1 on 8P.., and 4'E((PE \ {c}) n R) C R, we then get R \ [-1, 1] c 4'E((PE \ (c)) n R_). Put ZE(w)

.

1 + 2I

w

fap, 4E

IwI > 1

(5.3)

V. Contractible functions and metrics for the annulus

157

(where the boundar?+ 8 PE is taken with the positive orientation). It is well known

that ZF(w) = #(z E Pe: 4)E(z) = w) (with multiplicities). It is seen that Zt is continuous and Z, (w) -+ I as w - oo. Consequently, Z, - 1 and hence (5.3) implies that 4)F(PE n R..) C [-1, 11. In particular, I4)E(ro)I = (p(xo)/Mf < 1. Letting s -> 0 we get the required result. (b) We may assume that gp(xo) = 1. Let 4) be as in (a). It suffices to prove that

4 - 1 (since Re ip is a harmonic function on P, Re rp - 1, and so (p - 1). If has no singularity at c, then the result is trivial. Suppose that 4) has a simple pole at c. In particular, 4' is an open mapping. Let U be an open neighborhood of xo such that U C P, U nR_ = 0. Since 4)(U) is open, there exist too E (1, +oo) and zo E U such that 4)(zo) = wo. Let PE, M, and 4)E be as in (a). If s is sufficiently small, we have 4E(zo) = wo/ME > 1. Hence, by (5.3) and the fact that ZF - 1, we get zo E R_; a contradiction. Proof of Proposition 5.5. In view of Remark 5.4, it suffices to prove the inequality

Ih(z)I < lek(a, arg z, z)I, z E P, for any h E O(P.E) with ordp h > k. Fix zo = e`BIzoI E P \ (a) and h. Put q := h/ek(a, 8. ). Then (p satisfies all the assumptions of Lemma 5.6, and therefore IV(zo)I < 1.

According to our plans we continue studying the Kobayashi case.

Proposition 5.7. For any a E (1/R. R) we have (a)

r2 + I - 2.r cos(7r (s - t)) kP(a. z) _ [x2 + I - 2x cos(Zr(s + t))]

1/2

'

z E P.

where s := s(a). t := s(lzl) (cf. (5.1)), x := exp (Zi k). 9(z) E arg z n (-Jr, Jr]; xP(a; 1) =

4a(log R) sin(irs(a))

Proof Fix a E (1/R. R). The mapping 1

P-3 z-* z E Q:={zEC: Ra
R

a

is biholomorphic. In particular,

zEP,

and

rp(a;1)_!xQ(l;1).

(5.4)

158

V. Contractible functions and metrics for the annulus

Let ri := 1/Ra, r2 := R/a, and S := (w E C: log r, < Re w < log r2}. Define

a :=

ni

Ao := exp

log r2/r,

27ri log r,

log r2/ri

exp(aw) - I.,

H(w)

w E S.

exp(aw) - Ao

(5.5)

Then H : S -). E is biholomorphic and the mapping h := exp off ' : E -3 Q is a covering of Q with h(O) = 1. Consequently, by Theorem 3.3.7 and Exercise 3.8 we conclude that kQ(1, z) = inf(I),I: A E h-'(z)} = inf(IH(w)I: w E S, exp(w) = z) = IH(log IzI + i8(z))I for z E Q. 9(z) E argz n (-7r, rr) (the details are left to the reader), and

XQ(l;1) = Ih'(0)I

=

H'(0) 1.

0

Now, it suffices to use (5.4) and (5.5).

Corollary 5.8. For any a E (0, 1) we have

(a) (b)

02 + (log IzI - log a)'

kF (a, z) = 102 + (log IzI

j

1/2

loga)2]

0 = 9(z);

1

ME. (a; 1) = - 2a log a '

Proof. Use Propositions 5.7, 3.3.5(a), and Exercise 3.5 or use the covering method.

0 Proposition 5.9. Let a E (1 / R, R). Then (a)

mp)(a, z) = [gp(bk(a), -Izl)]'4kgp(a, z), y k)(a; 1) = [gP(bk(a), -a)] Il kAP(a; 1)

with gp(R, ) - 1.

zE

V. Contractible functions and metrics for the annulus

159

In particular,

cp(a, ) < gp(a. ) in P \ (a), YP(a;1) < Ap(a; 1). (b) For anv k > 2

C ,(a, ) < m p) (a..) in P \ (a), YP(a; 1) < y,, (a; 1). (c) For any k E N the following statements are equivalent:

(i) n

p)(a,

.) = gp(a. ).

(ii) m p) (a, z0) = g p (a, zo) for some Zo E P \ (a);

(iii) ypk)(a;1) = Ap(a; 1); (iv) ks(a) E N (cf. (5.1)). (d) For any k. k > 2,-k 36 k, the following conditions are equivalent: (i) m pk) (a..) = to p) / .) = gp(a. ); (a,

(ii) there exist 0 E R and a set I C (I/ R, R) such that I has an accumulation point in (1 /R, R) and

mp)(a,e'"x) =inp)(a.e`"x), .r E I. (e) For any k, k > 2, k # k and for any zo E P \ 1[i;+ there exists a E (I /R. R) such that mpk)(a. zo)

for any k, k > 2, k

= ntp)(a. zo);

k, there exists a E (1 R, R) such that Y (k) y(Pk)

(a; 1).

A p as k - +oo. (g) gp(a, ) < k,(a, ) in P \ (a), Ap(a; 1) <.;.-,,,(a; 1). (f) m pk) -* 9P, Ypk)

Proof. (a) follows directly from Propositions 5.2 and 5.5. (b) Suppose that c'p(a. zo) = m p)(a. zo) for some Zo E P \ {a} (resp. yp(a;1) _ y, (a;1)). Let xo := Iz I (resp. r():= a). Define

V(Z)

LR'f (Rz)'

f (b.

k 17)

EP

160

V. Contractible functions and metrics for the annulus

where b := bk(a), l := lk(a) (cf. (5.2)). Then 'p E O(P \ (-b)), 'p has at most a single pole at -b, and JVJ = 1 on 8P. By Proposition 5.5, I'(xo)I = 1. Hence, by Lemma 5.6, ' - const; a contradiction. (c) By Remark 5.3(b), gp(bk(a), -xo) = I for some xo E (1/R, R) if ks(a) E N. Thus the result follows from (a). (d) Suppose that (ii) is satisfied. Put 'f `bk(a), -e-' '0 Z) *k (Z)

GO

f I

k

'

(R . e-i6 z)lk(a) -e-i0

(R(e)'°z),,(a) ),

k

z E P.

,

By virtue of Proposition 5.5, *k (e'Bx) _ ik (e'Bx ), x E 1. Hence l/rk = *k, and therefore bk(a) = bk(a) = R and klk(a) = klk(a). Consequently, ks(a), ks(a) E N and we can apply (c). (e) Suppose that k > k. Define E(x) := mp)(x, zo) - mpk)(x, zo),

1/R < x < R.

By (c), E(RI-Ilk) > 0 and e(R1-2/k) < 0. Hence there is an a E with e(a) = 0. For the proof of the second assertion put

E(x):= yP)(x;l)-yp (x;1),

[RI-2/k,

RI-2/k]

1/R <x < R,

and use the same argument. (f) is a consequence of (a). (g) This follows directly from the fact that P is not simply connected; cf. Propo0 sition 4.2.13(a).

Now, we turn to the problem of characterizing c',.

Proposition 5.10. For any a E (1/R, R) and Z E P we have:

c''p(a, z) = cp(a, z) iff z E (1/R, R). The result directly follows from the following

V. Contractible functions and metrics for the annulus

161

Lemma 5.11. (a) If 1/R lr is extremal for c p (a. Zn) ), then

I *'(a) I < yp(a; 1).

(b) For any I / R < a < b < c < R the following equality holds

cp(a, b) + cp(b. c) = cp(a, c).

(5.6)

Proof of Lemma 5.11. (a) In view of Lemma 2.5.6, by virtue of Proposition 5.5 it is enough to prove that

f(!,-a), 0<6<2r. a

a

Let SP (Z) :=

f(1,-e-"17) f (al.

ZEP\

-Z)

a

Then SP E O(P \ (-I/a)), rp has a simple pole at -1 /a, IW I = 1 on 3P, and rp # const. Hence, by Lemma 5.6, Irp(a)I < 1. (b) First observe that (5.6) is equivalent to c* (a, b) + c,(b, c) +cpR(a,b)cW = c p(a, c). p(b,c)

(5.7)

1

For 1/R < x < y < R put *(x, y) := e, (x, 0, y). Then (5.7) may be written as

*(a, b) + *(b, c) _ f(a. c). 1 + 0, (a, b)ifi(b, c)

(5.8)

Now, we need some classical facts from the theory of theta functions; see {Rah) for details.

For fixed t E C with Im r > 0 let

q := e', C :_

fi(i - q'`"). II=I

162

V. Contractible functions and metrics for the annulus

Set w = w(z) := e2iri, z E C. Then the theta functions 90, ... , 03 are defined by the formulas:

wg2n-t)(1

9o(z,q) := C fl(1 -

-

1 qzn-t), W

n=1

81 (z, q) := -iq 1/d a

r

9o(z + 2, q),

02(z,q):=9t(z+Z,q), 03(z, q) := 9o(z

- -, q).

Further, put

c n (z )

:

F 92(z, q)

=

k 9o(z, q)' 1

sn( z )

91(z, q)

,lk- 9b(z,q)' 7O3(z, q)

dn( z )

Oo(z, q)'

where k := 021(0, q )/93 (0, q) and k' := 002(0, q)/93 (0, q) are the Jacobi nwdt di; recall that k2 + k 2 = 1. The functions cn, sn, do are called cosinus amplitudinis, sinus amplitudinis, and delta amplitudinis, respectively (with the periods a = 1/2, c02 = r/2). They satisfy the following relations cn2(z) + sn2(z) = 1,

k2 sn2(z) + dn2(z) = 1.

Moreover, the following addition-formulas are true: cn(u)en(v) + sn(u)sn(v)dn(u)dn(v) 1 - k22sn2(u)sn2(v) sn(u) cn(v) dn(v) ± sn(v) cn(u) dn(u) I - k2sn2 (u)sn2(v)

dn(u) dn(v) ± k2 sn(u) sn(v) cn(u) cn(v)

I - k2sn2(u)sn2(v)

V. Contractible functions and metrics for the annulus

163

can be easily expressed in terms of cn, sn, and dn. Namely, if The function x = e2n", y = e2n"s, then cn(t + s) sn(t - s)

Now, if we put

a =:

e2nfa

b =: e2nsR.

c =:

e2n;y

and if we introduce new variables := a - y, q := a - fl, and formula (5.8) is equivalent to the following identity

cn( + ) sn(q)

+ y, then

dn( - q)

+ cn(t;) sn( - q) dn(q) - cn(q + ) sn(:;) dn(q) dn(' - q) + k2

sn( - q) = 0.

cn( + ) cn(q + )

(5.9)

Define

Xo := k2,

XI := cn(i ),

X2 := sn(s)

X4 := cn(q),

X5 := sn(q),

X3 := X6 := dn(q),

X7 :=

X8 :=

X9 :=

Using the above notation and the addition-formulas we can write (5.9) in the form

R(X) = 0 with R := RI + R2 + R3 + R4, where RI(X)

(1 - XOXSX2)(XIX7 - X2X3X8X9)(X3X6+XOXIX2X4X5)X3X5,

R2(X)

(I - XOX8X8)(1 - XOX2X8)(X2X4X6 - XIX3X5)X3X6X7,

R3(X)

(1 - XOX2X8)(X5X6X8X9 - X4X7)(X3X6 + XOXIX2X4X5)X2X6,

R4(X) (X4X7 - X5X6X8X9)(X I X7 - X2X3X8X9)(X2X4X6 - X I X3X5)XOX2X5X7.

Of course, the variables Xo, ... , X9 are not independent. They are connected by the relations:

S2j+I(X) := X3j+I

+ X3j+2 - 1 = 0,

S2j(X) := XOX2j_1 +

1 = 0.

j = 0, 1, 2, j = 1. 2. 3.

164

V. Contractible functions and metrics for the annulus

So, finally, the problem reduces to the following implication:

for any X =(Xo....,X9)EC10ifSj(X)=0, j=

6, then R(X) = 0. In fact, this is true ! We advise the reader to use a computer; cf. Remark 5.12.

0 Remark 5.12. The polynomial R belongs to the ideal in C[Xo,... , X9] generated by S1. ... , S6; it is of the 16-th degree and consists of 24 monomials. If suffices to substitute all X31 +,'s by I - X31+2, j = 0, 1, 2 and all X3 1's by I - X°X3/_11

j = 1,2,3. We move to the problems related to c- and y-isometries of P.

Lemma 5.13. For any k E N and for any zo, w° E P, zo 0 wo, (resp. Zo E P) the extremal function form"' (zo, wo) (resp. 4k'(zo; 1)) is uniquely determined up to rotations.

Proof. We may assume that zo = a E (1/R, R). We already know (cf. Proposition 5.5) that the function cpo := ek(a, 9, ) (wo = Iwole'°) (resp. WO := ek(a, 0, -)) is extremal for m(k) (a. wo) (resp. y(ik)(a;1)). Let h E O(P, E) (orda f > k) be another extremal function for mPkl (a, wo) (resp. y, i(a; I)). Set cp := h/cpo. Then cp satisfies the assumptions of Lemma 5.6 and Ico(wo)I = I (resp. I(p(a)I = 1). Consequently, h = e1acpo for some a E R. Note that, in particular, the lemma says that all the extremal functions are proper.

Proposition 5.14 (Schwarz-Pick Lemma for P). Let F E O(P, P). Then the following conditions are equivalent: (i) cP(F(zo), F(wo)) = c*p (zo, wo) for some zo, WO E P. zo wo; (ii) yp(F(zo); F'(zo)) = yp(zo; 1) for some zo E P;

(iii) F E Aut(P). Proof. Assume that (i) (resp. (ii)) holds. The only difficulty is to show that F is proper; cf. Appendix HF 5. Let cpo be an extremal function for c* (F(zo), F(wo)) (resp. for yp(F(zo); F'(zo)). By Lemma 5.13, coo o F is proper as an extremal function for c*p(zo, wo) (resp. for yp(zo; 1)), and therefore the function F itself is proper.

Proposition 5.15. Let F : P - P. Then the following statements are equivalent (cf. Proposition 1.17): (i) F is a c-isometry; (ii) F is C' and F is a y-isometry; (iii) F E Aut(P) U Aut(P).

V. Contractible functions and metrics for the annulus

165

Proof It is easily seen that (iii) - (ii) and (iii) -* (i). (i) -* (iii). If F : P -* P is a c-isometry, then one can easily prove that F is continuous, injective, and proper; therefore F is a homeomorphism. Moreover, by Proposition 2.5.1(d), li m

F(z) - F(zo)

-zo

YN(zo; I)

Yr(F(zo),1)

E ( 0.

+ oo),

-.o E P .

Hence, by the Bohr theorem (cf. Appendix HF 6), F is either holomorphic or antiholomorphic. Now, we can apply Proposition 5.14. (ii) -- (iii). As in the proof of Proposition 1.17 we easily get that F belongs to CA(P) U 0(P), and so we can apply Proposition 5.14 once again. 0

0

we do not know any formulas Remark 5.16. We would like to emphasize that for the invariant objects on n-connected domains with n > 3

0

Notes

Proposition 5.5 for k = I was proved in [Bur 11. [Sim] and for k > 2 in [JarPfl 6,7]. Proposition 5.10 is taken from [Jar-Pfl 8]. Comparison results for yp, A p, and mp are given in [Azu 3]. Explicit formulas for cp are given in [Jar-Pfl 9].

V. Contractible functions and metrics for the annulus

166

Exercises

5.1.For AEPlet .F(1k) := {z E P: 3F E O(P, P): F(1) = 1, F(z) = A). Then

1/A) U (z E P: cP(1, z) ? k,(1,A)). Hint. Complete the following sketch of the proof. Fix 1A0 E P \ (1).

(a) If zo E P is such that t_ := c P* (1, zo) >- k,(1, Ao) =: t+, then we can take

F(z) := V where

v(t+)

(-*(z))

,

z E P,

E O(P, E), V E O(E, P), *(1) = 0, t/i(zo) = t_, cp(0) = 1, and _ XA0.

(b) Let Zo E P and F E O(P, P) be such that F(l) = I and F(zo) = Ao.

Suppose that S and H are the same as in the proof of Proposition 5.7 (with r, := 11R, r2 := R). Then there exists F E O(S, S) such that F(O) = 0 and exp of = F o exp. In particular, there exists ko E Z such that

F(w + 2 ri) = F(w) + 2ko7ri,

WES.

By the Schwarz Lemma applied to the function H o F o H-1 we conclude that

JH(2koni)I < IH(27ri)j, and hence ko E {-1,0, 1). If ko = ±1, then either F(z) = z or F(z) = 1/z. If ko = 0, then there exists >/i E O(P, E) such that *(1) = 0 and H o F = >G oexp. Consequently, k , (1, A0) < c* (1, zo).

5.2. Prove that if k , (1, A) < c* (1, z), then

max{c ,(l, z), c 1(1, A)) = c, p((l, 1), (z, A)). 5.3. Let G1, G2 c C be c-hyperbolic domains (cf. Proposition 2.3.1). Using the methods of the proof of Proposition 5.14, prove that any c- or y-isometry F : G I -+ G2 is either holomorphic or antiholomorphic. Observe that the c-hyperbolicity is essential.

Exercises

1 67

5.4. Let Gj C C be a bounded vs-connected domain which is regular w.r.t. the G, Dirichlet problem, j = 1, 2. Prove that any holomorphic mapping F: GI with

= c* c* F 0(°0)) GZ (F(-').

z'

for some `O -' " E G I z'0 -0

N)

is a proper k-fold ramified covering with v1 = k v,. In particular, if v, = v2, then F has to be biholomorphic (cf. Proposition 5.15), and if v, t v1, then there is no such a mapping.

Hint. Use the following general result of H. Grunsky (cf. [Gru 1,2]): if G C C is a bounded v-connected domain regular w.r.t. the Dirichlet problem, then any holomorphic function f : G -* E extremal for c* (z', z") (z', z" E G, z' z") is a proper v-fold ramified covering.

5.5. Let F(z) :=

z(z + 1), z E P, L := F(P). Prove that F: P --> L is a 2-fold

ramified covering such that

c*(F(l ), F(-1)) = c* (1, -1); cf. Exercise 5.4. Hint. For any f E O(P. E) the function

[f(w+ Vw2-I)+f(u>-

wEL,

belongs to O(L, E). 5.6. Modify the domains P and L from Exercise 5.5 to get a 2v-connected (resp. vconnected) domain P, (resp. L,.) which is regular w.r.t. the Dirichlet problem and such that F : P. L. is a 2-fold ramified covering with

cl,(F(l ), F(-1)) = c',.(I, -1). 5.7 (cf. [Jar-Pfl 1]). Let G := ((z1, z2) E C2: IZI I < 1, 1221 < 1, IZIZ21 < 1/2); cf.

Remark 2.7.8. Take R := 2 and let P := P(2). Define F = (F1, F2): P -+ C2 by the formula F(A)

2(f(I, -),f(1,X)),

X E P.

(a) Prove that F is injective, F(P) C G, and F(8P) C aG.

V. Contractible functions and metrics for the annulus

168

(b) Using the contraction-property of F: P -+ G and Proposition 5.5 prove that cc(O, z) = 21z,z21,

z = (z,, z2) E r,

where

r:= {(zi, z2) E C2: 3 t E (-2, -1/2): IzjI = F; (t), j = 1, 2}. (c) Using (b) and the maximum principle prove that cr(0, z) = 21z,z21,

= (Z,, Z2) E r. .

where

:= IAz

= (z,. Z2) E r, I < IRI < (2Iz,z2I)-`/");

cf. Fig. 2.1.

5.8. Let G := E2 \ (rE)2, where 0 < r < 1. Prove that for arbitrary e E (0, 1 - r) we have

gG((O,1),(O.s))$gE2((O,t).(0,s)) forr
Hint. Fix t E (r, 1) and use the function u defined for (z, w) E G as follows:

)

max(IzI. rgp(t. w)}

I

Iz1

where P := (A. E C: r < IAI < 1}.

if Iwl > r if IwI < r

c6pllo VI

The Bergman metric

The constructions of the metrics studied so far were based on the set of bounded holomorphic functions, on special plurisubharmonic functions, or on analytic discs. The Bergman metric will be based on the Hilbert space of square integrable holomorphic functions. Because of the 8-theory of Hormander it turns out that the use of this metric becomes considerably simpler while, on the other hand, the contractibility with respect to all holomorphic mappings is lost.

6.1 The Bergman kernel Let M be an arbitrary set; by Abb(M, C) we denote the set of all C-valued functions

on M. Moreover, we suppose that a linear subspace H C Abb(M, C) carries a scalar product ( . )H such that H becomes a Hilbert space. As usual, we write IIXIItI :_

(x, -x) m, x E H.

A function K: M x M-+ C is called a kernel function of H if

VEM.

(i)

(ii)

f(y) = (f K(..Y))H,

f E H, yEM.

Remark 6.1.1. Let M, H be as above and suppose that K is a kernel function of H. Then the following properties of K are easily seen: (i) 0 < K(y. y) = II y)II;,, y E M:

(ii) K(x, y) = K(y, x), x, y E M; (iii) IK(x, y)12 < K(x. x)K(y, y), x, v E M;

(iv)If(y)I<_Ilflltt K(y,)I), fEH,yEM. Observe that (ii) implies that the kernel is uniquely determined and that (iv) gives the continuity of the linear functionals H 31 f -* f (y) E C (y E H). Lemma 6.1.2. Let M and H be as above. If we suppose that any linear functional H 9 f -> f (y) E O. y e M, is continuous, then H carries a (unique) kernel K.

VI. The Bergman metric

170

0

Proof. Use the Riesz representation theorem.

To obtain explicit kernels a fundamental role is played by orthonormal bases. In the sequel we will always assume that M is a topological space with an infinite countable dense subset M' and that H C C(M, C). If H has a kernel K, then H y) when y E M'). Hence H is a separable Hilbert space (take the functions

has a countable orthonormal basis ((vj)jEJ with J = {1,..., jo) or J = N, and therefore K can be represented as

K(.,Y) = 1: cj(y)coj.

(6.1.1)

jEJ

Observe that property (iv) of Remark 6.1.1 implies that the series of functions in (6.1.1) converges uniformly on those subsets of M on which the function x -> K (x, x) remains bounded. Moreover, the series is independent of the order of summation.

Remark 6.1.3. Suppose H (as above) is equipped with the kernel K. Moreover, assume that for a fixed point Yo E M there is at least one function f E H with f (yo) 0. Then the following properties show how K relates to various extremal problems: (i) K(Yo. Yo) > 0;

(ii) min{Ilf(lH: f E H. f(Yo) = 1) = (iii) max(If(Yo)I: f E H, 11f 11H = 1) = (K(., Yo)/ K(Yo, Yo))(Yo) Moreover, the function g := K(., yo)/K(yo, yo) is the only function in H solving

the extremal problem in (ii).

So far, we discussed the kernel in a very abstract setting. Now, we turn to the concrete situation we are interested in during this chapter. Let G C C" be any domain and let

L'(G) := If E O(G,C):

jIf(z)12dA(z) < oo).

The space L2 (G) with the scalar product

(f g)L'(c) = (f, g) =

fc

f(z)g(z)dA(z),

f, g E L,(G),

is a complex Hilbert space, the Hilbert space of all square integrable holomorphic functions on G. Using the Cauchy integral formula it follows that for every z E G

6.1 The Bergman kernel

171

f -- f(z) is continuous. Thus, by Lcmina the evaluation functional L2 (G) 6.1.2, L2(G) carries the kernel KG which is called the Bergman kernel of G. Lh(G) is a closed subspace of the Hilbert space LZ(G) of all complex valued square integrable functions on G. By Pa: L2(G)

L'(G).

(PGf)(z) :_ (f.

z)),

we denote the orthogonal projection; PG is called the Bergman projection.

Remark. (a) For a bounded domain G the dimension of is infinite because all polynomials belong to L2(G). (b) For a plane domain G the following properties are equivalent (cf. [Skw 2], [Wie]):

(i) KG(z, z) > 0 for all z E G; (ii) L2 (G) 0 (0); (iii) dim L2 (G) = oo; (iv) K (., zi ), ... , K zk) are linearly independent for any k pairwise different

points zi,... , zk in G, k > I. (c) In the higher dimensional case the situation changes as the following result shows (cf. [Wie]): for every k E N there exists an unbounded Reinhardt domain G C C2 with dim L2 (G) = k; see Exercise 6.1.

Recall that Ko(z, z) > 0 if there exists f E L2 (G) with f (z)

0. Then,

according to Remark 6.1.3, we get KG (Z, Z) = sup{

I

If (z )I2

: f E Ln(G) f 0 0

IIIIIL2(G)

In particular, if G' C G, then KG (z, z) <- KG'(Z, z) for all z E G'. Moreover, the Hartogs' theorem on separately holomorphic functions (cf. Appendix HF 8) implies that the function G x G E) (z, w) -> KG(Z, w) is holomorphic

as a function of 2n variables. (Of course, here C is not the closure of G.) In particular, the kernel KG is continuous on G x G.

Remark 6.1.4. Here we give another useful description of the Bergman kernel. Denote by gp: C -* R+ a rotation-invariant CO0-function with supptp C E and fc rp(z)dk(z) = 1. Let G be a domain in C" and let w = (w i, ... , E G with

w+(rE)" CG (r >0). Put

*.(z) := (l/r")Q((zi - wi)/r) ... w((z,, - w")/r).

Z = (zI.... , Z") E C".

VI. The Bergman metric

172

Then for a holomorphic function f : G -+ C we obtain

f(W) =

f it.(z)f(z)da(z) G

via the Cauchy integral formula. In particular, this formula holds for every f in L2(G). which implies: PG

,, =

w).

(6.1.2)

Observe that the same formula remains true if *w is assumed to be invariant only under rotations with center at w.

Example 6.1.5. (a) We mention that the functions (n + -Jot I)! a

a !7r"

'p- (Z)

z E B,,. a E (Z+)",

form an orthonormal basis of L 2 (lB,,). Use induction over n to evaluate II za II r.= (B.)

Then the Bergman kernel of B,, can be calculated as follows: K8. (1z. w) =

r

(n + lal)! a -a alxn Z w

fix-. (n + v)!

1

n

1-0

n

7rn dxn

7a, a

I`

°O (n + v)!

1

V!

I

(z w)

-X) Ix=(z,w)

Thus we get n!

Kn. (z, w) = nn (1 - (z, W))-(n4-1)

wE H

(6.1.3)

(b) The Bergman kernel of the unit polycylinder E" can be obtained along similar calculations as n

1

n

Kf^(z. W) _ H KE(zj. wj) _ 7r-, [J(1 i=1

j=1

-

zjw;)-2.

(6.1.4)

6.1 The Bergman kernel

173

(c) Let P := {A E C: r < 1A < 1). Then an orthonormal basis is given by

_

+ kJ, 7r(1 -r2i+2)

j

jE

-1.

1/(-27r logr) . A

(p-I(),) = Hence 1

Kp(z. w)

27rzw log r

+

1

7rzw

j(zu)j. r-) jez 1 -

Z. W E P.

(6.1.5)

!#O

(d) Observe that L2(C") = {0}, which implies that KT-

0.

Also for more complicated domains there are explicit formulas for the Bergman kernel. We discuss here the kernel for the Thullen domains.

Example 6.1.6 (cf. [Ber 1 ]). For p > 0 put DJ, := (: _ (ZI, Z2) E C2: IZI I21p + Iz2I2 < 1).

Integration by parts shows that the following functions +1

JPu.t (z) _

1/2

J=o(p(l2 + 1) + j) it pvi

z1/1Z2

0<-v, u. ZEDJ,.

form an orthonormal basis of Li (Dp). Then by (6.1.1) we obtain KD,,(z, w)

- n2

E (zl wl)y(Z2w2)` 1II(p(lz+ 1)+j)

P J1.,-o

I

j=O

00

'2E00

=

(zlwl)``(µ+l)(p(u+1)+1) N=o 1

7r2(1

_

\

Zl iv- l

Z2w2)p+'N=0 \(1 - Z2u:2)p 1

7x2(1

(1+E

Z2@2 )p+2

p+l 2pq (1 -q)2 + (l -q)3

i,=I

Z2 W2

V!

11((/2+ 1)p+l +k) k=1

((µ + l)(p + l) + pa(µ + l)) .

VI. The Bergman metric

174

where q := zjw,/(l - z2w2)p, IqI < 1. We mention that the binomial series were applied at (*). Moreover, (I - z2w2)p is chosen as the principal branch. Thus, we finally arrive at the following formula:

2 (I_ z2w2)p-2 (P + 1)(1 - z2W2)' + (P - I)zjw 1 -Zlwl)3

KD,(z, w) =

7r

((I -z2iV2)p

(6.1.6)

Next, we are interested in the behavior of the Bergman kernel under biholomorphic or even under proper holomorphic mappings.

Proposition 6.1.7. Let F : G -i- D be a biholomorphic mapping between the domains G, D C C". Then we have

KD(F(z), F(w)) det F'(z)det F'(w) = KG(z, w),

z, w E G.

Proof If g E Lt2,(D), then obviously (g o F) det F E L2(G). Therefore, the properties of the Bergman kernel yield

f(g o F)(w) det F'(w)KG(W, z)dA(w) _ (g o F)(z) det F'(z) =

JD g(i;)K0(i

,

F(z)) det

= j(g o F)(w)KD(F(w), F(z))I det F'(w) 12 det F'(z)dA(w). In particular, if for f E L'(G) we put g := f o F-'t det(F-t)', then it follows that

0=

Jc

f (w)(KG(w, z) - KD(F(w), F(z)) det F(w) det F'(z))dA(w),

which gives the formula in Proposition 6.1.7.

0

It is clear that the domain D2 of Example 6.1.6 is the image of l2 under the proper holomorphic map F: P,2 -* D2, F(z) = (zi, z2). Observe that Proposition 6.1.7 fails to hold for F. Nevertheless, the following more developed transformation rule for the Bergman kernels remains true also for proper holomorphic mappings.

Theorem 6.1.8 (cf. [Bell 1]). Suppose G and D are bounded domains in C" and that F : G -p D is a proper holomorphic mapping of G onto D of order in. Let u : = det F' and let I1, ... , 4). denote them local inverses to F defined locally on

6.1 The Ber man kernel

175

{F (z): z E G, u(z) = 0}. Moreover, let Uk := det V. Then the D \ V, where V Bergman kernels transform by the rule

K0(z. 4)k(w))Uk(w) = u(z)KD(F(z) w) k=1

for allz E G and all W E D \ V. Proof. As the first step we derive the following transformation rule for the Bergman projections

Po;(u h o F) = u (PDh) o F,

h E L22(D).

(6.1.7)

Obviously, (6.1.7) holds whenever h E L2 (D). So it is sufficient to show that PG(u h o F) = 0 for all h E L2(D), h orthogonal to Lh2(D), i.e. h E Lh (D)l. Let g E Co (D \ V) and f E L2 (G). Then

JG

.f(z)u(z)dw o F(z)dA(z) i

- ID

f o tk(w) u o 4'k(w)IUk(w)Ik=1

8g

8u'i

(u,)dk(w),

1<j
Hence integration by parts gives PG (u -2L o F) = 0. 1 < j < n. It remains to verify that

if:=

a wj

V ),

is a dense subset of L2(D)1. So, let h E L2(D)-l fl Hl. Then h satisfies the Cauchy-Riemann equations

on D \ V (in the sense of distributions) since f o h(w) -(w)dA(w) = 0 for all g E C0 --'0(D \ V), l < j < it. Therefore (cf. Appendix HF 12), the function h E L2(D) is holomorphic on D \ V. By Appendix HF 13 we conclude that h extends holomorphically to the whole D which gives It = 0. We turn to the proof of the formula in Theorem 6.1.8.

VI. The Bergman metric

176

Let u, E D \ V with uw + (r E)" c D \ V (0 < r) and choose the corresponding * as in Remark 6.1.4. i.e. PD>]r = KD(., w). Applying (6.1.7), for z E G we obtain

u(Z)KD(F(Z), w) = u(z)(PD7)(F(z)) = PG(u * o F)(z)

fu()

i o F()KG(, z)dA()

m

= fD

_

A_1

Uk(w)KG(4k(w), Z), k=1

which finishes the proof.

Example 6.1.9 (cf. [Hah-Pfl 2] and Exercise 6.2(a)). The transformation rule from Theorem 6.1.8 provides a useful tool to establish explicit formulas for the Bergman kernel. Observe, for example, that the domain D2.2 := (z E C2: Iz1I + Iz21 < 1) is the proper holomorphic image of the Thullen domain D2 via F(z) = (z1, zZ). Then, straightforward but tedious calculations lead to the following formula: 2

3(1 - (z, uw))2(l + (c. u,)) +4z1z2wlu'2(5 - 3(z. w)) )3

((I - (z, w))'- -

Besides creating new explicit formulas for the Bergman kernel, Theorem 6.1.8

serves as a fundamental tool in the investigation of the boundary behavior of proper holomorphic mappings. Here we restrict ourselves to the following simple situation.

Theorem 6.1.10 (cf. [Bell 1]). Let G 1 and G2 be bounded complete Reinhardt domains in C". Then any biholomorphic mapping F: G1 G2 extends holomorphically to an open neighborhood of G1. In particular, there exists a biholomorphic

map F between open sets Uj D G j, j = 1, 2, with FIG 1 = F. Proof. Let K7 : Gl x G1 -+ C denote the Bergman kernel of G7. j = 1, 2. Observe that the monomials Z", of E Z , form a complete orthogonal system for L2 (G J ). Therefore

Ki(z,

c,,(j)z'

UEV

where cu(j):= IIZ°llc:(G1) >0

",

Z,

E G7,

(6.1.8)

6.1 The Bergman kernel

177

Put u(z) := det F'(z), Z E G, and U(w) := det(F-')'(w), w E G2. Then the transformation formula for the Bergman kernel (cf. Proposition 6.1.7) leads to

Ki(z, F-I (w))U(w) = u(z)K,(F(z), w),

z E G1, W E G,.

(6.1.9)

If we differentiate (6.1.9) a-times, Of E Z , with respect to the variable w, then we get act

-[K,(z, F-'(-))U]1

_o

=a!ca(2)u(z)Fa(z),

z E G1.

(6.1.10)

Choose e > 0 such that F-1(0) E 1+,f G cc G. Then, because of (6.1.8), we have the following equalities

zEG, E

KI(z,

1+EG.

Thus, if i; E 1+E G, then any derivative a;, Ki (z. ' ), a E Z+, extends to the domain (1 +E)G D G as a holomorphic function in z. From (6.1.10) we conclude now that every function uFa, a E Z ., extends as a holomorphic function ga to a certain domain G, containing G1. In particular, with a = (0, ..., 0) we find that there is a holomorphic function u on G, with ii I c = u. is already holomorphic If i(z) : 0 for a point z' E 8 G, , then F := (F, , ... , near z'. In the other case, when u(z') = 0 (z' E 8G) we write near z': N

N,

u = I p;;

and

g(o.....I.....0) = [I qj.V .

respectively,

V=1

V=1

, 1 < j < n, 1 < v < Nj, respectively, are irreducible elements in the UFD-ring d`, of germs of holomorphic functions at z'. Moreover, they are pairwise not associated via units. Now let j be fixed, 1 < j < n. Near z' we write

where the germs pv,z,, l < v < N, and q j.v.z

u-m-I g(o.....m..... 0) (N

f

v=

1,

-= 'm-I u

(uFjm) == g(0.....I.....0)

in E N,

)tfl_I divides

mEN.

V1. The Bergman metric

178

Since p,,,., are prime, it follows that (m - 1)l, < m E N. Hence 1, < ki,,,ivi, and so N

with I < x(v) < Nj,

N)

lip. divides f J';' v=1

v=1

i.e. uh j = g(a,...,1,,...0) with a suitable holomorphic function h j near z'. Therefore,

Fj is holomorphically extendable via hj to a neighborhood of z'. Altogether, F admits a holomorphic continuation F to a neighborhood of G 1.

Remark. The above proof and Theorem 6.1.8 immediately show that the first part of Theorem 6.1.10 remains true even for proper holomorphic mappings (cf. Exercise 6.2(b)). In this form the theorem is formulated in [Bell fl. We mention that a similar result is true for proper mappings between circular domains which contain the origin (cf. [Bell 2]). The boundary behavior of holomorphic automorphisms of

bounded Reinhardt domains was also studied by W. Kaup (cf. [Kau]) but his method of proof relies on the Lie theory.

We recall the form of the Bergman kernel in Example 6.1.5(b). The formula there appears as a special case of the following general result due to Bremermann (cf. [Bre]).

Theorem 6.1.11. The Bergman kernel of any domain of the form G 1 x G2 C CI I X C"2 is given by

KG,.G_((ZI, z2), (wl, w2)) = KG,(ZI, wl) . KG2(Z2, W2),

zj, wj E G j.

For the proof we will need the following preliminary result.

Lemma 6.1.12. Suppose that f E L2(GI x G2). Then for every Z2 E G2 the function f Z2) belongs to L, (G I). Proof. Use, for example, the mean value inequality at Z2 for the plurisubharmonic functions 1 f (z1, .)12, zl E G1.

Proof of the theorem. Suppose that and (I/',j,LEz+ are orthonormal bases of L, (G 1) and L, (G2 ), respectively. By (6.1.1) it suffices to show that the functions G1 x G2 9 (ZI,Z2)

form an orthonormal basis of L, (GI x G2).

Vv (ZI)*µ(Z2)

6.1 The Bergman kernel

179

obviously form an orthonormal system in L2 (G, x G2), it Since remains to prove its completeness. So, let f E L2(G, x G2) be an arbitrary function with f(Z1, Z2)WPv(ZI)*,i(Z2) dh(zj. z2) = 0.

V, µ E Z+.

c, X G2

Then by the Fubini theorem we get

f

c,

(Pv(Zi)

f

f (Zt, z2)*,(z2)d .(z2)d k(zi) = 0,

c2

where f (zi, ) E L2 (G2) for all zi E G, according to Lemma 6.1.12. To obtain f = 0 it suffices to show that the functions

f( zi,z2)*, (z2)d?(z2)

gu(zt) 1G2

are in L2 (G,). Choose a sequence (DJ)jEN of subdomains of G2 with Dj cc

Dj+t cc G2,

G2 and define on G,:

g,1.i(zi) := j f(zi, z2)*,,(z2)dA(z2) ,

The functions g,,.j are holomorphic on G, and we get Ig,.(zt)

- g,L.i(zt)12 < f ,

2\D,

If(ztz2)12dA(z2) J _\D, I*,,(z2)I2d), (z2)

< C(z°) f

6, xG2

If(zt, z2)12dA(z,, Z2) J

I*N(z2)I2dX(z2) 2 \A,

if z, E z°+(rE)" cc G,. Here we use the mean value inequality for the functions If(', z2)12, Z2 E G2 \ Dj. Hence g,,,jj Wig,,, i.e. g is holomorphic. Moreover, the Holder inequality implies the following estimate

f GIg,,(Zt)I2dA(zi)

J t f If(zl, z2)1dX(Z2) f GI0(Z2)I2d)(z2)JdA(zi) ,

G2

- II111L2(G,xGA II*NIIL2(G2), which finally shows that g,, E Lh (G, ).

VI. The Bergman metric

180

Remark 6.1.13. (a) Note that the use of the kernel function also gives us a direct proof of Theorem 6.1.11. (b) Let G be a domain in C"' X C"2 and let 7r: C"' X C"2 -+ C" be the projection onto the first n, coordinates. For f E Lh(G) put

S(f) :_ {zi E ,r(G): f(z,, ) t Lr2,(G fln-'(zi))}.

According to Lemma 6.1.12, S(f) = 0 if G is of the form G = G, x G2, Gi C C But in general the set S(f) can be a quite arbitrary set of Lebesgue measure zero; cf. [Jak].

Using analogous ideas as in the proof of Theorem 6.1.11 (cf. [Nar 1]), one can derive the following theorem (which we will need in the future).

Theorem 6.1.14. Any holomorphic function f on a product domain G, x G2 can be written as a locally uniformly convergent series 0C

f(zi, z2) _ Eg., (zi)hv(z2) v=I

with g,. E O(G,. C). h,, E O(G2, C).

Now, we want to describe how the Bergman kernels behave under other simple set theoretic operations.

Theorem 6.1.15 (cf. [Ram 1,2], [Skw 1]). Let G be a domain in C" that is the union of an increasing sequence of subdomains G. Then KG(Z, w) _ lim,-,, KG, (z, w), (z, w) E G x G, and the convergence is locally uniform. Proof. Suppose the claim of the theorem fails to hold. Then, without loss of generality, we may assume where

Now let F be an arbitrary compact subset of G. Then F C G, if v > vo for a suitable vo, and therefore for z, w E F we have KG, (Z, w)I

KG 7(z, z)

KG,(W, W)

KG,ti (z, z)

KG,ti (w, w) < sup{ KG,,

0: 2; E F) < oo.

6.1 The Bergman kernel

181

Hence the sequence (K,),,, where K, := KGB., is locally bounded on G x G which provides a subsequence (K,,)j with limj--.,c K,,,(z, w) _: k(z, w). Here the k(z, w) is convergence is locally uniform; so the function G x G 9 (z, w) holomorphic. Moreover, if D is any relatively compact subdomain of G with G,,1 D D, j > jo,

and if w E D, then w)Ili2(D) _

jlxJ IK,.,(z, D

< liminf j

f

IK,,(z, w)I2dA(z) = liK,.,(w, w) = k(w, w). J-+00

G ,.I

w) E L (G) for every

Since this estimate is independent of D, we obtain

WEG. w) reproduces the functions of i.e. As the last step we show that w) = KG(', w). Let f E Lh (G) and suppose that w E G,,, if j > jo. Then for j > jo we get

f(w) = fG,1f f(z)K,,,(z. w)dA(z) r

= J f (z)k(z.

w)

f

(z, w)dl(z) ., \n

- f fin(z)k(z, w)dX(z)

whenever D CC G and D C G,,,, j > jo. The third summand can be estimated as follows:

G,., \n

f(z)K,y(z, w)dk(z)

II!IIL2(G\n) Vi 711(U) W) 1

ill 1IL2(G\D)(k(w, w) + l]'/2

if j>> 1.

If D and j are sufficiently large, then the last three summands become arbitrarily small, i.e. we obtain

f (w) = f f (z)k(z, w)dA(z). Hence (K,.,) converges locally uniformly to KG on G x G which contradicts assumption (*).

VI. The Bergman metric

182

On the other hand, the situation for a decreasing sequence is more complicated (cf. [Skw 1 ]).

Theorem 6.1.16. Let

be a decreasing sequence of domains G, C C" with KG, (Z, w) locally uniformly on G x G G,, D G, V E N. Then KG (Z, w) =

if and only if for every w E G, limKG, (w, w) = KG(w, w). Proof. Only the sufficiency of the above condition has to be verified. Similarly as in the proof of Theorem 6.1.15 it follows that the sequence (KG,.) is locally bounded on G x G. So, without loss of generality, we may assume that KG,. (z, w)

k(z. w) locally uniformly on G x G. As before it turns out that Ilk(., w)II':(G) < slim KG,(w, w) = k(w. w) = KG (w, w),

If k(w, w) = 0, then

w) = 0 =

w)

k(w, w)

w). But if k(w, w) 3& 0, then

Ilk(', w)II LZ(G) LZ(G)

k(w, w)

w E G.

k(w, w)

1

k(w, w)

=

W)

KG(w, W) L2(G)

Therefore, by Remark 6.1.3 it follows that

w) = KG(', w). For a more detailed discussion on how the Bergman kernels behave under intersections see [Skw 11.

The boundary behavior of the Bergman kernel will play a fundamental role in the discussion about the Bergman completeness (cf. Chapter VII).

Theorem 6.1.17 (cf. [Pfl I]). Let G be a bounded pseudoconvex domain in C". Suppose that a point z' E 8G fulfills the following general outer cone condition: of points w° 0 G there exist r E (0, 1 ]. a > 1, and a sequence with lim w° = z* and G fl B(w°, rllw' - z*ll") = 0. V

00

Then for any sequence (z"),.EN C G with lim,,.y,,. z° = z' we can find a function f E L2 (G), Ilf IILZ(G) = 1, with sup(I f (z`')I: V E N) = oo. Moreover, lim:..- KG (Z, z) = 00.

6.1 The Bergman kernel

183

Proof. The second assertion is a trivial consequence of the existence result. To establish this one we need the following result due to H. Skoda (cf. [Sko], [PA 4]) which we state here without proof.

Theorem. Let ci be a pseudoconvex domain in C", * : Il -+ [-00, oo) a psh function, >) > 1, and p E N. Set q := min(n. p - 11. Then for any holomorphic functions g 1, ... , gp E O(11) and f E O(11) with

A

If 12

.n

9q-I

(1:P Igj 12)

exp(-tr )dk < 00

r_1

there exist functions h1, ... , hp E 0(Q) satisfying i,

f = 1: higj.

(i)

1=1

f(IhJ12) (Igjl2)

(ii)

P

P

i=1

r-1

exp(-*)d), <

q

A.

We are going to prove Theorem 6.1.17 by contradiction. So, let (z')V,EN be a

sequence in G with lim,,,,, z" = z* such that every function f E L,(G) is bounded along this sequence, i.e. I f (z") I < C(f), v E N. This means that the evaluation functionals S,.,: L2 (G) -- C, 8,,(f) := f (z"), are pointwise bounded. Hence, by the Banach-Steinhaus theorem, there exists a uniform bound C: V E N, f E L1,(G).

If (z")I < ClI f IIL:((-,).

(6.1.11)

Now, we apply Skoda's theorem for the following special data: Il = G, f - 1,

0, p=n,gj(z)=zj

I such that 2n <M:=2>i(n-1)+2<

2n + I la. So we are able to find functions h',..., h' E O(G) with the following properties: of

1 = Eh(z)(zi - w'),

(i)

(ii)

f

c

z E G. v E N.

1=1

Ih (z)I211z - w"II-2q(n-1)d ,(z) <

1

1

f

c

Ilz -

w"II-2n("-1)-2 dA(z).

VI. The Bergman metric

184

With r := r it z` - w" IIa we obtain from inequality (ii) the following estimate

Ilz - w" II-MdA(z) + f

Ilh j II'2(G) -< C U\B(zl,

(o.I)\a(o.r.)

Ilzll-MdA

< C'(I + rV M+(2n-i)+i ),

(6.1.12)

and therefore V E L2(G). Observe that S := M - (2n - 1) < 1 + 1/a. Then (i), (6.1.11), and (6.1.12) lead to the following chain of inequalities: n

I
C*IIz" - w"II +

< C'(1 + Ilz - w"IIa('-a))Ilz" -

w"Ila(t-a)+1

> 1, which gives the desired contradiction.

Remark 6.1.18. For a more precise description of the boundary behavior the reader should consult the following series of papers [Die-Her-Ohs], [Her 1,3], [Ohs 2], [Pfl 3].

We conclude this section with a few observations about the zero set of the Bergman kernel.

A domain G C Cn is called a Lu Qi-keng domain if KG(z, w) 0 0 for all z, w E G. It is clear that any bounded simply connected domain in C is a Lu Qi-keng domain; cf. Example 6.1.5.

On the other hand, the annulus P :_ (A E C: r < IAI < 1), 0 < r < e-2, is not a Lu Qi-keng domain (cf. [Skw 1]). In fact, according to Example 6.1.5(c) the Bergman kernel of P is given as

Kp(z, w)

00

_

1

1

22rzw log r

7rzw

E

mzmwm

1 - r2m

MOO

n

1

00

(

[ logr2 +n,=o \(1

7r X

where A := zw with r2 < IXI < 1.

Ar2m

r2m+2/A

ll

(1 -r2m+2/A)2l]

6.2 The Bergman pseudometric

185

For h we obtain the following facts: h is continuous,

h(A)ERifAER,r2
h(A)>Oifr2 3 there exists a p-connected bounded domain in C, which is a Lu Qi-keng domain. (c) Any bounded domain in C with smooth boundary that is a Lu Qi-keng domain is already simply connected. On the other hand, a recent example (cf. [Boa]) shows that in higher dimensions simple connectivity is no longer sufficient for being a Lu Qi-keng domain, even in the class of strongly pseudoconvex domains.

6.2 The Bergman pseudometric Recall that the Bergman kernel KG of a domain G C C" is a C0-function on G x G and that, on the diagonal, it can be represented as KG(z. Z) = SUP( If(z)}I`': f E Li (G) Ilf IIL'(G) = 1),

Z E G,

if L2(G) # (0). Therefore, the function G 9 z -+ log KG(Z, z) is psh on G. In order to have this function of class C2 on G we assume for the rest of this chapter that all domains G we will deal with have the property KG(z,z) > 0,

z E G.

(6.2.1)

For example, all bounded domains share (6.2.1) and, moreover, whenever the then (6.2.1) is true for G. coordinate functions belong to Then KG leads to the following positive semidefinite Hermitian form n

a2

logKG(Z,Z)X1,X,

BG(Z;X):= > aZ°aZN

ZEG, X EC".

VI. The Bergman metric

186

The pseudometric Z E G, X E C"

fG (Z; X) := VFBG (z; X),

induced by BG is called the Bergman pseudometric on G. Observe that

Z E G, A E C, X E (ii) PG (Z; X1 + X2) < PG (Z; XI) + PG (Z; X2), Z E G, X1, X2 E C", (iii) fiG : G x C" R+ is continuous. (1)

fG(z;), X) = I), IPG(Z;X),

C",

Set bG(z', z") := (f PG)(z', z"), z', z" E G; bG is called the Bergman pseudodistance on G. (iv)

bG is continuous on G x G.

Example 6.2.1. f8, (z; X) = (-,In + 1)y9q(z; X), (a)

Z E B, X E C",

(6.2.2)

rt

(b)

ZEE", X E C";

fE. (z; X) = V 2 > yE (Z j; Xj )2,

(6.2.3)

j=1 2

(c)

fipy(Z.X) _

>2 TijXiX j,

z E Dp, X E C2,

where T11(z) =(1 - Iz212)"

(C (Z) + D2(z)

T12(z) 421(z) = pI IZ2(1 -

Iz212)°-1

(z) Z72

T22(z)=

2-p

+ D2(z))

'

+ 3p(C(z) + p1Z112Iz212)(1 - Iz212)°-2

(I - IZ21) C 2(Z) _ p(p + 1) (D(z) - p(p -1)Iz1I2IZ212)(1-

Iz212)P-2

D2(z) with C(z) := (1 - Iz212)° - Iz112

and

D(z) := (p + 1)(1 - Iz212)° + (p

-

1)Iz112

6.2 The Bergman pseudometric

187

(cf. Examples 6.1.5, 6.1.6 and [Ber 1]).

(d) Let P = ]-k E C: 1/R < III < R) (R > 1). Then there exists a positive real analytic function X : (-2 log R, 2 log R) -+ IR>0 with the following properties: (i)

(ii)

x(t) = x(-t) whenever 0 < t < 2 log R, x'(t) > 0 for

- 2log R <1 <0,

if t=0.

x'(t)=0 (iii)

0<

X(t)

< o0 lim 1/2 log R (t - 2log R)2

The Bergman metric for P is then given by IXI

op0,;x) IAI

VI'X

1A12)

Details of how to derive this formula can be found in [Her 2].

In particular, we see that the General Schwarz-Pick Lemma does not hold for

the Bergman pseudometric: simply take F: E2 - E2, F(zi, Z2) := (zl, zi), and X := (1,0). Then PE2(F(0, 0); F'(0, 0) X) = 2 >

=,BE'((0, 0); X).

On the other hand, this metric is invariant under biholomorphic mappings.

Theorem 6.2.2. If F : G -> D is a biholomorphic mapping between the domains G, D C Cn, then (a) #n(F(z), F'(z)X) _ IOu(z; X), z E G, X E C", (b) bo(F(z'), F(z")) = bc(z', z"), z', z" E G. Proof. According to Proposition 6.1.7 we get log Ko(F(z), F(z)) + log I det F'(z) 12 = log KG(Z, z).

The function det F' is a nowhere vanishing holomorphic function, and therefore log I det F'12 is pluriharmonic. Hence the transformation rule for the Levi-form under holomorphic mappings immediately leads to (a) and (b) is a simple consequence of (a). O

Moreover, it turns out that the Bergman pseudometric is not monotone with respect to inclusions (cf. [Ber 1 ]).

VI. The Bergman metric

188

Example 6.2.3. Put GI

(z E C2: (49/50)11-.1 122/3 + IZ212 < 1),

G2

{z E C2: (49/50)Iz112/1 + (1/49)1z212 < 1).

The Gk's are Reinhardt domains and, because of 49(1 - (49/50)x2"s) > I - (49/50)43

whenever

0 < x, <

(50/49)3/2,

we have G, C G2. On the other hand (cf. Example 6.1.6),

GI = FI(D3) with F,(z1, z2) = ((50/49)3122,, z2). G2 = F2(D5)

with

F2(z,, z2) = ((50/49)51221.7z2).

Hence we obtain (0; X) = 2 (49/50)31X1122 + 51X212,

Bc, (0; X) = 3 (49/50) 51 X,12 +

1X212.

In particular,

#G, (0; ( 1, 0))

9Y

2 ( L5O

.cc

(L3'

(L509y= #G2(0; (1, 0)).

According to our general remarks in Chapter IV, the system ((1/f),G)GE6, where

6:= (G E t : G satisfies (6.2.1)), is a contractible family of pseudometrics but only with respect to biholomorphic mappings.

Remark 6.2.4. We also mention that as a consequence of Theorem 6.1.11 we have fG,XG,((ZI,z2);(X1, X2)) = whenever Z j E G; C C") 3 X 1.

fi 1(21;X1) + fcz(z2;X2)

(6.2.4)

6.2 The Bcrgman pscudometric

189

a

The following representation c aracter»es the lergman you attwirle Am by means of a variational problem.

Theorem 6.2.5. Suppose that G is a domain in C" with property (6.2.1). Then 1

PG(Z.X) =

supil f'(z)XI: I. E Lh(G), IIf II1.=(G) = 1, f(z) = 01,

KG(Z, Z

ZEG,XEC. Proof. Fix z° E G, X° E C. Then H" := If E L2 (G): f (z°) = 0, f'(z°)X° _ 0} is a closed subspace of H' := If E L2 (G): f (z°) = 0) whose orthogonal complement in H' is at most one-dimensional. Let dim H"1 = 1. Then there exists an orthonormal basis ((pj)jE2i. of L,(G) with

(pj (z°) = 0, j > I,

and

(pj(z°)X° = 0, j ? 2.

Simple calculations lead to

x 2

0

O

G( 'X)

Kc(z ,z°) I(P1 J=O

o

02_

)X 1

r"C

KG(Z°.Z°)'-I

o

o

°

)X wf(z )12'

j=()

from which we obtain #G(Z°: X°) =

I(pi (z°)X°1

VKG (z°, z°)

(*)

On the other hand, any f E L2 (G), f (z°) = 0, has the representation f(z) = D(f. 00 (Pi)L2(G)(Pi(z). i=I

which implies that

If'(z°)X°I

= I(1.(p0)L2(G)11(pi(z°)X°I < 11f11L2(G)1(Pi(z°)X°1.

This together with (*) yields the claimed formula. In the case H' = H" we only mention that 91 does not occur.

VI. The Bergman metric

190

Corollary 6.2.6. Let G be as in Theorem 6.2.5, z E G. Then fG (z; ) is positive

definite if and only if for any X E C", X

0, there exists f E Lh (G) with

f'(z)X 0 0. Because of the coordinate functions, the Bergman pseudometric for a bounded domain is indeed a metric. A domain G for which (6.2.1) holds and for which j8G is a metric is called 0-hyperbolic. Remark 6.2.7. Let MG denote the numerator of the formula in Theorem 6.2.5, i.e.

MG(z; X) := sup{If'(z)XI f E

IIf IIL2(G) = 1, f(z) = 0).

The description of fG in Theorem 6.2.5 has the advantage that (MG)GE6 is monotonic in the following sense:

if G C D, then Mo(z; X) < MG(z; X),

z E G, X E C".

Moreover, the system (MG)GE6 satisfies the following transformation rule:

if F : G -- D is biholomorphic, then

MD(F(z); F'(z)X)I detF'(z)I = MG(z;X),

Z E G C C" 3 X.

6.3 Comparison and localization The representation of QG in Theorem 6.2.5 reminds us the definition of the Carath6odory-Reiffen pseudometric. With this in mind we obtain the following comparison (cf. [Bur 2], [Hah 1,2], [Look]). Theorem 6.3.1. If the domain G C C" satisfies (6.2.1), then (a) yG(z;X) <.G(Z; X) forall z E G, X E C", (b) CG (Z', Z") < CG (Z', z") < bG (z', z"), z', z" E G.

Proof. Fix Zo E G, X E C", with yG(zo; X) > 0 and choose f E (7(G, E) to be extremal in the sense of the Carath6odory-Reiffen pseudometric, i.e. f (zo) = 0 and I.f'(zo)X I = YG(zo; X).

6.3 Comparison and localization

191

zo), belongs to Then the function g: G -> C, g(z) := f (z)KG(z. zo) / Vf'K-G LI(G) with g(zo) = 0 and IIgIIL2(c) < 1. Hence by Theorem 6.2.5 we get Ig'(Zo)XI

I

$G(zo;X) >

Kc

zo, zo) IIgIIL2(c)

If'(zo)XI =YG(zo;X).

To obtain (b) use Remark 2.5.3.

Remark. In the case that yc (z; X) > 0 we even have yc (z; X) < Oo (z; X)

-

On the other hand, there is, in general, no chance to compare the Bergman and the Kobayashi pseudometrics; cf. [Die-For 21, [Die-For-Her].

Theorem 6.3.2. There exist: a bounded pseudoconvex domain G C C3 with realanalytic smooth boundary, a sequence C G with Z, =: z` E 8G, and a vector X E C3 such that flc (z,,, X) xc (zL; X) V--,00

Remark 63.3. In Chapter VII we will see that there also exists a bounded balanced pseudoconvex domain in C3 whose Minkowski function is continuous such that is unbounded for suitable the sequence (fic(ZI; CG and (X'v)IEN C C3. Prof. With q(z) := Rez1 + Iz,12 + IZ2112 + IZ31L2 + IZ214IZ312 + Iz2I2Iz316, Z E C3,

we put G := (z E C3: q(z) < 0). Then it is easy to see that G is a bounded pseudoconvex domain with real-analytic smooth boundary. Moreover, G C 2B3.

We putz(t):=(-t,0,0)EG,0 G by g,(A) (-t, 0,1). Observe that g,(0) = z(t) and g,'(0) = (0,0, 1) which implies that

xc(Z(t);X) = xo(g,(0);9;(0))

/2 x(3)'112E(0;1) = I l

X,

1/12

Hence we obtain in2 6c(z(t);X) 2) _ xc(z(t);X) Bearing in mind the representation of PG in Theorem 6.2.5 we estimate the Bergman kernel KG(Z(t), z(t)) from above. We mention that the polycylinder >pc(z(t);X)(t

VI. The Bergman metric

192

PE(t) := B(-t, Et) x B(0, (Et)I/5) x B(0, (Et)1110) is contained in G if 0 < E << 1. So we get

(z(t ), z(t)) =

KG (Z(1), Z(0)

7r-3(Et)-2(Et)-2/5

(Et)-2/10

=

7r-3(Et)-1315

The last step in the proof consists of a lower estimate for MG(z(t);X), 0 < t < 1/2. Put d := (z E C3 : Re Z1 + IZ2141Z312 + Iz2121z316 < O). Obviously,

G C (2B3) fl G =: G' and G' is connected. Then, according to Remark 6.2.7, we have MG(z(t);X) ? MG.(z(t);X). Now for 0 < t < 1 /2 we define f, (z) := 412Z3/(t - ZI)2,

Z = (Z1, Z2, Z3) E G'.

Then f, is holomorphic on G', and moreover we obtain the following estimate for the L2(G')-norm of fr:

f J

f Iz312

I

f

r(:2.z3)

J-oo

JR ((t -

yi)2dx1dA(Z2,

Z3),

where r(z2, z3) = Iz214Iz312 + IZ212Iz316. The substitution v(t - x I ) = y, then yields r(z2.<3)

'112 II

dv

1

2(G,) < )6t4 f&2 1Z312

(t - X1

2

< 16t4

J

IZ312

)3dx1 dA(z2, Z3)

1

121<2 (t + r(Z2, Z3))2

Iz,1<2

JR (I + v2)2

dl(z2)dA(z3)

f2

r2

= 16t447r2 J s2 J (t + r4s2 + r2s6)-2rdr sds. 0

0

Finally, the substitutions r = t 1 /5 p and a t I / 10 = s lead to

'

[_I/IO

4

11111112(G') < C11 0

o

00

< c1

=:

L

C2t14/5

I

00

J0

or

3pt8/10

t2(1 +P2,21P2 a3p dpdv

+ p2Q2(p2 + Q4))2

+Q41)2dpdo

. 114/5

6.3 Comparison and localization

193

with positive constants C1, C2, which are independent oft. Because of f,(z(r)) = 0,

f, (z(t))X = 1, we find that MG(z(1);X) > C2 112 . t-7/5.

and therefore

G(z(t); X) /G(z(t ); X) > Ct

i/i2-1/io _.,

oo.

-o+

Remark 6.3.4. The above example shows that, in general, the Bergman metric is not majorized by a multiple of the Kobayashi-Royden metric. On the other hand, for the punctured disc E. we have

I _t2 _ YE.(t; l) fE.(t; l) 00', 1) = 22tllogtl r-o+) XE.(t;1)

00.

Thus, in general, there is no relation between these two metrics. Nevertheless, if we restrict our considerations to good classes of domains, e.g. the strongly pseudoconvex domains, we will find positive results; cf. the formula for the ball.

We conclude this section with a general localization result for the Bergman metric; cf. [Die-For-Her], [Ohs 2].

Theorem 6.3.5. Suppose that G is a bounded domain of holomorphy and let zo E aG. Then for neighborhoods U, = U, (zo) CC U2 = U2(;.o) of zo there exists a positive constant C such that for all z E V fl U1, X E C", where V denotes any connected component of G fl U2, we have (i) (ii) (iii)

(1/C)Mv(z1X) < MG (Z; X) :5 MV (17; X),

(1/C)Kv(z, z) Z) < Kv(z, z), (l/C)l3v(z; X) < 9G(z; X) < CfJv(z; X)-

Proof. Obviously (iii) is an immediate consequence of (i) and (ii). Since the proof of (ii) is very similar to that of (i), we restrict ourselves to (i). Remark 6.2.7 gives the second inequality in (i); so only the first one has to be verified.

Choose intermediate neighborhoods U, CC U3 := U3 (zo) CC U4 U4(zo) CC U2 and a cut-off function X e CC (U4), 0 < X < 1. with Xlu, = 1

VI. The Bergman metric

194

Now fix z' E v n U,, X E C", and f E L2 (V) with IIf11L=(v) = 1, f(z') = 0. Moreover, put v(z) := 2(n + 2) log Ilz - z'II and define the (0. I)-form

"a

a:=a(f.X)=fEaXXdzj j=1

a'J

(by trivial extension) as a a-closed C"'-form on the whole G. Because of L lot 12 (z) exp(-wp (17))d), (z)

= f If(z)I2

lax (z)1-exp(-V(z))dk(z) < C, < oo

j

(Cl independent of f and V), the form a belongs to Leo 1) (G. exp (-(p)). the Hilbert space of (0. 1)-forms that are square-integrable with respect to the weight function exp(-(p). Observe that V is plurisubharmonic. By Hormander's theory (cf. Appendix PSC) there exists a Cx-function g : G -+

C with ag = a and Ig(z)12(I + IIzII`) `exp(-(P(z))dX(z) < C1.

Because of with IIgIIL'(G)

IIzII-)

> C2 > 0 on G, it follows that g E L2(G)

C1/C2 _: C3.

Put h := Xf - g: then h is a C°`-function on G which satisfies the CauchyRiemann equations ah =- 0, i.e. h is holomorphic. Moreover, we obtain IlhIIL'(G) < 11f 1IL2(V) + 1191IL2(G) < I + C3 =: C.

According to the choice of X, we have 5(Xf) = 0 on U3 n G, which implies that g is also holomorphic on U3 n G and satisfies Ig(<)1'

IIz -

cc.

''III' (1 +

Now, the existence of this singular integral means that

g(z') = 0 =

S

a <.j

(z'),

I < j < n,

9.4 T hi M Ml AIN O W8J1 to

191

and therefore

h(z') = 0,

If'(z')XI = Ih'(z')XI < IIhilo(G)

h(`)

X < C MG (Z'; X).

IIhIIL1(G)

Observe that we may assume that h # 0. Since f is arbitrary, it follows that Mv(z';X) < CMG(z';X).

6.4 The Skwarczynski pseudometric Let (H, ( , )H) be an arbitrary Hilbert space. By PH we denote the associated projective space, i.e. PH := H \ {0}/.r, where x y if x = Ay for a suitable I E C., x, y E H \ (0). Then PH is a complete metric space with respect to the distance

dH([x), [y]) := dist([xJnSH,

[y] E PH,

IIxiIHII)IIH

(6.4.1) where

This general observation was used by M. Skwarczynski (cf. [Skw 11) to introduce

another pseudodistance on domains in C. Let G be a domain in C" with (6.2.1). i.e. KG(Z, z) > 0 for all z E G. The following map

r: G - 1P(L2(G)).

r(z) := [KG(., z)]

enables us to introduce the following continuous pseudodistance on G x G pr, (z', z")

1

dL2(G) (r W), r(z")) =

I-

IKG(zz")I KG (z'. z') KG(z

Z")

PG is called the Skwarrryriski pseudodistance. Observe that the following conditions are equivalent:

(i) r is injective; (ii) for each two distinct points z', z" E G the functions linearly independent; (iii) PG is a distance.

This is, for example, the case when G is bounded.

Remark 6.4.1. (a) PE(z', Z") = m(z', z"), z', z" E E.

z'),

z") are

VI. The Bergman metric

196

(b) By Proposition 6.1.7 we conclude that if F : G -* D is biholomorphic, then PD(F(z'), F(z")) = PG(Z', z"), z', z" E G. (c)

PExE((0, 0), (A, A)) = (1 - (1 -

IAI2)2)1/2

> (1 - (1 - 1Al2))1/2 = PE(0, A),

0 < IAI < 1,

i.e. the Skwarczytiski pseudodistance fails to fulfill the General Schwarz-Pick Lemma.

Nevertheless, according to Chapter IV, the system (PIOG16 is an in-contractible family of pseudodistances with respect to all biholomorphic mappings.

Proposition 6.4.2. For any domain G we have top PG = top G if L2(G) contains the coordinate functions.

Proof Obviously, top PG C top G. Now fix Z' E G and a sequence (z1 ) jEr C G with lim j. oo pG (z, z j) = 0. Because of (6.4.1) we may assume that e'8JKG(-, zj)

tim

j-oo

-+

tz

i.e. zj) Since 1 E Lie, (G),

K ((zj zj)

KG Z', z')

KG(., z') with Cj := e49

lim Cj = lim (1,

j1, 00

_ KG(', z')

j-+oo

= 0,

L2(G)

KG(z', z')l

KG(Zj, zj).

zj))L2(G) = (1, KG(-, Z'))L2(G) = I-

If we denote by 2rk the k-th coordinate function, then we obtain lim rrk(Zj) = lim

1

-(rrk, Cj

= j-+oo lim (Irk,

Zj))L2(G) = 7rk(Z'),

i.e.

lim zj = Z'.

j moo

Hence the two topologies coincide.

Recall that (PG)GEe is not an m-contractible family of pseudodistances in the strong sense. So it is not clear how to compare pG and c*. To give an answer we need the following estimate of the Skwarczyfiski pseudodistance by an expression similar to that of Theorem 6.2.5 (cf. [Skw I ]).

6.4 The SkwarczytSski pseudometric

197

Theorem 6.4.3. Suppose that G C C" satisfies (6.2.1). Then PG z(', z") <

MGZ( ', Z")/

KG (z', z') <

Z")

2 PG(,, z

Z',

z EG,

where 1NG(z', z") := SUP( If (z')I : f E L' (G), 11f 11V(G) = I. f (z") = 0).

Proof. Write KG g(z")

z") +g with g E

z') =

and a = KG(Z Z,)/KG(Z z").

z")) = 0

= (g,

0, then

Therefore, by definition, if II9IIL2(G)

Ig(z')I

MG(Z',

_

z")]1 and a E C. Then

I(g,

z'))L2(G)1

= 11811E=(c)

IIgIIL2(G)

II811L2(G)

On the other hand, if f E Lh(G), Ilf IIL2(G) = 1, and f (z") = 0, then

If (z')I = I (f,

z") + g)L'(G)I 5 IIgIIL2(G)

Z'))L2(G) I= I (f

Hence with II81IL2(G) = MG(Z', z") we find that z") KG (z', z') =

(

MG(Z',

KG ' , z ' )

l KG(z'+ z')

=Pc(z,z)

1

-

KG (Z', Z")12

IKG(Z'+ z")I

+

KG (Z

Z

)

KG (Z 11,

KG(z',Z')

Z") KG(z"

By virtue of the Schwarz inequality the claimed inequality follows.

0

Theorem 6.4.4 (cf. [Bur 3]). Let G satisfy (6.2.1). Then r < N/2-PG-

Proof. Use Theorem 6.4.3 and proceed as in the proof of Theorem 6.3.1.

0

Remark 6.4.5. (a) The proof in [Bur 3] is different from the one presented here. It is based on the following inequality IKG(z'+Z") I'

KG(z',

z')KG(z

<

0 - If(z')I2)(l I I - f (z')f

If(z )I2),

f E O(G, E), z', z" E G,

and it gives the more precise inequality (2 - pc )p2 > cc

VI. The Bergman metric

198

(b) In Chapter VII we will construct a bounded pseudoconvex balanced domain in C3 whose Bergman distance and then, in particular, its Skwarczy6ski distance, is not majorized by any multiple of its Kobayashi distance (cf. Theorem 6.4.6).

We conclude this section with calculating the inner distance associated to pG (cf. [Maz-Pfl-Skw]).

Theorem 6.4.6. Let G E 6. Then pG = (1/..f2_)bG. Proof. According to Lemma 4.4.3 we only have to prove that PG is a C `-pseudodis-

tance with VpG = (1/J)fG. So, fix a E G and take (z;,),,N, C G with z for all v and limy-.,(z,, - zl;)/Ilzv z;, = limvo z;,' = a, X E C". Then we write zv )

PG (z

z' z" KG(Zv, Zv)

KG (Z,' Zv)(

KG(z', zv)

KG(z, zv) + IKG(zv, z)I)

Here 4)(z, w) := KG(Z, z)KG(W, w) - KG (Z, w)KG(w, z), z, w E G. is a C°O-

function with 4) > 0 and 4)(z, z) = 0, z r= G. Therefore, the Taylor formula up to second order and the holomorphicity properties of the Bergman for 0(., kernel lead to PG (z;,, z' )

lim -00 IIzv - zV"II

_

1

2KG (a., a)

a2KG

azvai

(a,a)KG(a,a)- aKG(a,a)aKG(a,a)\ X"kA azv az, )

_ (1/2BG(a;X))1 2.

Corollary 6.4.7. (a) For G E 6 we have cG < LPG < ,f2-pG' = bG > cc,. (b) If G is bounded then top G = top bG = top p'.

Notes

In the twenties the study of square integrable functions was initiated by S. Bergman in order to solve the classification problem for domains in C". Recall that the

Notes

199

Euclidean ball and the bidisc in C2 (which obviously are homeomorphic) are not biholomorphically equivalent. i.e. there is no analogy for the Riemann mapping theorem in higher dimensions. These investigations had led Bergman to introduce his kernel and his metric which became to be known as the Bergman kernel and the Bergman metric. For a complete list of Bergman's papers see [Ber 21. In 1976, M. Skwarczytiski introduced his pseudodistance which is based on the Bergman kernel.

For a long time it was difficult to work explicitly with the Bergman metric. But with Hormander's 5-theory an important new tool entered the theory of L2holomorphic functions. Deep results were found in the theory of the boundary behavior of biholomorphic or proper holomorphic mappings. A glance into the Mathematical Reviews over the last 20 years easily proves the importance of L2holomorphic functions in many branches of complex analysis.

200

VI. The Bergman metric

Exercises 6.1 (cf. [Wiel). Put Stk := S2 U f)4k, k e N, where

fl:_{(z,w)EC2: Iz1 <2e,IwI <2e} U {(z. W) E C2:

IzI > e, IwI < i/(IzI log IzI)) U {(z, w) E C2: IwI > e, IZI < I/(IwI log IwI)} and

SZm :=

{(Z. w) E C2: IZI > 1, IwI > 1, I IzI - 1101 1 <

Prove that dim

(IzI +IIwI)m } '

m E N.

(SZk) = k or more that Lh (1lk) = span{ I. _W..... (zw)k-1 }.

Hint. Verify first that a monomial znwy belongs to L2((1) itT p = q. Then show that zewP E Lt2, Mk) ifi: p < k.

6.2. (a) Prove the formula in Example 6.1.9. (b) Prove the first part of Theorem 6.1.10 only under the assumption that F is a proper holomorphic mapping. Hint. Use Theorem 6.1.8 instead of Proposition 6.1.7. 6.3. Use Remark 6.2.4 to prove that

bc, xc((zi, Z2'), z (i, z2 )) :

V;G. (iz , zi) + bc2(zi, zi

6.4. Let D(k) := {z = (z,, ... , z") E C" : Izi I2* +

zJ, zj E Gi CC C"'.

Izi 12 < 1), k > O. Prove:

n+I

Ko(k)(z, w) = kx-"g(z.

w)-"-Ilk Ebfg(z. w)i/k((1 _ g(z. w))Iik - zlwt)-'. J=2

where g(z, w) := I - E =2 zit'ui (cf. {d'An}). Hint. Use an orthonormal basis.

Exercises

201

6.5. Let D := D(p1..... p,,) := IZ E C": IZII-/P' +... +

Iz"I'-/P. < 11. pj E N.

Prove that

Kn(z, w) =

I 11

pl ...

P'

an

I

PI,

ax l

where ) j j = 41"E1,. I < j

L ...

... a .r

1 i=1

P.

I

1 - j,,1 - ... - 'j..n j.=l

i =: n.

1
pi, I < i < it. here Ej,; are all the p; -th roots of

unity (cf. IZin]).

6.6. Let D(k) be as in 6.4. Prove (cf. [Yin]) that fin(A) S Sv(k) on D(k), where

a-'logg(Z) Sn(A)(z; X)

X'Xj' II:II'-)-(rek+1l/k.

R(Z) :_ (1

t» I.

(zl.z)ED(k). XEC".

6.7. Let G be a bounded domain in C" with a transitive group of automorphisms. Prove (cf. [Look]) that there exists a positive constant k = k(G) such that for any bounded domain D C C" and any F E O(D. G) the following inequality is true:

PG(F(z): F'(z)X)

kfD(:.: X).

E D. X C C".

6.8. Let G C C be a bounded domain with smooth Cx-boundary. Denote by gc(a. ) the classical Green function of G with pole at a E G; see Appendix GR. Recall from the classical potential theory that gG is a CI-function of both variables

on0xG\((:.Z):ZEG). Prove that

n

jTu= (z. w) = K(;(z. w).

z. u' E G. z

96

w.

Hint. Use Stokes' formula to show that the function

2 a' 7r Odyaz

has the reproducing property (with respect to the L2-scalar product) for all h E C"(G)nC)(G). Finally. observe that the space C'(G)nO(G) is dense in L2(G).

Chapter VII

Hyperbolicity and completeness

In the previous chapters we have already observed that the Kobayashi hyperbolicity does have a deep influence on the topology induced by k. On the other hand, we have an example of a domain G which is c-hyperbolic but top G 4 top cc. First in this chapter we study the relations between the different notions of hyperbolicity.

7.1 Global hyperbolicity

Remark 7.1.1. Let G C C". Consider the following conditions: (i) G is bounded; (ii) G is biholomorphic to a bounded domain; (iii) top G = top cG; (iv) G is c-hyperbolic; (v) G is k-hyperbolic; (vi) G does not contain non-trivial entire curves, i.e. every holomorphic map f : C -). G is constant; (vii) for any f : C -+ C", f (k) = a + Xv, v 36 0, the image f (C) does not lie in G, i.e. G does not contain any affine complex line; (viii) no complex fine through 0 stays inside G.

Then, obviously, (i) + (ii) -+ (iii) -- (iv) -+ (v) -+ (vi) -+ (vii) -9, (viii), A domain G satisfying the property (vi) is sometimes called Brody hyperbolic. We will see that for domains of more restrictive shape some of the converse implications are true.

Theorem 7.1.2 ([Kod]). Any k-hyperbolic balanced domain G in C" is bounded.

Proof. Suppose that B(0, R) CC G (R > 0) and assume that G is not bounded. Then G contains a sequence (ZW)i,EN of points z,, with R < IIz,II / oo. If we V- 00

U Clobal hyperbolicity

put

M'(

E E, then we get

%zV,

kG(O, Rz,/Ilz,ll) < p(0, Rlllz,,ll) - 0. V-4O

has an accumulation point a with Ilall = R, it follows that Since kG(0, a) = 0, which contradicts the assumption of hyperbolicity. 0 By Theorem 7.1.2 we see that there is no difference between k- and c-hyperbolicity in the class of general balanced domains. For balanced pseudoconvex domains in C2 the following characterization of the Brody hyperbolicity is due to J. Siciak.

Theorem 7.1.3. Let G = (z E V: h (z) < 1) be a balanced pseudoconvex domain. Then the following properties are equivalent: (i) G is Brody hyperbolic; (ii) G does not contain complex lines through 0;

(iii)h(z) >0ifz54 0. Proof. The implications (i)

(ii) -+ (iii) are trivial and remain true even in

arbitrary dimension.

Now, we assume that (iii) holds. Suppose that there exists an entire curve f = (fl, f2) : C -> G with, for example, f, not identically constant. Put u := h o f . Then u is a subharmonic function on C with u < I which implies that u - C for a certain constant C E 10, 1); cf. Appendix PSH 25. If C = 0, then our assumption leads to f - 0 contradicting our choice of f,. Thus we may assume that 0 < C < 1. In particular, f2(A) # 0 if f, (.k) = 0. Put v := - log C + log h (l, ) on C. Then v is a subharmonic function satisfying the following estimate

-oo < v(X) < C, + log(l + I)4).

A E C.

(7.1.1)

Observe that for A E C \ fi 1 (0) we have

loglfl(A)l = -u(m(A)),

(7.1.2)

where m := f2/fl E O(C \ fi ' (0)). Because of (7.1.2), m is not identically constant. Applying Picard's Little The-

orem it follows that there are points , , ii2 E C such that D := C \ (c, , 2 } C m(C \ fi 1(0)). Now, we study v on D. Fix o E D. Using the local invertibility of m we conclude via (7.1.2) that v is a harmonic function in a punctured neighborhood of ia. Moreover, v is locally bounded which implies that v is harmonic in a full neighborhood of a. Hence we have v harmonic on D with estimate (7.1.1).

VII. Hyperbolicity and completeness

204

On the other hand, we note that v cannot be harmonic on the whole C. Otherwise

v = Re g, g E 0(C), and therefore we find from (7.1.1) that leg(') I = e°(;L) < ec-(1 + IXI) for A E C.

So Liouville's theorem implies that the entire function e8 is of the form ,% -+ a+bA, which obviously leads to ex and also v being identically constant. By (7.1.2) this contradicts the assumption that f, is not identically constant. Thus we may assume that v is not harmonic in a neighborhood of fit. Using the Riesz decomposition theorem, v can be written as

v(A) = vo(A) +

' 2-r

JIx-Cil,

log IA

- Id u(A),

IA - t I < r <<

r); where A is a nonnegative measure given by Av and vo is harmonic in cf. Appendix GR 10. Because of the harmonicity of v outside , it follows that µ is concentrated on hence

v(),)=vo(A)+alog

IA - lI < r<< I

with a > 0. Recall that a generalized function whose support consists of a single point can be represented as a linear combination of derivatives of the Dirac delta distribution. Now, µ > 0 allows us to exclude derivatives in that combination. Because of the definition of v, the constant or has to be zero. Hence v is harmonic near 41; a contradiction.

Example 7.1.4 (cf. [Azu I)). We start with the following subharmonic function (p: C -+ [-no, no) rp(A) := maxilog Al, IE k2 log IA - iI),

A E C,

k>2

and on C2 we define

h(z)

f

Iz2I exp V(zl /z2)

if

z2 00

(zI I

if

z2 = 0

Observe, that h is plurisubharmonic on C x C. and upper semicontinuous on C x C. By Appendix PSH 21 it follows that h is plurisubharmonic on C2. Therefore, h defines a balanced pseudoconvex domain G := {z E C2: h(z) < lj in C2, which is not bounded. For example (1/2, k/2) E G, k > 2. Hence G is not k-hyperbolic.

7.1 Global hyperbolicity

205

On the other hand, Theorem 7.1.3 leads to the Brody hyperbolicity of G. This example shows that the property "Brody hyperbolic" is strictly weaker than k-hyperbolicity, even in the class of balanced pseudoconvex domains; see also Exercise 7.1.

Remark 7.1.5. If a balanced domain G has a continuous Minkowski function h, then the condition (viii) of Remark 7.1.1 implies that G is bounded. Observe that, in particular, all complete Reinhardt domains belong to this special class of balanced domains.

Corollary 7.1.6. For a complete Reinhardt domain G the properties (i) - (viii) of Remark 7.1.1 are equivalent. Proposition 7.1.7. For a balanced pseu doconvex domain the conditions (vii) and (viii) are equivalent.

Proof Without loss of generality we may assume that n > 2. Let f a + .kb, a, b E C", b # 0, be an affine complex line inside G. If a, b are linearly dependent, then the line f (C) contains the origin. Now suppose that a, b are linearly independent. Since A.a +pb = A(a + 4) E G

if 0 < JX < I we obtain E. a + C C. b c G. Moreover, since 0 E G we have eE . a + eE b C G for a suitable E > 0. By hypothesis G is a domain of holomorphy. Hence the Kontinuitatssatz implies that C b belongs to G.

0

In the class of convex domains containing the origin the following result due to T. J. Barth (cf. [Bar 3]) is true.

Theorem 7.1.8. Any convex domain G C C". 0 E G, that satisfies condition (viii) in Remark 7.1.1 is biholomorphieally equivalent to a bounded domain.

Proof. Let b, := (1, 0...., 0). By assumption there exist a complex number a, and a unit vector a, E C" such that the following is true:

a, b, r: aG and G C {Z E C":

0

oil

Now, if n > 2, choose a unit vector b2 with (b2. a,) = 0. Then the same argument leads to a number a2 0 and a unit vector a2 with a2b2 E aG

and G c (z E C": Re(;. - a,b2, a2) < 0).

206

V11. Hyperbolicity and completeness

Obviously, the vectors a,, ay are linearly independent. Continuing this construction

we obtain linearly independent unit vectors a,...., a,, such that "

GC n(ZEC": Re(z-aibi.aj) <0) i=I with suitable numbers a1 and vectors bj. Therefore, the mapping

r:C"-'C", n(z):=((z-a,b1,at),.. gives a biholomorphic map from G onto a domain D C (z E C" : Re zj < 0, I < j < n). The latter is biholomorphically equivalent to a bounded domain. O Remark 7.1.9. There are examples of domains, most of them easy to obtain, which satisfy the condition (i + t) of Remark 7.1.1 but not (i), I < i < 7 (cf. lAhl-Beul and Theorem 2.4.3). Here we only mention that C \ (0. I) is k-hyperbolic but not c-hyperbolic. This result follows from the well-known fact that the unit disc is the universal cover of C \ (0. I } and the following theorem.

Theorem 7.1.10. Let n : G -* G be a holomorphic covering. Then we have: G is k-hyperbolic iff G is k-hyperbolic.

Proof Since for any k-hyperbolic domain the k-topology and the II-topology coincide, the implication. "-+" is an immediate consequence of Theorem 3.3.7. Now let us assume that G is k-hyperbolic. Of course, it suffices to show that II

k6(1, y) > 0 whenever x, ,v E G are different points with Yr(i) = n(,v") =: x. Since any II It-open subset of G is also kG-open, there exist a kc;-ball Bc, (.r, r) =:

U and open disjoint neighborhoods U, := U,(i) and UZ := J2(y) such that R If),: Ui -+ U is biholomorphic. Thus any continuous curve of in d connecting .r and v has the kb-length greater or equal than 2r which, by Proposition 3.3.1, implies that k,; (x, y) > 2r. Hence d is k-hyperbolic. 0 So far we have discussed the notion of hyperbolicity on the level of the Caratheodory or Kobayashi pseudodistance. Now, we ask whether it is possible to express the property "hyperbolic" in terms of the associated metrics (cf. also Chapters 11 and IV).

7.2 Local hyperbolicity

207

7.2 Local hyperbolicity Let G be any domain in C" and let S = SG : G x C" [0, oo) be an arbitrary pseudometric on G. We say that G is S-hyperbolic if for any Z E G there exist a neighborhood U = U (:o)) C G and a positive real number C such that Sc; (z. X) _

S(::X)>CIIXII.:EU.XEC"(cf.§4.1). Remark 7.2.1. (a) Any bounded domain is y-hyperbolic, and therefore S-hyperbolic. A-hyperbolic, and x-hyperbolic. (b) Because of Proposition 2.3.1 the notions of "y-hyperbolic" and "c-hyperbolic" coincide for plane domains. But fl nothing is known about their relations in higher dimensions

0

On the other hand. we have the following complete description of Kobayashi hyperbolicity.

Theorem 7.2.2. For a domain G in C" the following properties are equivalent: (a) G is k-hyperbolic: (b) top G = top kG: (c) for any domain G' C C". any rd E G', any :' E G. and any neighborhood U = U (:') C G there exist neighborhoods V V (w') C G' and U = U(:') C U

such that if f E O(G'. G) with f (w') E U, then f (V) C U: (d) condition (c) is true for G' = E and w' = 0 E E; (e) G is x-hyperbolic; (f) for any :' E G there exists a Kobayashi-hall around z' with finite radius r, which is a hounded subset of C":

(g) any point :' E G has a neighborhood U = U(:') C G such that, for:, w e U. kG (:, w) > M II: - u' II, where M is a suitable positive constant.

Proof We already know that (a) implies (b).

(b) - (c). Let G'. w'.:'. and U be as in (c). Without loss of generality we may assume, because of (b), that U = BA,; (:'. r). Set V := BA,;. (uw'. r/2) and U := r/2). Now take f E O(G', G) with f (w') E U. Then for w E V we obtain

k;(:' f(w)) -< kG(:'. f(rr'")) +k(;(f(u'') f(w)) <
(d) - (e). Fix a point :' E G with U := B(:'. r) CC G. For 0 E E and :' E U there exist neighborhoods V = V(O) C E and U = U(z') C U such that, if

VII. Hyperbolicity and completeness

208

rpEO(E,G)with q(0)EU.then rp(V)CU.Thus. ifB(0,s)CV(0<s
XG(z:X)>sxt!(z:X)>CIIXII (C>0) whenever z E U and X E C".

(e) - (f). Let Z' E G. By (e) we may assume that we have the following inequality on U := B(z', R) CC G: xG(z:X) > CIIXII (C > 0). Take now any point z with kG(z', z) < CR. By Theorem 3.6.4 it follows that z E B(z': R), i.e. the ball BkG(z'. CR) is a bounded subset of G. (f) -+ (g). Again we start with a fixed point z' E G. Observe that for any JP E O(E, G) with V(0) E B(z'. S) CC G we have the following estimate: kG(IP(A) z') < kG(co(A). W(0)) + kG((p(0). z') < p(A. 0) + kG((P(0) :')

If, in addition, S << 1, then kr;((P(A), z') < r for III < S, where r denotes the radius according to M. So we conclude: 11 p(A)II < M if Al I< S. where M is independent of V. Then, applying the Cauchy-Riemann inequality we obtain xG;(z.X)?M11X11

with M>0 (zEB(z',S),XEC").

Now, we consider two points z', z2 E B(z'. S/2). By Theorem 3.6.4 we get

xG(a(t);a'(t))dt: a a piecewise C'-curve in G connecting z' and z2}

inf( 0

> MHz' -:211. (7.2.1)

The last inequality is clear as long as we consider curves a inside B(z'. 8). On the

other hand, the xG-length of any curve that leaves B(z', S) is at least 2M 8/2, which immediately implies (7.2.1). (g) -+ (a). Take two different points z', z" in G. By assumption we can choose

a ball U := B(z', R) CC G with z" f U and a positive constant M such that kG(Z, w) >- MIIz - wII whenever z, w E U. Hence the kG-length of any continuous curve a in G that connects z' and z" exceeds M R/2. Since kG is inner, this implies kG(z'. z") > 0. Hence G is k-hyperbolic.

Another description of the k-hyperbolicity of a domain G in terms of topological

properties of the embedding O(E. G) -+ C(E. G*), where G* denotes the onepoint compactification of G, is given in [Aba 4J.

209

7.2 Local hyperbolicity

Remark 7.2.3. Observe that there exists a domain G in C2 which is not khyperbolic and thus not x-hyperbolic but such that xG (z; X) > 0 for every z E G and X E C" \ (0) (cf. Remark 3.5.11). Therefore, in general, we have to distinguish carefully between the S-hyperbolicity and the pointwise S-hyperbolicity.

The property (d) of the above theorem can be used to verify the following sufficient criterion for hyperbolicity.

Corollary 7.2.4. Any taut domain G in C" is k-hyperbolic.

Proof Suppose the contrary, which means that condition (d) is violated. This

implies the existence of a point z' E G, a neighborhood U = U(z'), a sequence A, -) 0 in E, and a sequence (gyp,) C O(E, G) with z' but

f U. For taut domains such a combination is impossible.

In general, it seems rather difficult to calculate the Kobayashi-Royden metric explicitly. On the other hand, plurisubharmonic functions are very flexible. So the Sibony metric may serve as a tool to find a large class of S-hyperbolic, and therefore also k-hyperbolic domains in C".

Theorem 7.2.5 (cf. [Sib 4]). Suppose that there is a bounded plurisubharmonic function u on a domain G in C", that is C2 and strictly plurisubharmonic near a point zp E G. Then SG(z; X) > C11 X II, Z E V, X E C", for a suitable C > 0 and a suitably chosen neighborhood V = V (zo) C G. Proof Without loss of generality we may assume that

(i) -1 < u < 0 on G, (ii) u is strongly plurisubharmonic and C2 on U := B(zo, 3R) CC G, (iii) (Gu)(z; X) > a1IX II2, z E U, where a > 0. To prove the required inequality, we are going to construct a function of the Sibony class SG(z'), Z' E V := B(zo, R) (cf. Remark 4.2.1). Put

v,(z) := X (Ilz RZ'112) exp(tu(z)), 2

z E G, t > 0,

where x: R -+ [0, 11 denotes an increasing CO0-function with X(t) = t if 0 < t < 1/2, and X(t) = 1 if t > (3/4)2. Obviously, v, is of class C2 near z'. v,(z') = 0,

and0> I:

VII. Hyperbolicity and completeness

210

First observe that

logv,=tu log y = log

on

G\B(z',(3/4)R),

II

Rz' 112 + to

on B (z', R/4).

Moreover, for 1 /2 < Hz - z' II / R < I we easily obtain

(G1ogv,)(z;X)>(-m+at)11X112> 11X112 ift>to=(m+1)/a. We point out that to can be chosen independently of z'. Hence for any z' E B(zo, R) we have constructed the function to SG(z'). Therefore,

SG(z';X)

v,; belonging

I(ICvb)(z';X)Jtn = 11X11(1/R)exp(ttu(z'))

? IIXIH(I/R)exp(-ti) with ti := to/2.

0 Corollary 7.2.6. Any connected component of the set (z E C": * (z) < 0), where * is a strictly psh C2 function on C", is S-hyperbolic, and so also k-hyperbolic. Remark 7.2.7. Suppose that we are in the situation of Theorem 7.2.5 except for the C2-condition. Then, as in the proof above, we may define the function v,, for z' E B(zo, R). Similarly as above, the reader may verify that v,a is a log-psh function on G with 0 < v,0 < I and v,o(z') = 0.

Now let X E C" and tp E O(E, G) with ,(0) = z' and a(p'(0) = X (a > 0, X 0 0). Then v := v,0 o (p is log-subharmonic on E with v(0) = 0, 0 < v < 1, and v(A)/1A12 is bounded near the origin. Therefore, the extension theorem for subharmonic functions and the maximum principle lead to 2

1 > limsupv(A)/17112 1-0

11X112

Q R

exp(tou(z')),

i.e. xc(z';X)

0

11X11(1/R)exp(-to/2) It is unclear whether this inequality still holds for SG.

0

Remark 7.2.& Let G denote the unbounded component of (z = (zt, z2) E C2 : Re(z3 + z2 + zi) + Izi 12 + IZ212 < 01

(cf. Corollary 7.2.6).

7.2 Local hyperbolicity

21

In (Suz 21 it is shown that G is y-hyperbolic. Using a similar argument one can also prove that G is even c-hyperbolic. a We do not know of any example of a domain like the one in Corollary 7.2.6 that is neither y- nor c-hyperbolic

0

Under the hypothesis that G is a k-hyperbolic domain the following localization result was formulated by H. Royden (cf. [Roy], also [Gra 11).

Proposition 7.2.9. Suppose that G is a k-hyperbolic domain in C" and let U C G be our subdomain. Then

xt!(z:X)
cothka(z. w) = sup{ I/r > 1: 3cp E O(E. G): q.(0) =

V(r) = w}.

Z, wEG. z# w. Now fix Z E U and X E C" and let tp E O(E, G) be an arbitrary analytic

disc with (p(0) = z t). a(o'(0) = X with a > 0. Moreover, choose s > 0 with I Is > coth ka (z(,. G \ U). Observe that ka (zo. G \ U) > 0 since top ka = top G by hyperbolicity. Hence every * E Q(E.G) with VI(0) = zo maps sE into U. In particular, if we put 0(A) := cp(sA). A E E. then cP E (9(E. U) with Cp(0) = ;(). (a/s)o'(0) = X. i.e. a/s > xt- (;.o: X). Since the choice of a and s was arbitrary, the claimed inequality follows.

0

In a forthcoming chapter we will discuss more details of the localization of the Kobaya hi-Royden metric. We will then exploit Proposition 7.2.9 We conclude this section with a generalization of the Big Picard Theorem (cf. Chapter I). Recall that this theorem can be stated as follows.

Any holomorphic function f : E. -+ C omitting at least two complex values extends as a holomorphic or a meromorphic function to the whole unit disc. The following generalization of this formulation of the Picard theorem is given by M. H. Kwack (cf. [Kwa]).

Theorem 7.2.10. Let f : E. -* G be a holomorphir map, where G is a k0 the hyperbolic domain in C". Assume that for a sequence (Ac )R i C E. with kc sequence (f (Ac) )' , con verges to a point Z s E G. Then f extends to a holo» corphir

map f : E -+ G.

212

VII. Hyperbolicity and completeness

Proof. Without loss of generality, we may assume that the sequence rk := IxkI is strictly decreasing and that zo = 0 E G. We are going to prove that the function

f given by f (k) := f(;Q, A E E., f (O) := 0 is continuous on E. Fix s E (0, dist(0, 8G)). Since G is k-hyperbolic, we have BkG(0, 8s) C B(0, e) with Sf being a suitable positive number. By assumption there is ko E N such that for k > ko we have f 00 E Bk, (0, 8E/2). Applying Corollary 5.8 for A E E,, I), 1 = rk, we obtain

kG(O f(A)) < kG(0, f(At)) +kG(f(Ak), f(A)) 8£

< kG (0, f (,Xk)) + kE. (Xk, A) <

2

+

n - log rk

< SF

if k > k, > ko.

Therefore, f (aB(0, rk)) C B(0, e) if k > k1. Now, it suffices to show that if rk+I < Al I< rk, k >> 1, then f (a,) E B(0, s). We proceed by supposing on the contrary. Then we may assume that there are numbers rk+i < bk+i < ak < rk < bk, k > 1, such that f ({A E E: at < IAI < bk)) C B(0, e), 0, f (8B(0, ak)) n aB(o, e) 0. f(aB(0, bk)) n aB(0, e)

Similarly as above, we may also assume that there are points z', z" E aB(0, s) with

klif(aB(0, ak)) = z'

klim

f(aB(0, bk)) =

z

(otherwise, take an appropriate subsequence). Choose a holomorphic funct g: C' -± C with g(0) = 0 and g(z')g(z") 36 0. Consequently, we have

g o f (8B(0, at)) k -oo g(z') 0, g(z") # 0, g o f (3B(0, bk)) k-oo

gof(Ak)-0. k-ac Let B' (resp. B") be a small disc around g(z') (resp. g(z")) such that 0 0 0 B"). Then for sufficiently large k we obtain

gof(dB(O,ak))CB'c8' 0gof(Xk), gof(dB(O,bk)) C B" C B" -1gof()Ik).

B' (resp.

7.3 Completeness - general discussion

213

Application of the Cauchy theorem leads to

f

dA

I

(g o f)'(k)dk

where c = a or c = b. From here the argument principle implies that the function

g o f - g o f (kk) is without zeros in the annulus (k E C : ak < Ikl < bx }: a

0

contradiction.

7.3 Completeness - general discussion Let G be an arbitrary domain in C" equipped with a continuous distance dG. e.g. dG = cG when G is c-hyperbolic. In general. the dG-topology may be different from top G. Therefore, we have to distinguish very carefully between different notions of completeness. We say that G is weakly dG-complete if the metric space (G. dG) is complete in the usual sense. Since we are mainly interested in the interplay between topdG and top G. the notion of "dG-complete" seems to be more convenient for our purpose.

A domain G is called dG-complete if any dG-Cauchy sequence (z,,),.EN C G converges to a point z() E G with respect to top G. i.e. IIz, - zoll - B. Moreover, there is also another important notion which has been borrowed from differential geometry. Namely. we call the domain G dG--fmitel1 compact if all dG-balls are relatively compact (w.r.t. top G) inside C. Of course, here we may also formulate the "weaker" property weakly dG-finitely compact. If we deal with the Carathr odory, Bergman, or Kobayashi distances (!) we will always omit, for simplicity, the subscript G. e.g. we just say that G is c-. b-, or k-complete etc.

Remark 7.3.1. (a) For any dG-finitely compact domain G the topologies top G and topd0 coincide. (b) Moreover, the following implications (i) -+ (ii) -). (iii) are obvious, where (i) G is dG-finitely compact; (ii) G is do-complete: (iii) G is weakly dG-complete. (c) Already here we would like to point out that for the Carathdodory distance the question whether (iii) implies (i) is still unsolved; cf. Exercises 7.3 for the case of complex spaces.

VII. Hyperbolicity and completeness

214

As is known from differential geometry. a theorem of H. Hopf asserts the equivalence of Cauchy-completeness and finite compactness. This result was generalized by W. Rinow (cf. (Ring) and S. Cohn-Vossen (cf. [Coh-Vos]) to the situation we are interested in.

Theorem 7.3.2. Let dG denote a continuous inner distance on a domain G c C". Then top G = top dc; and, moreover, the following properties are equivalent: (i) G is do;-finitely compact; (ii) G is weakly dG-finitely compact; (iii) G is dc;-complete;

(iv) G is weakly dG-complete;

(v) any half-segment a: [0, b) - G (i.e. any II II-continuous curve a: 10. b) -+ G 0 < t' < t" < b < oo) has a continuous extension with dG (a(t'), a (t")) a : [0, b] -* G. To prepare the necessary tools for the proof of the main implication (v) --> (i) we derive first the following two lemmas. Lemma 7.3.3. Under the assumptions of Theorem 7.3.2 the dG-closure of a dG-ball can be described as

a E G, r > 0,

BJ1.; (a r)u' ; = Be,; (a, r) n G = Bd,; (a. r) U Si1,,. (a, r),

where SJ,;(a,r):={zeG:do;(a.;.)=r). Proof. The only thing we have to verify is that each point in Sd,; (a. r) belongs to Bd (a, r )d" . So, we fix ZO E Sd,; (a, r). Because of d( ', (a, zo) = dc; (a, W) = r there exists a sequence of II II-rectifiable curves a,,: [0, 1] -> G with a,.(0) = a. 011'(1) = zo, and Ld,; (a,,) \ r. -7x is continuous on [0, 1]. Thus for suitable t,. E (0, 1) we Recall that dG(a. conclude that dG(a, a, (t,)) = '' 1r. i.e. a, (t,,) E Bd,;(a, r). On the other hand, the relations

v-I v

imply that lim,,

r +dG(a,.(t,.), za) <

v_a.

,dG(a,,(t,.), zo) = 0. Hence Z4i E B-d,, (a, r)

r .

Remark 7.3.4. The reader should recall that for the equality in Lemma 7.3.3 it is essential that dc; be inner; cf. Example 2.5.8.

Lemma 7.3.5. Let (G. dc;) he as in Theorem 7.3.2. Moreover, suppose that do do Bd,; (a. r) is a compact subset of G. Then for any point b E Bd,; (a. r) , a 96 b.

7.3 Completeness - general discussion

215

there exists a dG-geodesic a (i.e. a is a curve a: [0. 1] -> G with a(0) = a. a(l) =h, and L,j,,(a) =d(;(a.b)). Proof. If dc; (a. h) < r we choose a sequence of curves a,.: [0. 11 - G connecting a and b with r > \ dG(a. b). i.e. a,. ([0. 11) C 84(a. r ). Reparametrization w.r.t. the do;-arc-length and the use of the fact that L,,,; (a,.) < r < ac. v > 1, enable us to assume that the family (a,.) is equicontinuous. Thus the ArzelaAscoli theorem leads to a subsequence (a,,,, which converges uniformly to

a curve a: [0. I]

G with a(0) = a. a(l) = b. and dG(a.b) < Ld,,(a) <

lint int',,_. L,,,, (a,,,,) = du(a. b). i.e. a is a dG-geodesic in G from a to b.

In the case where dG(a. h) = r. we approximate b by points b,. E Bd,.(a. r). We choose dG-geodesics a,. in 8,1,; (a. r) from a to h,. such that the family (a,.) is again equicontinuous. Then the same argument as above finishes the proof.

Proof of'Theorem 7.3.2. Obviously, the only non-trivial implication is (v) - (i). To establish it we argue by contradiction. i.e. we suppose (i) to be false. Hence there exists a dG-ball Bd,, (a. R) whose dG-closure is not compact. dj; Put ro := sup(r > 0: Bd,,(a.r) is compact 1. Because of topG = topdG we obtain 0 < r( < R. d" We claim that B,,,, (a. rj)) is not compact. (*) For otherwise, by a simple compactness argument we would be able to find a finite covering ,v

Bd,.(a.ro)

C U Bd,;(z.i.r,)

U.

i=i

where :j e B,,,, (a. r(i) t and where Bd,; (Zj. rj)`t" are compact subsets of G. In particular. U CC G. Suppose that:

there are points III,. E G \ U. v E N. with dG(a, w,.) <

(**)

1/v.

Then we will find curves a,.: [0. 1] -+ G with a,,(0) = a. a,.(1) = u,,. and Ld,;(a,.) < r + I/v and real numbers 0 < t,. < I with dG(a,a,.(t,.)) = r,,, i.e. a,. (t,.) E St,, (a.

Without loss of generality we may assume that a,.(t,. )

:' E Sd,, (a. and z' E Bd,, (zj,,, r;,, ). Thus a,. (t,.) E Bd,, (z,,,, Combining all our information so far we obtain

I/i' > LI,,(a,.)

u),.).

i.e. lim,._. dc(a,,(t,.), w,.) = 0 and d(;(w1.. zj,,) < rj,,. v >> I. Thus w, E U if v >> 1. which contradicts (**).

for almost alI v.

VII. Hyperbolicity and completeness

216

Hence we can conclude that a larger ball Bd,, (a, ro+s) is contained in U, which contradicts the definition of ro. Summarizing, we have found a ball BdG (a, ro) whose dG-closure contains a sequence (z,) without an accumulation point in G. (* * *) By Lemma 7.3.3 we may assume that all of the z,'s belong to BdG(a, ro) and

that dG (a, zv) =: r, / ro. Applying now Lemma 7.3.5 we find dG-geodesics av : [0, r,] --> Bdo (a, ro) for a and z, which are parametrized by the arc-length, i.e. the a are segments in the sense of (v). A successive exploitation of the Arzela-Ascoli argument then leads to a chain of subsequences

(aj.v)v>j C (aj-I.0v>j-1 C "' C (al.v = av)v>1 j Then P j: [0, r j] G is a continuous curve with flj(0) = a and i8j+I I[o.r,] = YjSet fl : [0, ro) -+ G as P (t) : _ fi j (t) if 0 < t < r j . Then fl is a segment in G and according to the property (v) it can be continuously extended to #: [0, ro] G. such that a j, v l (o,rj

v

Put z' := f (ro) E G. Observe that for µ > j we have dG(Z*, zj.µ) = dG(Z`, aj.a(rJ.a)) do (z*, flJ (rJ)) + do (lJ (rJ ), aJ.,. (rj)) + dG (aJ., (rJ ). aJ.,,(rJ.u) )

< dG (z*, , (ri)) + dG (8J (rJ ), cr., (rJ)) + (ro - r1).

Because of aj.,, (r j) * f j (r j) we see that z' is an accumulation point of the

-

sequence (zv)vEN contradicting (* * *).

Since the Kobayashi distance is inner we immediately obtain

Corollary 7.3.6. For any k-hyperbolic domain G C C" all the properties (i) - (v) in Theorem 7.3.2 are equivalent for dG = kG.

7.4 Caratheodory completeness In this section we study relations between the notions of Carathdodory completeness

and well-known properties of domains of holomorphy. Moreover, we present a sufficiently large class of c-finitely compact domains.

7.4 Caratheodory completeness

217

Theorem 7.4.1 (cf. [PH 21). For a c-hyperbolic domain G C C" the following statements are equivalent: (i) G is c-finitely compact;

(ii) for any a) E G and for any sequence (:,.),.EN C G without accumulation points (w.r.t. top G) in G there exists f E O(G. E) with f (zo) = 0 and sup«EN If (z,.)I

=1. Proof. It suffices to establish the implication (i) -- (ii). Because of the fact that (z,.) does not accumulate in G we may assume, without loss of generality, that (Zo, 1 -1/2='', v E N. Thus we are able to choose functions f,, E O(G. E)

with f,.(zo) = 0 and f,,(:,,) > I - 1/22". Put g,(z) := (1 + f,,(z))/(I - f,,(z)), E G. Then g,. E O(G, H+). g,,(zo) = I. where H+ denotes the right half plane. Now, we define g(:) := F,'.` i(1/2'')g,,(z), z E G. Observe that on any compact subset K C G the functions f,. are uniformly bounded by a number R = R(K) < 1. With Ig,.(z)I < 2/(1 - R).: E K. it follows that g E O(G. H+). Moreover, we obtain Ig(z,.)I

Reg(z,)I ? (l/2")g,,(z,.) >- 2" - oo. n-00

Hence with f (z) := (g(:) - 1)/(g(z) + 1),: E G. we get a function f E O(G, E) with f (zo) = 0 and

If(Z")I > Ig(z,,)I - I > Ig(z,.)I + I

2"

- 1) I. -oc

21' + I

0

Example 7.4.2. (a) Any bounded strongly pseudoconvex domain in C" and any bounded pseudoconvex domain in C2 with real analytic boundary is c-finitely compact. Use the existence of peak-functions (cf. [Bed-For 11, [For-McN], [ForSib]).

(b) Any bounded convex domain is c-finitely compact. Again argue by means of peak-functions obtained via convexity.

(c) A convex domain G C O" is c-finitely compact iff G is k-hyperbolic (cf. Theorem 7.1.8).

Proposition 7.4.3. Amy c-complete domain G in C" is an H'-domain of holomorphy.

Proof. Suppose the contrary. Then there exist :o E G and 0 < r < R with the following properties: B(zo. r) C G. but B(zo. R) rz G and for any f E H'0(G) the restriction f 18(,,.r)

extends to a holomorphic function f on B(zo. R).

VII. Hyperbolicity and completeness

218

Let Z' E B(zo. R) \ G and choose the point z' on the segment

with zuz' C

G U {z') but z' E 8G. Then if z,, := zo + (I - 1/(2v))(z' - zo). V E N. we get (cf. Remark 2.1.4)

cc(z,..z,,)

r;(a,.R)(z,..z,,) --) 0,

i.e. (z,, ),BEN is a cc-Cauchy sequence with

z,, = z' E 8G, which contradicts

0

the hypothesis.

Corollary 7.4.4. Any c-finite!v compact domain G in C" is necessarily H"°(G)convex and it is an H4O-domain of holomorphy.

Remark 7.4.5. (a) 0 It is not known whether the property "c-complete" implies that G is H"0(G)-convex.

0

(b) Observe that although the notions "H'-domain of holomorphy" and "HO"(G)convex" coincide for bounded plane domains (cf. [Ahe-Sch)) they are not comparable in higher dimensions (cf. (Sib 11). (c) In Theorem 7.5.7 we will construct a bounded pseudoconvex balanced domain G in C3 with a continuous Minkowski function which is not c-complete although it is an HO0-convex H°°-domain of holomorphy. Other examples of domains sharing these properties were given by P. R. Ahem & R. Schneider (cf. [Ahe-Sch)) and N. Sibony (cf. [Sib fl). (d) It is well known that any bounded pseudoconvex domain with smooth C0Oboundary is H"0-convex and it is an H'x-domain of Despite much efforts no example has been constructed of a domain of this type not being c-finitely compact.

0

Now, we present a class of domains which are all c-finitely compact (cf. IPfl 2)).

Theorem 7.4.6. Any bounded pseudoconvex Reinhardt domain G containing the origin is c-finitely compact.

Proof. Without loss of generality we may assume that G is contained in the unit polycylinder. Assume that the theorem does not hold, i.e. there exists a sequence z° E JG and r G(0, ,.") :5 R < oo, z"EN C G with lim,.°i; z" = z° _ -0 v E N. Suppose for a moment that z....... Z" 0: Then the real point x° := (log Izi I. . . . . log ;;Iz'I) becomes a boundary point of the convex domain log IGI. Its convexity implies the existence of a linear functional

L: R" -+ R satisfying L(x) _ F,j=i i fxi < L(x°) _: C on log ICI. Since G is a complete Reinhardt domain we obtain E, > 0, and therefore, by the above assumption, C < 0. Assume that ti,.... , 4a, are positive while the other ti's vanish. Then applying the Dirichlet pigeon hole principle (cf. [Har-Writ) we

219

7.4 Carathdodory completeness

choose natural numbers &N, ... , &k,N E N and I < qN < Nk such that

<

I

.

NENlarge.

NqN

qN

Put

fN(z) := e-CqNZi,YIN.

Y4N

and

zi,

gN(Z):=fN(z)/IIfNIIG

Thus we have gN E O(G, E) with gN(0) = 0. If Z E G, z near z°, we establish the following inequality: k

log I fN(z)I = -CqN +

Nv.N log

(7.4.1)

V=1 k

#".A,

= -qN(C - L(logIziI....,logIz,, 1))+gN

qN

> -Nk(C - L(log Izi I, ... , log

(k/N) M.

kM/N).

fN(z)I ? exp(-N"(C - L(log Izi I..... log Now let z = (zi, ... ,

- i, ) log I zi, I

E G with zi 540, 1 <j < n. If FY-1 & N log IZi,1 <

CqN, then log I fN (z) I < 0; hence I fN (z) I < 1. Otherwise the inequality rk q Q Nv.N log IZi, I > CqN implies that Yv.N I log IZi, I < I CIgN. I < v < k. Therefore, using 1B,,,N/qN ? l;i,,/2, N >> 1, we find that I

log Izi,.II :5 ICIgNI18v.N < 2ICI/(min(l;i : I < µ < k}).

Thus (7.4.1) gives

I fN(z)I < exp(M/N),

N >> 1,

(7.4.2)

where M denotes a suitable positive number. By continuity we conclude that the estimate (7.4.2) holds everywhere on G. This implies that if N >> 1, v >> 1, then

Ig,v(z°)I ? exp(-Nk(C - L(log Izi I, ... , log Izv, 1)) - (kM + M)/N). Choosing N first and then taking sufficiently large v we obtain holomorphic functions gN with values almost one at z which contradicts CG (0, z") -< R < oo. It remains to consider the case that z° . . . z° 0 0 and the other z° = 0. Projecting C" onto the space Ck by (z i , ..., z 1 . . . . . zi,) we obtain a new Reinhardt 71,

220

VII. Hyperbolicity and completeness

domain n(G) =: G' sharing all properties of G. Then the above result together with CG-(0, n (z")) <_ CG (0, z") finishes the proof.

The only case in which the relation between "c-complete" and "c-finitely com-

pact" is well understood is the plane case which was studied by M. A. Selby (cf. [Sell) and by N. Sibony (cf. [Sib 1]). Theorem 7A.7. Let G C C be c-hyperbolic. Then G is c-complete iff G is c-finitely compact.

Proof. It remains to verify the implication "-+". Unfortunately, so far. the only available proof requires a detour into the theory of function algebras. For the convenience of the reader we will quote all the results we need but without going into proofs.

The first result due to A. Browder (cf. [Brol, also Th. 26.12 in [Stol) presents a metric condition for a point zo E K, K a compact subset of C, to be a peak point for the uniform algebra R(K). Here R(K) denotes the subalgebra of those f E C(K) that can be uniformly approximated by rational functions with their poles outside K.

Lemma 1. Let K C C be compact. Suppose that Zo E K is not a peak point for R (K ). Then

lim a-o

+n2(PF(zo) n B(zt). S))

=

1.

nS2

where P,(zo) := (z e K: if f E R(K). 11f 11K < I, then 1f (Z) - f(zo)I < s1 and where m2 is Lebesgue measure.

For the domain G in C we denote by Spec = Spec HO0(G) the set of all non-zero continuous algebra homomorphisms ip: H°°(G) -> C. Spec is called the spectrum of H°D(G) and, endowed with the Gelfand topology, it is a compact topological space. A So E Spec is said to belong to the fiber over z, z r= 8G, if for any f E HI(G) that extends holomorphically through z we have -p(f) = f (z). This fiber is called Spec..

In order to prove it suffices to see that for any boundary point z E 8G the fiber Spec, is a peak set for H°0(G), the algebra of the Gelfand transforms f, f E H0O(G). For the future let us assume that Spec, is not a peak set for H'°(G) with a fixed

e 8G.

(*) To continue we need the following theorem of M. S. Melnikov (cf. VIII. 4.5 in [Gam]).

7.4 Caratheodory completeness

221

Lemma 2. Let K C C be compact and let Zo E K. Fix 0 < a < 1. Then Z() is a peak point for R(K) iff E' i aj y (E j \ K) = oc. where

Ej := (;. E C: aj+1 < Iz -:c1 < a'), y(A) := sup(I f'(oc)I: f E O(C \ L). L C A compact. II1II < I. f (oo) = 0). We cal I y (A) the analytic capacity of A. Moreover (cf. [Gam-Garl), we have

Lemma 3. Let G be a c-hyperbolic domain in C. ZOt E dG, 0 < a < 1. Then

x Spec., is a peak set for H" (G)

if

a' y (Ej \ G) = oc. j=1

Thus. if we replace :.o by , then (*) gives F_x , ajy(Ej \ G) < oo. Now, we with choose a compact set K C (B a) fl G) U aj y (E j \ K) < cc. So is not a peak point for R(K). Even more, using the characterization of peak points for uniform algebras in terms of representing measures (cf. II. 11.3 in [Gaml) we find a positive measure 1

with f f(x)da(x). f E R(K). By means of the localization operator the following approximation result can be

a on K \

derived (cf. VIII. 10.8 in (Gaml).

Lemma 4. For any f E H" (G) there exists a sequence (f j) jEN C H" (G) with e) ), E > 0, and, 1711 f IIG such that fj - f uniformly on G fl (C \ IIfj IIG moreover, f j extends holomorphically to a neighborhood of 4 .

Therefore. (*) together with Lemma 4 give an element E Spec, which is represented by a. According to Lemma I we can choose points E K \ (c ), him,..,;, z,. _ . such that If f (z,.)I < 1/t' for all f E R(K) with Of II K < 1. In particular, if f E O(G. E) extends holomorphically to with f (i;) = 0. then If (:,,)I < I /v. Now let f E O(G. E) with f (4o) = 0. Then Lemma 4 delivers a sequence of functions fj E O(G. 17E) such that f j extends through 2; and him j....x f J (z,.) f (Z,.), v E N. Because of the properties of 0o we also obtain 4U(f j) = f j

Then gj E O(G. E), g) (fj - fj(i;))/(I7(l + extends to . and gj(4) = 0. Therefore, Igj(z,.)I < 1/v which implies

4'o(f) = 0. Put gj

If(z,,) I = jlim Ifj(z1.)I < Iimsup 17(1 +

1-'

Igj(z)I + lim

<7

VII. Hyperbolicity and completeness

222

choose f E O(G. E) with f(z,.) = 0 and define g := (f - f(4))/2. Then Ig(zn )I I7/pt or To estimate c*;(z,.,

If(z,,)I < 21g(z,.)I +21g(",)I

0.

Thus we have found that the sequence (z,.),,EN is a c-Cauchy sequence, which contradicts G being c-complete.

Remark. We emphasize that Theorem 7.4.7 fails to hold for one dimensional complex spaces (cf. [Jar-Pfl-Vig 3J). We give some hints for this counterexample

in Exercise 7.3. In the case of domains F?] it is still an open problem whether Theorem 7.4.7 remains true in higher dimensions

0

We end this section by presenting an example which shows that c-finite compactness is not a "local property" (cf. [Eas]). Example 7.4.8. We study the following example of N. Sibony (cf. [Sib 11):

G := (z E E x C: 1z21 < exp(-u(zi))). where u: E - [-oo, oo) denotes the following psh function on E

x

ajlog

u(A):=exp

IX

-t

2 111111

Here (a; )JEN is a discrete sequence of points in E such that every boundary point of E is a nontangential limit of a subsequence and aj are positive numbers with I aj log I > -oo (cf. Example 2.5.8(61)). Using Hartogs' series and the Fatou theorem it is easy to see that any bounded holomorphic function on G extends holomorphically to the bidisc E2, i.e. G is not c-complete (cf. Remark 2.1.3). On the other hand, let z' E d)G be fixed. In the case where Iz; I = I or Iz;I = I it is clear that there is no c(;-Cauchy sequence in G converging to z'. In the remaining

,j

case IziI < I and dist(zl'. [at: j > 1)) =: a > 0 put U := B(zi.a/2) x C and V := G f1 U. Observe that V is connected. Now, for any j we choose log L:2L

to be holomorphic on B(z,, a/2). If we define g(zi) = e" n j g(zi) E [0, oo) we obtain the following holomorphic function on V: f (zi, 1Z2) = z2 expg(..1)

with

f (z') = I

and

2with

f E O(V. E).

Again we conclude that there is no cv-Cauchy sequence in V tending to z'.

7.5 Kobayashi completeness

223

9.9 Itohoynslti completeness Since c < k, every c-complete domain is k-complete. So section 7.4 provides examples of k-complete domains. On the other hand, the following necessary condition shows that there are many domains which are not k-complete.

Proposition 7.5.1. Any k-complete domain is taut. Proof. Let (cp j )BEN be an arbitrary sequence of qp; E O(E, G). Assume that ((p;) jEr-

is not uniformly divergent. This implies that there are compact sets K C E and L C G such that, without loss of generality, rpi (Af) E L with k1 E K. Fix Z' E L and let 0 < r < 1 with K C B(0, r). Then for X E B(0, r) we obtain kG((pj(,k). z*) < kG((J(),), pi(),i)) + kc(Vj(AJ), z*) < p(A, A1) + sup{kc(z, z*): z E L} < C(r).

Hence U;EN (p1(B(0, r)) c BkG (z', C(r)) CC G. Therefore, Montel's theorem guarantees the existence of a subsequence (Vj,,) C (v j) which converges locally uniformly to a map in O(E. G). Corollary 7.5.2. Any k-complete domain is a domain of holomorphy.

Remark 7.5.3. (a) For a while there was the question whether tautness can imply k-completeness. The first negative example was found by J.-P. Rosay (cf. [Ros 21). Later in this section we will present two other examples. (b) Because of Remark 3.2.3(c) and Theorem 7.1.2, any k-complete balanced domain is bounded and it admits a continuous Minkowski function. There is a simple example of a domain which is not c-complete, but which is kcomplete, namely the punctured disc E. This observation is a direct consequence of the next result due to S. Kobayashi (cf. [Kob 3,41).

Theorem 7.5.4. If r : G -* G denotes a holomorphic covering between domains in C", then the following statements are equivalent: (i) G is k-complete; (ii) G is k-complete.

Proof (i) -+ (ii). According to Theorem 7.1.10, G is k-hyperbolic. Fix a ball BkG(zo, r) in G. By Theorem 3.3.7 it is clear that Bk,; (zo, r) C 7r(Bke(io, r)), where io is a point in G with 7r(zo) = zo. Then the assumption and Corollary 7.3.6 imply that Bk,) (zo, r) CC G, and therefore BAG(zo, r) CC G.

224

VII. Hyperbolicity and completeness

(ii) (i). As above, G is k-hyperbolic. Fix a kG-Cauchy sequence C G. Then obviously (7r(2,)),,EN is a kG-Cauchy sequence. By assumption, this sequence converges to a point zo E G. Using again Theorem 3.3.7 it is easy to con-

struct a subsequence (zvU )LEN of (zv) and points Zo.a E G with it (%.,,) = zo and k6 (2,,, io,,p) < 1 /µ. Thus kd(io,u, io,),) 0. On the other hand, there exist a a.µ- 00 neighborhood BkG (zo, r) of zo and neighborhoods U,, of io,,, such that n I UP : U,, ->

Bkc (zo, r) is biholomorphic, A E N. Put V,4 := tr(Bk,, (zo, r/2)) fl u.; then a V,,) > r/2. This observation together with kd = PG - shows that for a sufficiently large µo we obtain zo,,,q = zo.t A _> µo. Put z-o := z-o.t,,, Then a standard argument leads to limv... kd (iv, Zo) = 0. 0 k6 (2o,,,, ,

Example. C \ (0, 1) is a k-complete domain which is not even c-hyperbolic.

We will see that the property of k-completeness is a local one (cf. [Eas]) in contrast to Example 7.4.8.

Theorem 7.5.5. Let G be a bounded domain in C". Suppose that any boundary point zo E a G permits a bounded neighborhood U = U (zo) such that any connected component of G fl U is k-complete. Then G is itself k-complete.

Proof. Assume that there exists a kG-Cauchy sequence (zv)IEN C G with limv-,, Z, = zo E aG. Choose R > 0 so large that G U U(zo) C B(0, R) _: V. As a consequence of the equality of the topologies, U(zo) contains a kv-ball Bk,, (zo, 2s) (s > 0). Then, for 0 < s < s/3, take vE E N such that kc(z,, zµ) < s and z E Bkv (zo, s/3) whenever v, z > vE. Now fix such v, A. According to the definition of kG we find analytic discs coj E O(E, G), 1 < j < k, and points a, E E, 1 < j < k, with the following properties: k

(p, (0) = zv, (p j (ai) = Wj+1(0), 1 < j < k, Vk (ak) = z,,, and E p(0, aj) < S. j=1

Observe that, in particular, p(0, aj) < e < s/3. For A E Bp(0, s) C E we obtain the following inequalities: kv ((pj (A) zo) < kv (tpj (A), (pj (o)) + kv ((pj (0), z,,) + kv (zg, zo) k

< p(A, 0) + E kv (tGv(0) (pv(av)) + s/3 v=j k

<s+Ep(O,a,)+s/3 <2s, v=1

7.5 Kobayashi completeness

225

i.e. rpj (BP(0, s)) C U(zo).

If we denote by y : E -> Bp(0, s) the biholomorphic dilatation, then we have analytic discs cZ, := rp; 0y E O(E, UnG) with 01 (0) = z,., Oi(y-I (ai)) = 0i+I (0),

1 < j < k, Ok(y-I(ak)) = zµ, and F_=l p(O, y-I (ai)) < CEk=l p(O, ar) < ce, where c is independent of E. In particular, the above reasoning also shows that belongs to a connected component U' of U(zo) fl G. So we the sequence obtain a ku--Cauchy sequence with lim,,... z, = zo t U' which contradicts the assumption that U' is k-complete.

Example 7.5.6. (a) Applying Theorem 7.5.5 to Example 7.4.8 shows that the Sibony domain is k-complete. (b) The above theorem also provides a simple proof that any strongly pseudoconvex domain is k-complete using only the existence of local peak functions.

By Theorem 7.4.6 we know that any bounded pseudoconvex Reinhardt domain containing 0 is k-complete. Moreover, tautness, i.e. the continuity of its Minkowski function, is necessary for a balanced domain to be k-complete. Nevertheless, the following result shows that tautness, even in the case of a balanced domain, does not imply k-completeness.

Theorem 7.5.7 (cf. [Jar-Pfl 51). There exists a bounded balanced pseudoconvex domain G = Gh = {z E C": h(z) < 1 } (n > 3) with continuous Minkowski function h which is not k-complete.

Proof. It suffices to construct an example of G in C3 since taking G x Ei-3 if n > 3 leads to an example in C". Our construction will be based on a series of reduction steps: 11 reduction. It suffices to construct a continuous log-psh function g : C2 satisfying the following properties:

R+

(i) limjj,,p..g(z)/Ilzll = I E R, (ii) the set G := (z E C': g(z) < 1 } is bounded and it has a connected component G' which is not k-complete. Assume that g, 1, G', and G are given. We define a new function ho: C2 X C -+ [0, oo) by putting ho(z, Z3) :-

j IZ3Ig(ZI/Z3, Z2/Z3)

if

Z3 # 0

IIIIzII

if

Z3=0

Observe that ho is a continuous homogeneous function on C3 which is log-psh

on C2 x C.. By Appendix PSH 21 we conclude that ho is even log-psh on the whole C3. By (ii) we know that G C B(0, ro) for some positive ro. Define

V11. Hyperbolicity and completeness

226

h:C;= C2xC-+R+by h(z, z3) = max

h is a continuous log-plurisubharmonic homogeneous function with h-1(0) _

(0). Thus the domain

6 .=((Z.Z.i)EC2xC:h(z.z3) < I) is a bounded pseudoconvex balanced domain and G' x (I I is a connected component

of G fl (C2 x (1)) = G x (1). Therefore, G is not k-complete. 2"d reduction. It suffices to find a connected set X C B(0.r(j) c C' for some ri) > 0 and a sequence (zi )jeN C X, limi_a, zi = 0, with the following properties: (i) there are holomorphic maps )pi: E -> C", such that

Wi(E) C X,)Pi(lj) _

zi+) (11'j.A E E), andEp(Aj.A,) _: A < oo; 1=)

(ii) there exists gyp: C2 -+ R+, continuous and log-psh, such that (ii)) (P(0) = 1, (PI X < 1.

(i42 for suitable positive C. a, R we have the inequality (P(,-) :5 CjjzjjU for IIzII > R.

Drawing on these assumptions we will construct a function g as in the first reduction step. Taking V1120 instead of v we can assume that a = 1/2. Put

g(z) := max((y(z), IIzII/(2r0)).

: E V.

Obviously. g is continuous and log-psh on C" with limll,N--, g(z)/IlzII = I/(2ro)

Moreover. X C G := (: E C": g(:) < 1) C B(0, 2ro) and the connected component G' of G with X C G' is not k-complete because of kC,-(z).:i) < E =i p(A;,, A < oo and zi -> 0 E HG'. What only remains to produce is the data of the second reduction step.

Let ci := I/2i+i and aj := I/21(i+)}; put

Xi :_ ((z, w) E C-: w =:(ai+) +ai) -alai+) Izl 5 ri),

j E N.

(7.5.1)

Then Xj is a connected compact set with Xi C B(0, 1) and Xi - (0). Moreover, J-.m for :i :_ (aj, a;) we obtain z1. Zj+) E X. Let qi : E --> C2 be a holomorphic

7.5 Kobayashi completeness

227

map defined by tpj(A) := (cjA,cj),(aj+i +aj) -ajaj+i). Then tpj(E) C Xj, SOj(Aj) = Zj, and ,(A') = Zj+I, where Xi := aj/cj, Xi := aj+i/c E E. oo. If we put X := U; i Xj we Direct calculations lead to > i p(,,. have found all we wanted in (i). To construct the function tp as in (ii) put

Pj(z, w) := w - z(aj +aj+i) +ajaj+i and observe that 31111,

= (z, w) E

IIII > 1,

C22.

Next we choose sequences (r) j and (e) j of positive numbers satisfying the following properties:

rj > 3, rj >

x log 3/4 = Esj log(ajaj+t/rj)

.i}.

j=I

In particular, log(4/3) > E" E j =: a. Using these sequences we get the following psh function *: C2 -> [- 00, oo): 00

1,(

) :_ EEjlog IPr)I j=1

E C2.

i

Note that tib (0) = log 3/4, *Ix = -oo, * < 0 on B (O, 1), and >G ( ) < a log III II

ifIIII>>1. Let (D: C2 -i- 118+ be a Co function on C2 with support in 13, and satisfying 4 (z, w) = 4t(Izj, Iwl) and foe L( )dA( ) = 1. For E > 0 we put CF(A) :=

fC2

*( - En)(D(n)dA(i ),

E C2.

Thus we obtain a sequence of psh Cl-functions *E on C2 with *e

\,

>G. In

F- 0+

log 3/4 and * the definition of /re leads to particular, *E (0)

a log(2IIt11)

0 if III II < 1, 0 < E < co < 1. Moreover,

if 0 < s < I

and

>2.

VII. Hyperbolicity and completeness

228

Setting

0, := exp *,,/ exp *,.(0),

0 < e < co

(7.5.2)

we receive a family of positive log-psh CO0-functions on C2 having the following properties:

4/3

0,(0) = 1,

if

IIi

and

li < 1,

if

2,

where C is a suitable positive constant. Now, we follow Bishop's construction of peak-functions. Put Ui := B(0, 1/2) and choose Ei E (0, eo) such that (p, := cp£, is less than 1/3 on X \ U1. Suppose we have already constructed neighborhoods U1 D U2 D ... D Uk of 0 > Ek such that, with W1 := cpE,, we have and numbers El > cpj

1/3

on

X \ Uj,

1 < j < k,

Uf,

on

1<j
Define

Uk+l := { E Uk:

I

+2-(k+l)/3,

1 < j < k)

and choose Ek+l, 0 < Ek+l < Ek, such that the function cpk+l 1/3 on X \ Uk+l. Given the sequence (cpj)jEN we put

is below

OC

cP :_ E(1/2)'cpj.

i=l This series is locally uniformly convergent. Therefore, the function cp is continuous log-psh on C3 with cp(0) = 1 and cp(i;) < CIIIIII if IIIII >- 2. Moreover, for E X we have:

if E X \ Ul, if E X fl (Uk \ Uk+i),

then

cp(s) < 1;

then

cp(t) < 1 - 2-(2k-l)/3 < 1.

Thus, the function cp satisfies all the properties needed in (ii).

0

Remark 7.5.8. (a) H? It would be very interesting to know whether such an example could be also constructed in C2. n? We emphasize that the method used above does not work in the two-dimensional case.

7.5 Kobayashi completeness

229

(b) So far it is totally unclear how to characterize the k-completeness of a bounded balanced domain via the properties of its Minkowski function. For example, 0 is any bounded balanced domain with a CO0-Minkowski function k-complete (c) We would like to point out that the example of Theorem 7.5.7 is a taut HO0convex HI-domain of holomorphy; cf. [Sic 2].

0

Up to now 0 it is an open problem whether every bounded pseudoconvex domain with C'-smooth boundary is k-complete 0 . The strongest result in the negative direction is the following unpublished one due to N. Sibony; cf. [Sib 3]. We mention that a part of the construction of Theorem 7.5.7 was based on this work.

Theorem 7.5.9. There exists a pseudoconvex non k-complete domain G CC B2 given as a connected component of (Z E B (0, 3) : u (z) < 11, where u is a continuous psh function in B(0, 3), CO0 outside zero with grad u(z) 0 0 if z 36 0, and u(0) = I.

Proof. Following the construction of the proof of Theorem 7.5.7 we may assume (adding suitable multiples of IIz112 to (Qe, cf. (7.5.2)) that all the functions 0, are strictly psh on C2. Thus the function rp = (1 /2)gpi + E J°_, (1 /2J )Vj is strictly psh and, even more, 0 := V- All .112 is a continuous strictly psh function on C2, where A is a suitable positive number. By the approximation theorem of R. Richberg (cf. Appendix PSH 30), a C°°-strictly psh function u : C2 \ {0} -, R can be found with 0 < u <

u(0) = I

and

u(z) <- 1 - (A/2)Ilzll2,

Z E X.

Let p be any C'-psh function on B(0, 3) with p - 0 on B(0, 1), p(z) > 2 if IIziI ? 2, and limjjZp.3 p(z) = 00. Define u£(z) := u(z) + EIIz112 + p(z),

z E B(0, 3), 0 < e < A/2

Observe that

X C {z E B(0, 3): u,(z) < 1} _: Ge CC B(0, 3)

and

0 E BGF.

Using Sard's lemma for the function (1 - p(z) - u(z))/IIz112 on B(0, 3) \ (0) we are able to find a small so, 0 < so < A/2, such that the function uep is regular at every z E B(0, 3) \ (0) with u,. (z) = 1. Hence the connected component G := G,(, of Ge0 that contains X is the domain whose existence we claimed. 0

VII. Hyperbolicity and completeness

230

7.6 Bergman completeness In this section we restrict ourselves to bounded domains in C". Thus all domains which will be under consideration are automatically p- and b-hyperbolic. Moreover, since the Bergman distance is inner, all the completeness notions w.r.t. b coincide. Therefore, we will only speak of b-completeness. Observe that Theorem 6.4.7 implies that any p-complete domain is also b-complete. [] The inverse implication is still an open problem. Again we obtain the necessity of the pseudoconvexity.

0

Theorem 7.6.1 (cf. [Bre]). Any bounded b-complete domain G is pseudoconvex.

Proof. Suppose the contrary. Then there exist polydiscs 0 := zo + rE" C G and

0' := zo + RE" iZ G, R > r, such that for every f E 0(G) the restriction f ID extends holomorphically to A. In particular, by Hartogs' theorem, there exists a C with the following properties: function F : A' x A'

(i) FIA.A = KGIoxo, (ii)O'xA'9(z,w)-> F(z, tip) is holomorphic, where O':=(wEC": iv- E A,). By hypothesis there exists z' E 0' fl 8G such that the segment [zo, z') C G f1 O'. Since log KG (Z, z) = log F(z, z) near [zo, z'), it follows that any sequence zj = z' is a bG-Cauchy sequence; a contradiction. (z j )BEN C (zo, z') with lim j

0 Example 7.6.2. The converse of Theorem 7.6.1 fails to hold. For example consider the pseudoconvex domain

G := {z E C2: Izi I < IZ21 < l}.

Recalling that G is biholomorphic to E x E. we obtain I

KG(z, z) = flz (Iz2 1 2

z

z)z(1 - Izi I2 - Izz

Iz)z'

is a bG-Cauchy sequence with serve that H0(G) is not dense in e.g. the approximated; cf. Exercise 7.8. ((0,

-+

0, that is I/v) = (0,0) E G. Ob-

Then a direct calculation shows that bG((0, 1/v), (0, 1/µ))

"'P-+00

1/Z2 cannot be

Lemma 7.6.3. Let G be a bounded pseudoconvex domain in C" with z' E 8G. Suppose that there is a neighborhood U = U (z') with the following properties: (i) U fl G is connected,

7.6 Bergman completeness

231

(ii) any sequence (Zv),,eN C U n G with lim,,yx Zv = z' is not a PGnu-Cauchy sequence. Then there is no PG-Cauchy sequence converging to z'.

Proof. Let U' = U'(z') CC U. Then an argument similar the one used in the proof of Theorem 6.3.5, but simpler, leads to the following inequality: MGnu(Z,,z)

z, ,z E U, nG,

,
where

MGnu(Z', z") := sup(If (z') 1: f E L2(G n U), Ilf IIL=(Gnu) = 1, f

(z") = 0}.

Then, applying Theorem 6.4.3, we obtain PGnu (L , Z ) <

MGnu(z'. z")

KGnu(Z' z')

<

CMG(z', z")

<

V12-CPG

(Z', Z"), Z', Z" E U'nG,

KG (z', z')

0

which completes the proof.

Observe that the above lemma immediately leads to the conclusion that the Sibony domain in Example 7.4.8 is p-complete.

Lemma 7.6A. Let G be a bounded domain in C" satisfying (i) lim..aG KG(Z, z) = 00, (ii) H°O(G) is dense in L'(G). Then G is p-complete. Proof. Suppose the contrary. Then there exists a PG-Cauchy sequence (zv),.EN C G such that z, = z' E aG. Put

H := (f E L%(G) : lim

°OO

If (z»)i

= 0).

KG (z,,.

Then H is a closed linear subspace of L2(G). For if f; -- f in L2(G), then

I(f -fi)(zv)I + KG(Zv, Z.)

KG(Zv, Zv)

Ilf - filIO(G) +

If;(z")1 KG (Z., Zv)

I fj(Zv)I KG(zv, zv)

232

VII. Hyperbolicity and completeness

Therefore, by (ii) we have H = Lh(G). By the definition of PG and the completeness of 1P(L2(G)) there is a function f E L2 (G), IIfIILh(c) = 1, such that zv)

KG(Zv, zv)

ei9 - f

L2(G),

in

where 9, are suitable real numbers. Then we obtain

_

If(zO I

(fKG(,z,,)eie')L2(c)l I

G (ZV . zv.

2

c z, zv

0

which contradicts the fact that f E H.

After these preparations we formulate a rather general result on p-completeness (cf. [Ohs 11, [Pfl 31, [Skw I]). Theorem 7.6.5. Any bounded pseudoconvex domain with C 1 smooth boundary is p-complete, and therefore also b-complete.

Proof. Use the previous lemmas together with the local starlikeness and Theorem

0

6.1.17.

Remark 7.6.6. The above theorem is essentially based on the boundary behavior of the Bergman kernel function KG (Z, z). To understand the non-smooth case better it seems worthwhile to see 0 whether the statement of Theorem 6.1.17 is still true 0

for any bounded pseudoconvex domain G with G = G

0

We conclude this section with a positive result for pseudoconvex balanced domains (cf. [Jar-Pfl 2]).

Theorem 7.6.7. Any bounded pseudoconvex balanced domain G = Gh C C", whose Minkowski function h is continuous, is p-complete and hence b-complete.

Proof. Since H°O(G) is dense in L2 (G) (take ,(z) := f (vz/(v + 1)) for f E L2 (G), V E N) it suffices to see that limZ-,,. KG(Z, z) = co; cf. Lemma 7.6.4. More precisely, we will prove that KG(z,z) >- M/(1 -h2(z))2 by induction with respect to the dimension. In the case n = I we have h(z) = IzI/R and so Kc(z. z) =

7r R2 (I -

h2(z))2,

Z E G.

233

7.6 Bergman completeness

We turn to the step from (n - 1) to n. Let ;.' E G, z' 0 0. We may also assume

that z' _ (zi , 0, ... , 0). Define a hyperplane H := (z E C": z,, = 01. Then G' _ [i E C'-': (i, 0) E G} is a bounded pseudoconvex balanced domain in with Minkowski function h(,) := h( . 0). Now, we apply the following deep result of T. Ohsawa - K. Takegoshi (cf.

C"

[Ohs-Tak]) whose proof is beyond the scope of this book.

Theorem. Let S2 be a bounded pseudoconvex domain in C", tJr : Q -# [-oo, oo) a psh function, and H C C" a complex hyperplane. Then, for any holomorphic function f on 11 n H satisfying fit;,H I f 12e- 'dA."_t < oo, there exists a holomorphic

function F on Q with FIsinH = f and ff2 IFI2e-*dX,, < C fml,H If 12 e-Id,,_t, where C = 16207r(l + (diam 11)2)2, and where dXk denotes 2k-dimensional Lebesgue measure.

This theorem implies that there exists Co > 0, depending only on the diameter of G, such that any holomorphic function f on G' that is square integrable w.r.t. dal,, _t extends to a holomorphic function F on G satisfying IIFIIL2(G) Collf IIL2(G') So we get

If (I' __ Collf11L2(G,)

IF(z')I- < IIF11L2(c)

K G ( "" , Z' ) ,

which leads to the following inequality: KG(z', z') >- (l /CC)KG,(z', Z'). Finally, the induction hypothesis implies that KG(z', z') >- C/(1 - h2(z'))2.

Observe that C is independent of z'.

Remark 7.6.8. So far 0 it is not clear whether the assumption about the Minkowski function in Theorem 7.6.7 is indeed necessary

0

Along with Theorem 7.5.7 we obtain:

Corollary 7.6.9. There exists a bounded pseudoconvex balanced domain G in C3 for which any comparison of the type bG < CkG fails to hold.

234

VII. Hyperbolicity and completeness

Notes

Most of the general investigations on hyperbolicity have been initiated by S. Kobayashi [Kob 41. Particular cases were treated by K. Azukawa lAzu 11, T. J. Barth [Bar 31. N. Sibony [Sib 41, and J. Siciak (private communication). In case of domains hyperbolic with respect to various pseudodistances, a natural notion of Cauchy-completeness appears. In differential geometry this property can be described by the compactness of balls (see the Theorem of Hopf). But in general the situation becomes more difficult. It was Rinow (see 1RinJ) who put Hopf s result in this general context introducing the concept of inner distances. The example of finitely compact Reinhardt domains is taken from IN 21. The question whether "c-finitely compact" and "c-complete" coincide was solved by M. A. Selby [Sell and N. Sibony [Sib I ] for hyperbolic plane domains. The counterexample in the case of reducible one-dimensional complex spaces can be found in [Jar-Pfl-Vig 31. It is still not clear how to construct a counterexample for domains

in C",n>2. The pseudoconvex balanced domain with continuous Minkowski function is due to [Jar-Pfl 5]; so far, the situation in dimension two remains open. The discussion of completeness for the Bergman metric was started by S. Koba-

yashi in [Kob I,21, where he found a sufficient criterion. The results here are formulated in terms of the Skwarczynski distance. The final result is based on [Ohs I]. [Pfl 31. and [Skw I]. The fact that any pseudoconvex balanced domain with continuous Minkowski function is Bergman complete is contained in [JarPfl 21. The proof of this property uses a deep extension theorem for L2-holomorphic functions which was found by T. Ohsawa and K. Takegoshi [Ohs-Tak].

E.ercises

2,35

Exercises

7.1. Define

G:=(z=(ZI.Z2) EC2:IziI < 1.Izi1721 < 1}\F, where F := (z E C2 : zI = 0. Iz21 < 1). Prove that G is Brody hyperbolic but not k-hyperbolic. (A similar example is due to D. Eisenman and L. Taylor; cf. [Kob 4], page 130.)

7.2. Let G, C C2 be the domain introduced in Example 7.4.8. Put G = {z = (z', z3) E C3: Z' E G,, Iz31 < dist(z'. 8G,)). Prove that G is an H'(G)-domain of holomorphy which is not c-complete (cf. [Sib 11).

7.3. Complete the details of the following construction of a connected complex space X, which satisfies: (a) top X = top cx, (b) X is c-complete but (c) X is not c-finitely compact (cf. [Jar-Pfl-Vig 3]).

Sketch of the construction. Write Aj,k := (1 - 1/(j + 1))exp(2nik/(j + I)) E E, j > 1 and 0 < k < j. and put Dj := E, j E Z,. X is defined by identifying the points A j.k E D j (j > 1) with Aj,k E Do in Ux o D j. Then a holomorphic function f on X is a collection of functions f j E O(E) with fo(A j,k) = f j (A j,k ), 0 < k < j, j > 1. Hence the collection (g j )jEz. with g j (A) = jlj-o repre-

sents a holomorphic function g on X. Moreover, there is a subsequence g j. g

with g E O(E) and Ig(0)I = I/e. These facts lead to a proof of the equality top X = top cx and to the c-completeness of X. To see that X is not c-finitely compact observe that any (f j)j,z, E O(X) with fo(O) = 0 yields holomorphic functions h j E O(E) with h j(A) := (1/2)(fj(A) -

fo(A)), j > 1. Then hj(A,.k) = 0 for 0 < k < j. Therefore, by the SchwarzLemma, it follows that I f j (0) I < 2(1 - 1 /(j + 1))j+ I -+ 2/e, which can be read

j-x

as Oj E B1.,r (0c, 2), j > 1. Here O j denotes the origin of Dj in X. 7.4. Prove that any domain G in C" admitting a bounded psh exhaustion function

u (i.e. u: G -> [0, 1) with {z E G: u(z) -< a) CC G for every 0 < a < 1) is k-hyperbolic (cf. [Sib 4)).

7.5. Let (a j) be a sequence of points in E, a j a,, if j k, such that every boundary point of E is the nontangential limit of a subsequence of (a'). Choose

236

VII. Hyperbolicity and completeness

sequences (n j)jEr and (m j) jEN of natural numbers satisfying

! n),g to (/ j=I

lay 21

mj > n

> -00,

j

B(aj,3exp(-jmj))f1B(ak,3exp(-kmk))=0ifk 76 j, andB(aj,3exp(-jmj)) C E for all j E N. Then consider the following domain in C22 G :_ {z = (z,, z2) E +C2: Izi I < 1,

Iz21

expu(z,) < 1),

where

u(A) := 00 E(1/nj)max{log(Iz -ajI/2), -jmj}. j=1

Prove that u is continuous psh, that the pseudoconvex domain G is k-complete, and that any bounded holomorphic function f on G depends only of the first variable,

i.e. f (zi, z2) = 9(z,) with g E O(E); cf. [Sib 4]. 7.6. Let G be a k-complete domain in C". Prove that for f E H°°(G) the domain G' := {z E G : f (z) # 01 is again k-complete. 7.7. Let G j C C"' be a domain, j = 1, 2. Assume that G2 is k-complete. Let A C G, be relatively closed with HZ"--2(A) = 0. Then every holomorphic mapping F : G, \ A -+ G2 extends to a holomorphic map F : G, -+ G2. Hint. Use Appendix HF 10.

7.8. Prove that for G := (z = (z,, z2) E C2:

Iz, I < Iz21 < 1) the function f(z) := 1/z2 belongs to L2(G) but not to the closure of H00(G) in LI(G).

Chapm P. Ulll

Complex geodesics. Lempert's theorem

8.1 Complex geodesics The notion of complex geodesics is a natural generalization of the notion of extremal discs (cf. Propositions 3.2.4 and 3.5.13). Recall that if G C C" is a taut domain, then for any Z,, zQ E G, zo z (resp. for any Zo E G. Xo E C. Xo 0) there exists a holomorphic mapping *: E -+ G such that

zo = *(0), z _ *(a),

where a = z)) (resp. zo = *(0), X0 = a*'(0), where a = xc(zo; Xo));

(8.1.1)

(8.1.2)

such a mapping Or is called an extremal disc for (zo. zo) (resp. for (zo, Xo)); cf. Chapter III. Now let G C C" be an arbitrary domain and let i/i E O(E, G) be an extremal disc for (zo, zo') (resp. for (zo. X0)). Observe that if h E Aut(E), then the mapping V := * o h has the following property: 3ko, ko E E : zo = P(Ao), zo

'P (X"),

kc (za, Z0) = PVO. Au)

(8.1.3)

(resp. 31o E E, ao E C: zo = V(A0), Xo = ao(P'(Ao), XG(zo; Xo) = YE(,ko;ao)); (8.1.4) that is. rp is an extremal disc as well. Conversely, if rp : E -+ G is a holomorphic mapping satisfying (8.1.3) (resp. (8.1.4)), then for suitable h E Aut(E) the mapping := (p o h-' satisfies (8.1.1) (resp. (8.1.2)).

Let d = (dG)cEg, be an arbitrary contractible family of functions (resp. let S = (SG)GEgo be an arbitrary contractible family of pseudometrics). A mapping V E O(E, G) is said to be a complex dc-geodesic for (za, zo) (resp. a complex Sc-geodesic for (zo, Xo)) if 34,,X0 E E: z9 = (p(4),

4-0 = (P(Ao),

zo) = p(Ao, Xo)

(8.1.5)

238

VIII. Complex geodesics. Lempert's theorem

(resp. 3Ao E E, ao E C: zo = (p(Ao), Xo = aoip'(Ao), SG(zo; Xo) = YE(Ao;ao)) (8.1.6) In this terminology, extremal discs are simply complex kG- (resp.xe-) geodesics. Note that if (p is a complex SG-geodesic for (zo, Xo), then it is a complex SGgeodesic for any pair (zo, AXo), A 0. The notion of complex dG-geodesics extends in a natural way to m-contractible families of functions. It suffices to substitute in (8.1.5) p(Ao, X0") by m (X0', X0").

The following simple lemma gives a relation between complex geodesics and extremal discs.

Lemma 8.1.1. If (p is a complex dG-geodesic for (4, z() (resp. a complex SGgeodesic for (zo, Xo)), then dG(zo,zo) =kG(zo,zo) (resp. SG(zo; Xo) = xG(zo; Xo))

Consequently, tp is an extremal disc for (4, zp) (resp. for (zo, Xo)). In particular, if the pair (zo, zo') admits a complex kG -geodesic (resp. if the pair (z0, Xo) admits a complex xG-geodesic), e.g. if the domain G is taut, then (4, zp') admits a complex dG-geodesic i,,O" dG (4, zo') ° kc (4, zQ') (resp. (zo, Xo) admits a complex SG,geodesic if SG(zo, Xo) = xc(zo, Xo)).

Proof. Let g(A') = 4 and g(Ao) = z, , (resp. g(Ao) = zo, Xo = aogo (Ao)) as in (8.1.5) (resp. (8.1.6)). Then

P(Ao, A) = d0(zo, zo) < kc(zo, zo) S P(Ao, Ao) (resp. YE(Ao;ao) = SG(zo;Xo) < xu(zo;Xo) < YE(Ao;ao))

0 Remark 8.1.2. (a) If G C Ct is a taut domain, zo, 4o, zq' E G. to 714, then by Proposition 4.2.12(a) the following conditions are equivalent: (i) 3k E N: (4. z.4') admits a complex mkl-geodesic; (ii) 3k E N : (zo. 1) admits a complex YG 1-geodesic; (iii) (zo, zfl') admits a complex gG-geodesic; (iv) (zo, 1) admits a complex AG-geodesic; (v) G is biholomorphic to E.

8.1 Complex geodesics

239

In particular, if G C C is a taut multiconnected domain, then there are no complex cG- or yG-geodesics gyp: E -.* G.

(b) Let G = Gh C C" be a balanced domain of holomorphy (h denotes the Minkowski function of G) and let a E G be such that h(a) > 0. By Propositions 3.1.10 and 3.5.2 we see that the mapping E z) a. -+ Xa/ h (a) E G is a complex kG- and xG-geodesic for (0, a). Consequently, by Proposition 2.2.1(b) and Lemma 8.1.1, the following conditions are equivalent: (i) the pair (0, a) admits a complex cG-geodesic; (ii) the pair (0, a) admits a complex yG-geodesic; (iii) a) = yG(0;a) = h(a).

In particular, if G is not "convex at the point a/h(a)", i.e. if condition (v) from Proposition 2.2.1(b) is not satisfied, then there are no complex CG- or yG-geodesics

for (0, a). In the sequel we will discuss mostly the case of complex CG- and yG-geodesics (from a certain point of view this is the most interesting case). More precisely, we would like to discuss domains G C C" such that for any zo, za E G, zo 54 zo,

the pair (zo, ;.g) admits a complex cG-geodesic,

(8.1.7)

for any zo E G. Xo E C", Xo 0, the pair (zo, Xo) admits a complex yG-geodesic.

(8.1.8)

and/or

Recall that in the category of taut domains condition (8.1.7) is equivalent to the equality cG = kG and (8.1.8) is equivalent to the equality yG = xG; cf. Lemma 8.1.1. We know that if G is biholomorphic to 1B" or to E", then (8.1.7) and (8.1.8) are fulfilled. On the other hand, by Remark 8.1.2(b), if G is an unconvex balanced domain of holomorphy, then neither (8.1.7) nor (8.1.8) is satisfied.

Proposition 8.1.3. Let G C C" be an arbitrary domain and let cp: E -> G be a holomorphic mapping. Then the following conditions are equivalent: (i) 3X', xo E E, Ao # X": CG (V (Ao), V (A0)) = P(X', Ao), i.e. V is a complex CG-geodesic for ((p(Ao), (o(X")); (ii) VA', A" E E: cG(cV(A'), V(V)) = P(A', )'"), i.e. rp is a complex cG-geodesic for any pair (p(A")); (iii) VA E E: YG((p(A); V'(A)) = YE(A;1), i.e. cp is a complex yG-geodesic for any pair (cp(A), V'(A)); (iv) 3XO E E: yG(gq(Ao); V'(Ao)) = YE(Ao; 1), i.e. V is a complex yG-geodesic for (cp(Xo), V'(.Lo)).

VIII. Complex geodesics. Lempert's theorem

240

Consequently, any complex cc- or yc-geodesic w is an embedding (injective, proper. regular mapping) of E into G. In particular, (p(E) is a I-dimensional complex submanifold of G. Before we begin the proof we would like to mention the following

Remark. (a) A part of Proposition 8.1.3 remains true for more general objects like m(k),y(k) or g, A; cf. Exercise 8.1.

(b) The implication (i) -+ (ii) is not true for kc. In fact, let P C C' be an annulus (as in Chapter V). Suppose that gyp: E -+ P is a "global" kp-geodesic, i.e. kp((p(A'), V (.V)) = p(.', A"), A', A" E E. Then q would be injective and proper, and consequently bijective; a contradiction.

(c) Note that the implication (iv) -+ (iii) remains true for xc-geodesics in taut domains G C V. In fact, suppose that cp: E -+ G is a complex xc-geodesic for (zo, 1), where G is a taut domain in C (i.e. #(C \ G) > 2). We will prove that (p is the universal covering of G; then, by Exercise 3.8, xc(q (A);V '(A)) = 1/(1- I), 12) for all A E E, that is, cp is a "global" xc-geodesic.

We may assume that 0(0) = zo, W'(0) = 1/xc(zo; 1). Let n : E -+ G be the universal covering of G with 7r(O) = zo. Recall (Exercise 3.8) that xc (zo;1) = 1/In'(0)I. Therefore, we may assume that n'(0) = l/xc(0; 1). Let gyp: E -+ E denote the lifting of cp with 0(0) = 0. Since cp = n orp, we have 0'(0) = 1. Hence, by the Classical Schwarz-Pick Lemma, 0 = idE. Finally, V = n. Proof of Proposition 8.1.3. (i) -+ (ii). Define u(A)

FG(A))

(AO', A)

A E E \ {AO'}.

Then u is subharmonic, u < 1, and u(Ag) = 1. Hence, by the maximum principle, u =_ 1, which shows that SP(A)) = p(Ao, A).

A E E.

Now, we can repeat the same argument w.r.t. the first variable (for fixed ).* E E), which proves (ii). (ii) - (iii). Take Ao E E. Then, by Proposition 2.5.1(d), we have

yE(Ao;

1) = lim

n(Ao, A)

Xo IAo -,X1

cc((P(Ao), SP(A)) x..A0

IAo - AI

_

- yc(co(Ao);cP'(Ao))

8.1 Complex geodesics

241

(iv) ---> (i). Put

OX) :_

cV0Ao).cp(A))

.

A E E \ {Ac).

Then u is subharmonic and u < 1. Moreover, if we put u(Ao,) := limsup;,-, u(A), then it is subharmonic on the whole E (cf. Appendix PSH 21). In view of (iv) and Proposition 2.5.1(d). we get is (AO) = 1, and therefore, by the maximum principle,

it - I which shows that c'G(cP(Ao).

(A)) = p(Ao, A).

A E E.

Corollary 8.1.4. Let rp, >fi : E -+ G he complex CG- or yG-geodesics such that co(E) = Eli (E). Then there exists h E Aut(E) for which >' = cP o h.

Proof By Proposition 8.1.3, V :_ W(E) = >f(E) is a submanifold of G and the mappings V. /i : E -* V are biholomorphic. Hence h := cP-1 o iJr : E - E is an automorphism.

Proposition 8.1.5. Let G C C" he it rant dcm ain. Then the following conditions are equivalent: (i) kG and yG = x; (i.e. (8.1.7)+(8.1.8)): (ii) VG = kG (i.e. only (8.1.7)): (iii) for any Z('). ;,n E G with w)

14 there exist cP E O(E. G) and f E 7(G. F. ) such that Z(. y E V(E) and f o ip = idE: (iv) for any ;'. ;0 E G there exist a holoinorphic embedding cp: E - G and a holomorphic retraction r : G - op(E) such that z , ;0 E cp(E). Moreover, a,n holoinorphic mapping gyp: E -a G satisfying (iii) or (ir) is a complex for (-'. - ). Vice versa, for any complex ('G-geodesic (p for there exists f (resp. r) such that (iii) (resp. (iv)) is fulfilled.

Proof. (ii) -> (iii). Let V be a complex ca-geodesic for (;''. -) with 0(0) _ ;; Take f E 0(G. E) such that f (;') = 0 and f (;.;;) _ (P(Ao) Aii = (f is an extremal function for (;0 z0) ). Then 0 := fop : E --+ E is holomorphic. *(0) = 0. and G(Ai) = c

f(-") = 141

-f . cc)) e"k'c'(,

G

Hence, by the Schwarz Lemma. 0 = id,.

e'(41

1A,. = A 0I

cr

VIII. Complex geodesics. Lempert's theorem

242

(iii) -- (i). Take Zo, zp E G, zo j4 zo', and let gyp, f be as in (iii). Suppose that = W(Ao), z' = p(Ao). Then

p(Ao Ao) = p(f(w(Ao)), f((P(Ag))) = p(f(zo), f(z )) < eG(ZO, zo) <_ kG(ZO, zo) < kG(V(Ao), p(Ag))

p(Aa, 4)-

Consequently, cG = kG (Lemma 8.1.1). In particular, (8.1.7) is true.

To prove thatyG =xG,take ZOE GandX0EC",Xo j4 0.LetV,,: E-+ Gbe a zo, zo+(1/v)Xo (0 < a < 1), v >> I. Since G is taut, we may assume that E O(E, G),

complex cc-geodesic for (zo, zo+(1/v)Xo) with (po(0) = 0. Then cpo(0) = lim

v-.x

pv(av) - gyp"(0) av

1

= (lim -) . Xo. vary

On the other hand, (zo, zo + 1 X0) cc lim va = lim I vex v-.oo

- YG(zo; Xo)

V

Hence Xo = YG(zo; Xo)gpo(0) which proves that yG(zo; Xo) = xG(zo; Xo) (iii) (iv). In view of the first part of the proof of the implication (iii) -+ (i), if rp is as in (iii), then rp is a complex CG-geodesic for (zo, zo'), and therefore, by Proposition 8.1.3, cp is an embedding. Now, it suffices to put r := (p o f.

(iv) -+ (iii). Put f := rp-1 o r. Then f is holomorphic and f o (p = idE.

O

Using the same methods as in Propositions 3.1.10 and 3.5.3, one can easily derive the following

Proposition 8.1.6. Let G = Gh = {z E C" : h (z) < 1 } be a balanced domain of holomorphy (h is the Minkowski function of G), let a E G be such that h(a) > 0, and let rp: E -> G. Then the following conditions are equivalent: (i) the mapping w is a complex kG-geodesic for (0, a) (resp. (p is a complex xGgeodesic for (0, a)); (ii) the mapping 9 has, up to Aut(E), the following form

((A) = Ao(A),

A E E,

where 0 E O(E, C") is such that h o 0 - 1 and 0(h(a)) = a/h(a) (resp. 0(0) = a/h(a)).

8.2 Lempert's theorem

243

8.2 Lempert's theorem The aim of this section is to prove the following fundamental theorem which has been already announced several times in the previous chapters.

Theorem 8.2.1 (Lempert's Theorem) (cf. [Lem 2,31, [Roy-Won]). Let G C C" be a domain biholomorphic to a convex domain. Then

c'G = kc; = kG and yG = xG.

Prof gWe may assume that the domain G itself is convex and that n > 2 (the case it = I is obvious). Let (G,. ),. be a sequence of bounded convex domains such that G,. / G. Recall that CG, \ cG (Proposition 2.5.1(a)), kG, \ kG (Proposition 3.3.5(a)), yG, \ yc; (Proposition 2.5.1(a)). and xG, \ xG (Exercise 3.5). In particular, if the theorem is true for each G,., then it is true for G. Consequently. we assume that

G is a bounded convex domain in C" with 0 E G (n > 2).

(8.2.1)

Thus, in view of Proposition 8.1.5. the Lempert Theorem is equivalent to the following claim.

(*) If G is as in (8.2.1), then for any complex kc;-geodesic (P: E --+ G there exists

a holomorphic mapping f : G - E such that f o (p = idt:. Here and in the sequel "kG-geodesic" means "kG-geodesic for some (;o, z)". The proof of (*) will be given by a (long) sequence of lemmas.

Let H 1(E) denote the first Hardy space on E: cf. Appendix H. Recall that if lp E H 1(E ). then for almost all A1) E M the function (p has the non-tangential boundary value lp'(it))) = lim lp(k); 4

here and in the sequel "for almost all" means "for almost all w.r.t. Lebesgue measure on M". Recall that 10 = (;.. Uh)

zj w1,1.

!=1

= (z,..... Z,,). it, = (Il)l..... U)") E C".

VIII. Complex geodesics. Lempert's theorem

244

Lemma 8.2.2. Let D be an arbitrary bounded domain in C". Suppose that q E O(E, D) and h E H1 (E, C") are such that Re

((z - V*(A)) fh`(A)) < 0 for all z E D and for almost all A E 3E. (8.2.2)

//

Then there exists f E O(D, E) such that f o tp = ids. Proof of Lemma 8.2.2. For s > 0 we define I (z, A)

(z - co(A))

h(A) - EX,

Z E C", A E E,

Z E C", X E E. Note that cf. = (Do - eA and T, = To - E. The mapping f we are looking for will be a holomorphic solution of the equation (Fo(Z, f (z)) = 0,

z E D.

(8.2.3)

Let a := wp(0). Observe that if we put Wo(a, 0) := -rp'(0) h(0),

then'Po(a, ) E H' (E). Hence in view of (8.2.2) the maximum principle gives Re To(a, ,1) < 0,

In particular, Io(a, 0)

A E E.

(8.2.4)

0, and therefore

a4)(

a, 0) = To(a, 0) : 0.

By the implicit function theorem there exist: an open neighborhood U of (a, 0) with U C D x E, an open neighborhood V of a with V C D, and a holomorphic function F : V -+ E such that {(Z, Al) E U: Co(z, A) = 0} = {(z, F(z)): Z E V}.

(8.2.5)

A for A in a neighborhood of 0 E E.

(8.2.6)

Obviously,

8.2 Lempert's theorem

245

Suppose that f E O(D, E) satisfies (8.2.3). Then condition (8.2.4) implies that f (a) = 0. Hence, by (8.2.5), f = F in a neighborhood of a E D, and therefore, by (8.2.6), f o (p = idE. Thus the problem is to find a holomorphic solution of (8.2.3). It suffices to find functions ff E O(D, E), E > 0, such that Z E D.

(D, (Z, f.- (Z)) = 0,

(8.2.7)

For the proof observe that, by (8.2.4), f, (a) = 0, s > 0. Hence, using Montel's argument, one can select a sequence s \ 0 such that

fr, =.f E O(D. E). It is clear that f is the required solution of (8.2.3). Now, we turn to the construction of f, e > 0. Fix e > 0. It is enough to prove that

Re'f(z,A)<0, zED, r(e)
(8.2.8)

Suppose for a while that (8.2.8) holds. Then, for fixed Zo E D and for r(s) < r < 1, we have

#fA E E: 4f(zo,1l) = 01,=

.8

a) (zo, X)

1

27ri j)i=fr 1

f(zo.

/d

2,ri J

A

dl /'

I

+

).I=r

2ni J

IAI=r

?'' (zp' A) dX 'I' (zo, A)

+ 1o(e, zo, r).

where 1o(s, zo, r) denotes the index at zero of the curve

[0, 2,r} a 9 -- 'I'(zo, re'o). In view of (8.2.8), lo(e, zo, r) = 0, r(e) < r < 1. Thus for any Zo E D the equation 4)f(zo, A) = 0 has exactly one solution A =: ff(zo). Moreover, the function D 3 z -+ ff(z) E E is holomorphic since

ff(z)

I

A

l) I=r

dA,

ZED, r(s) < r < 1.

VIII. Complex geodesics. Lempert's theorem

246

To prove (8.2.8) it suffices to show that

Re W0(z. A) < Re

(Li0(z. 0) - A 40(z 0) j =. M(;,, A),

1), i. = L

Note that the function M is continuous (on C" x E,) and M(;,, A) = 0 for i. E E. So, if (8.2.9) holds, then for any c > 0 there exists 0 < r(e) I such that M(c, A) < e, z E D, r(E) < IxI < 1. which gives (8.2.8). To prove (8.2.9) fix Zo E D and let g := 4o(2o, ), bo 0) 0 We want to prove that

Re(g(x))
AEE*.

(82.10)

Put

E E, g (0),

A =0

(8.2' 11 )

Then g E H ' (E) and by (8.2.2) we have

Re

(A) < - Re

(bo)

Re(bo)A) for almost all A E a F.

Consequently, by the maximum principle,

Reg(1) < -Re(boA),

k E E,

which, together with (8.2.11), gives (8.2.10).

Let qG denote the Minkowski function of G. i.e.

yG(;j:=inf{t>0: 1ZEG), t

2EC".

Recall that

G = (zEC":gG(z)<1},

i)G={zEC":qG(z)=l}.

(8.2.l2)

8.2 Lempert's theorem

247

Let 4G be the dual Minkowski subnorm, i.e.

gc(w)

sup

Re(,

w)

E (C")" Jj

I

w): c E 8G},

= max{Re(z

W E C".

The functions qG and gG are Minkowski subnorms on C. Recall that if (F. II IIF) is a complex normed space, then a function Q : F -+ R+ is said to be a Minkowski subnorm on (F, II lip-) if co,>[i E F.

Q( V+f) <_ Q(V)+Q(TY),

Q(t*) = IQ(*).

I > 0, i/i E F,

f E F.

I II*IIF _< QW -< CII*IIF.

(8.2.13) (8.2.14)

(8.2.15)

where C = C(Q, 1111 F) is a positive constant. In view of (8.2.13) and (8.2.15), we get

I Q((v) - Q(*)I

CII(v -

(p, L E F;

IIF.

(8.2.16)

in particular, any Minkowski subnorm is continuous. In the general case the counterpart of the dual Minkowski subnorm gG is defined in the following way.

Let F denote the dual of F, F' := {µ: F -+ C: u is C-linear and continuous}, endowed with the standard sup-norm: 11µ11F- := sup

1µ0)I

:

?A E F, } .

II*IIF

µ E F'.

J

Then we define

Q(u) := sup

Re it (*)

Q()

:

u E F'.

>G E F,

(8.2.17)

11

One can easily prove that Q : F' -> R+ is a Minkowski subnorm on (F', II

Remark 8.2.3. Let

such that Re(zo

wo) = gc,(wo).

IIF')

VIII. Complex geodesics. Lempert's theorem

248

Then for any z E G we get

Re((z - zo) wo)

(9G(z) - I)9G(wo) <0.

This means that the plane

{ZEC": is a supporting plane for G at zo and that the vector wo is an outer normal vector to dG at zo. In particular, if the unit outer normal vector v(zo) to 8G at zo is uniquely

determined, then wo = pv(zo) for some p > 0. Lemma 8.2.4. Let G be as in (8.2.1) and let qp : E -> G be a complex kG -geodesic. Then

(8.2.18)

W* (.k) E 8G for almost all A E 8E

and there exists h E H' (E, C"), h # 0, such that Re (co*(A).

gG

(*)

for almost all k E M.

(8.2.19)

Note that Lemmas 8.2.2 and 8.2.4 imply (*). For observe that, by the identity principle for H'-functions, h*(A) # 0 for almost all A E 8E; cf. Appendix H7. Then, using Remark 8.2.3, we conclude that conditions (8.2.18) and (8.2.19) ensure that the mapping h from Lemma 8.2.4 fulfills condition (8.2.2) of Lemma 8.2.2. Finally, the existence of the mapping f is a consequence of Lemma 8.2.2. Lemma 8.2.4 will be proved via the next lemma.

Lemma 8.2.5. Let G be as in (8.2.1) and let co : E -+ G be a complex kG-geodesic zo'). Then condition for (zo, zo') with zo = cp(Ao), zo' = (p(Ag) (4, zo' E G, 4 (8.2.18) is fulfilled and there exists h E H' (E, C"), h 0, such that Re((p*(A)

u*(A)) = gG(u*(A)) for almost all k E 8E,

where

u(A) :=

h(k),

Ao

(), - Ao)(A -

JE E

00

)

Proof that Lemma 8.2.5 implies Lemma 8.2.4. For i; = rer' E E. put AC(A) := r +

I - (Ae-1e +

e'°),

A E C,.

(8.2.20)

8.2 Lempert's theorem

249

Then:

- the only pole of Oz is the simple pole at . = 0, - the only zero of 4)c in E is the simple zero at A

-ifA 8E, then 1

-2Re(Ae-ie)>r+

r

-2>0.

Define if ).,,. n." 0 0

if 'X)= 0

if Aa=0

It is clear that the function h extends holomorphically to E, h # 0, and h E H' (E, C"). Moreover, (D(A)u'(A) for almost all A E

E.

Using the fact that 4) > 0 on dE (and (8.2.14)), we obtain for almost all I E aE: Re (co'(A)

h*(A) I =

So far, we have shown that the proof of (*) reduces to the proof of Lemma 8.2.5. Before presenting the latter we need some auxiliary facts.

Lemma 8.2.6. Let G, cp, 4, zo, X' ,.k0 be as in Lemma 8.2.5. Suppose that E -* C' is a holomorphic mapping with *(Ao) = zo and

zg. Then

sup{grW,l)): I E E} > 1. Proof of Lemma 8.2.6. Suppose that gc(>(i(A)) < r < 1, A E E. Then *(E) CC G, and therefore there exists s > 0 such that i/r(E) + elm,, C G. Let X E Aut(E) be such that x (0) = kp and x (ao) = Ao for some ao E (0, 1). Note that p(0, ao) _ za'). Choose 0 < a < ao such that p(Ao, Ao) =

II4 - *(x(a))II < ca

Vlll. Complex geodesics. Lempert's theorem

250

and define

w(A) := *(X(A)) + k (Z - *(X(°))).

A E E.

Then O(E) C *(E) + EB" C G, 0(0) = *(A') = z , and 40(a)

Hence

kG(zil. zg) < p(0, a) < p(0, ao). This contradiction finishes the proof.

0

Let F := C(aE, C"). II*IIF := max(IIV'(A)II: A E SE),

Q(*) := max(yG(*(A)):

A E aE).

* E F, +Jr E F.

it is easily seen that Q : F -> R+ is a Minkowski subnorm on the Banach space (F. II 11F). Recall that

F'=JA =(tt1....N-n): µ( ) :_

µ 1, ... , A. are complex Borel measures on a E },

J * (A)dµi(A)

E !=1

_ (/ll..... µn) E F'. Y' _ ( 1..... Y'n) E F.

Let Q : F -> R+ be the dual Minkowski subnorm (defined by (8.2.17)). Denote by r the normalized Lebesgue measure on M. in other words.

J v(A)dr(A) :_

E2n ./111f

v(e'x)dt,

v E L' (aE. C; r).

Lemma 8.2.7. Let µ = ur = (u1 r.... , u,,r) E F' with u E L'(aE. C"; r). Then

Q(µ) = f.E 9G(u(A))dr(A). Proof of Lemma 8.2.7. Observe that for any v E L' (a E. C"; r) if we put v := vr. then by (8.2.16) we get IQ(v)-Q(µ)1
11 (t6 o v)dr 6b

f

r1E

(QG o u)drI <

JaE

CIIv-UII1I. CIIv - ulldr < Clly - ulll.I.

Hence, without loss of generality, we may assume that it E C(a E, C"), it # 0.

8.2 Lempert's theorem

251

Take >/i E F.. Then

Reµ(*) Q(*)

{

- Q(*) JaE Re(* . u)dt 1

QW I (9Go*)(9Gou)dT

<

(4G o u)dr. eE

Hence

Q(µ) < r (gG o u)dT. JaE

If fLE (4G o u)dr = 0 we are done. Now suppose that the integral is positive. Fix

0<e<

f

(4G o u)dr.

uE

Observe that for each 4o E dE there exists z(oo) E dG such that

4000))-

Re(z(4o)

By continuity, there exists a neighborhood Re(z(4o)

u(A))

of 4o in dE such that

4c(u(A)) - e,

AE

Let s..... N E a E be such that dE =

(8.2.21)

U U U(.N) and let X1, .. . XN be a continuous partition of unity for the covering {U(1), .... U(SN)}. Define N

X E 8E.

VG (A)

j=1

Then * E F and by (8.2.13), (8.2.14) we have N

9G(

(O)) :

1.

. E dE.

VIII. Complex geodesics. Lempert's theorem

252

Hence Q(*) < 1. Moreover, in view of (8.2.21), N

Reµ(>/r) =

f

c

j=1

Re(* u)dr = >2 JOE Xj(A) j_I

J

aE A'

IP

4G(>2 Xj()A)u(X))dr(A) - s = j_1

aE

iE

(4G o u)dr - E.

In particular, 'i/r $ 0 and finally,

ReA(*)

o u)dz - s. Q() > faE(4G

0

With e -+ 0 we get the required result. Let

A := {* E F: 3 fr E C(E,C") nO(E,C"): tJaE = Vi); we will always identify i/r with >

.

Obviously, A is a closed subspace of F. Put

Al := (A E F': LIA = 0). By virtue of F. & M. Riesz Theorem (cf. Appendix H 8) we have

Al = (?J1 r: h E H'(E,C")).

(8.2.22)

Proof of Lemma 8.2.5. Let gyp: E -+ G be a complex kG-geodesic with V(Ao) = zo, (v()0") = zo. X0, 0 X0". Put

Vo:=(1 EA: *(),0)=i/r(Ag)=0). Observe that Vo is a closed complex subspace of A. Fix qpo E A such that (p0(A) _ zo and (po(JAo) = zo, e.g.

(POW - zoI 'X,IX0(zo-zo), 0

0

AEE.

8.2 Lempert's theorem

253

By Lemma 8.2.6 we have

Q(Wo +,)

(8.2.23)

E Vo.

Let us define a functional 1 by

trpo+(ER. Obviously, 1 is well-defined and R-linear. Moreover, in view of (8.2.23), for any

t > 0 we have E Vo.

Hence 1(trpo + >1i) < Q((Po + ').

t E R, * E Vo.

By the Hahn-Banach theorem the functional 1 extends to a R-linear functional

L : F -+ R such that L < Q on F. Note that L is continuous (use (8.2.15)). Define

A(*) := L(*) - iL(ii,r).

* E F.

Then 1L: F -+ C is C-linear and continuous, i.e. µ E F. It is easily seen that µl vo = 0. Hence by (8.2.22),

µ

h *T

= (A - Aa)(A -

for some h E H)(E,C"). Since Reµ(Wo) = L(Vo) = I(con) = 1, we have h Moreover, Re µ = L < Q on F. which shows that Q(µ) < 1.

(8.2.24)

(In fact Q(µ) = 1.) Put u(A)

Ah(A) (A - A0)(A - A0 )

0.

A E E \ {A0, Ao)

VIII. Complex geodesics. Lempert's theorem

254

Observe that the function (gyp - cpo)

u extends holomorphically to E as a function

of the class AH'(E). Consequently, fa (v*(),)

- (po(A))

u*(A)dr(A) = 0,

and in particular Re

r

(p'(),)

u`(,l)dr(A) = Re µ((po) = 1.

aE

Finally,

I = f Re(cp'(A) u`(A))dr(A) E

yG(W'(A))gG(u'(A))dr(A) (use the definition of gG) arE

<

gG(u'(X))dr(L) (use the fact that gyp: E

G)

aE

= Q(µ) (use Lemma 8.2.7 ) < 1 (use (8.2.24) ). This shows that

gG((p'(A)) = 1, i.e. cp*(A) E 8G for almost all A E 8E (cf. (8.2.18)), and

u*(A)) = gG(u'(A) for almost all A E 8E,

Re(V*(A)

which finishes the proof.

The proof of the Lempert Theorem is completed.

In view of the Lempert Theorem (and of Proposition 8.1.3), if G C C" is (biholomorphic to) a convex domain, then one can introduce the notion of the complex geodesic in G (without prefix CG, kG, yG, KG, etc.). In this terminology Lemma 8.2.4 reads as follows (cf. the remark after Lemma 8.2.4):

Corollary 8.2.8. Let G C C" be a bounded convex domain with 0 E G. Then a holomorphic mapping rp : E - G is a complex geodesic if V*(A) E 8G for almost all A E 8E

(8.2.18)

255

8.3 Uniqueness of complex geodesics

and there exists h E H 1(E, C"), h $ 0, such that Re I cp*(A)

1h*(A) I = gG

(!h*(x)) for almost all A E 8E.

(8.2.19)

Remark. The boundary behavior of complex geodesics in convex domains will be summarized in Miscellanea.

8.3 Uniqueness of complex geodesics We are going to discuss the problem of uniqueness of complex geodesics in the case of convex domains. Here "uniqueness" means "uniqueness modulo Aut(E)". If cp: E -* G is a complex geodesic, then we identify cp with cp o h, where h is any automorphism of E. Recall that complex geodesics cp, i1i : E -+ G may be identified if V(E) = *(E); cf. Corollary 8.1.4. We will consider the following two types of questions: - given z,, Z()' E G with ILO

1-o, decide whether there exists exactly one (modulo

Aut(E)) complex cG-geodesic for (;0. z ): - given zo E G, Xo E (C")*, decide whether there exists exactly one (modulo Aut(E)) complex yG-geodesic for (zo, Xo). First, we will prove (Proposition 8.3.2) that for convex bounded domains G the above problems are in some sense equivalent. We will need the following version of the maximum principle.

Lemma 8.3.1. Let G E C" be a bounded convex domain and let cp : E --> C" be a bounded holomorphic mapping. Suppose that cp*(A) E U for almost all A E 8E. Then cp(E) C G; in particular, either cp(E) C G or (p(E) C i)G. Proof. One can assume that 0 E G. Let qr denote the Minkowski function of G. Suppose that zo := cp(Ao) V G for some Ao E E. Let L : C" -* C be a C-linear form such that Re L < qG and L(zu) = gG(zo) > 1. Observe that Re(L o cp)*(A) = Re L((p*(A)) < gG(tp*(A)) < 1 for almost all k E 3E.

Hence, be the classical maximum principle, Re Locp < 1. In particular, gG(zo) < I and so we have a contradiction. 0

256

VIII. Complex geodesics. Lempert's theorem

Proposition 8.3.2 (cf. [Gen 1,2]). Let G be a bounded convex domain. Suppose that gyp, ill : E -+ G are complex CG-geodesics for (4, 4), zo $ za' (resp. yG-geodesics for (zo, Xo), Xo # 0) such that rp 0 * mod Aut(E). Then -for any Ao, Ao E E, X0' # Ao, the complex CG-geodesic for (cp(Ao), (p(Ao)) is not uniquely determined, - for any Ao E E the complex yG-geodesic for (tp(Ao), tp'(Ao)) is not uniquely determined.

Proof. We can assume that zo = 0 E G (resp. zo = 0 E G) and that (p(0) _ >/i(0) _

0, co(o) = *(a) = za (0 < or < 1) (resp. co(0) _ i(0) = 0, w'(0) = *'(0) = Xo). Fix Ao, Ap E E, Ao :,4 Ao with (,X', Ap) # 10, o) (resp. Ao E E.). Let 4' be a function

meromorphic in a neighborhood of E without poles on 8E such that 4'(A) E (0, 1],

A E aE.

(8.3.1)

Suppose that the function

X(A) :_ 4'(A)*(A) + (1 - 4'(A))co(A)

extends holomorphically to E and that X(Aa) _ ((Ao) and X(Ao) = ((Ao) (resp. X(Ao) = w(Ao) and X'(Ao) = (p'(Ao)). Note that X E H°D(E, C"). By (8.3.1) we have

X*(A)EG for almost allAEaE. Consequently, by Lemma 8.3.1, X is a complex CG -geodesic for (W (AO'), (p(Ao)) (resp. a yG-geodesic for (cp(Ao), (p'(Ao))). Moreover, X,54 (p mod Aut(E).

It remains to find such a fi. We will use the functions 4Pt from (8.2.20). One can easily prove that it suffices to take

where M > 0 is a constant such that 4) < 1 on E.

0

Now, we move to the case in which the complex geodesics are uniquely determined.

8.3 Uniqueness of complex geodesics

257

Proposition 8.3.3 (cf. [Din]). If G is a strictly convex bounded domain, then the complex CG- and y(; -geodesics are uniquely determined.

Here strictly convex means "geometrically strictly convex", i.e. if a, b, (a+b) E

;

a G, then a = b.

Proof Suppose that (p, *: E -> G are complex co-geodesics for (4, 4) (resp. (resp. z, (p(a) = ((a) yo-geodesics for (zo, X0)) with V(O) = *(0) (P (0) _

(0) = ZO, V (0) _

(0) = Xo). Put X:= 2(V+Vr)

Since G is convex, X (E) C G. Moreover, X (O) = -', X (a) = ;a (resp. x (0) = zo, x'(0) = Xo), and therefore x is also a complex cG-geodesic for (zo, z) (resp. yageodesic for (zo, Xo)). Since G is bounded, p, *, x E H00(E, C"). By Proposition

8.1.3, (p*()A), **(A), x*(A) E aG for almost all A E 8E. Now, by the strict convexity, (p* = Vr* a. e. on aE and finally co = i.

0

Example 8.3.4. If G is not strictly convex, then complex geodesics need not be uniquely determined. For let G = E2, rp(A) := (A, A/2), *(A) := (A. X=), A. E E. Then to and f are both complex cE2-geodesics for ((0. 0), (1/2. 1/4)) but p(E) # if,(E), and consequently qp and >' are not equivalent; cf. Example 8.3.8.

In the case of balanced domains of holomorphy the uniqueness of complex geodesics may be characterized more precisely, namely we have

Proposition 8.3.5. (a) Let G = G,, C C" be a balanced domain of holomorphy (h is the Minkowski function of G) and let a E G be such that h (a) > 0. Then- the following conditions are equivalent:

(i) the mapping E 3 X 4 Aa/h(a) E G is the unique (modulo Aut(E)) complex kr,- or xG-geodesic for (0, a); (ii) the point b := a / h (a) is an "extreme point for G ", i.e. there is no non-constant

holomorphic mapping f : E - G with b = f (0). (*) (b) (cf. I Ves 11) Let G = G,, C C" be a convex balanced domain ( q is a seminorm ) and let a E G be such that q(a) > 0. Then the following conditions are equivalent: (i) the mapping E

A.

),a/q(a) E G is the unique (modulo Aut(E)) complex

CG- or yG -geodesic for (0, a),

(ii) the point b := a/q(a) is a "complex extreme point for G ", i.e. there is no E (C"), with b + El; C G. Remark. Let G C C" be an arbitrary domain. Suppose that G is strongly pseu-

doconvex at a point b E 8G, i.e. there exist a neighborhood U of b and a

258

VIII. Complex geodesics. Lempert's theorem

function r E C2(U, lit) such that G fl U = jr < 0), 8G fl u = jr = 0), dr(z) 5k 0, and (Gr)(z; X) > 0, z E U, X E (C"),; cf. Appendix PSC. Then b is extreme in the sense of (*). For suppose that f : E -* G fl U is a holomorphic mapping with b = f (0). Since r o f is a psh function, the maximum principle gives r o f =_ 0. Consequently, 0 = (L (r o f))(A;I) = (Cr)(f (A); f'(A)), A E E.

Hence f - b. Proof of Proposition 8.3.5. (a) (i) -+ (ii). We already know that the mapping rpo is both a kG- and mG-geodesic for (0, a); cf. Remark 8.1.2(b). Suppose that f : E -* G is a holomorphic mapping with b = f (0). By the maximum principle for psh functions we have h o f - 1. We consider the following two cases. 1) The mapping (po is the unique complex ku-geodesic for (0, a).

Let g E Aut(E) be such that g(h(a)) = 0. Define *(A) := Af (g(A)), A E E.

Obviously, * : E -+ G and *(0) = 0, *(h(a)) = h(a) f (0) = a. Hence * is a complex kc-geodesic for (0, a). In view of the uniqueness, there exists X E Aut(E)

such that i/r - X b. Clearly, X (0) = 0 and X (h (a)) = h (a). Consequently, X = idE, and therefore f o g - b, which shows that f - b. 2) The mapping po is the unique complex xG-geodesic for (0, a).

Put >]r (A) := Af (A). Then * : E -> G, *(0) = 0, and t/r'(0) = f (0) = b. Hence * is a complex xG-geodesic for (0, a). By the uniqueness, * = X b for an automorphism X E Aut(E). Then X(0) = 0 and X'(0) = 1. The end of the proof is the same as above. (ii) -+ (i). This implication follows from Proposition 8.1.6 and (*). (b) This is an immediate consequence of (a) and of the following lemma. Lemma 8.3.6 (Harris). Let f : E -+ C" be a holomorphic mapping. Then

f (0) + )?(f (A) - f (0)) E conv(f (E)) for all q E C and A E E such that 21gA1 < 1 - Al. I

f (0) + 2 E f'(0) C conv(f (E)). Consequently, if G C C" is a convex domain and if f : E --> G is a holomorphic mapping such that the point f (0) is a complex extreme point for G, then f =_ const.

Proof (cf. [Gra 3]). We may assume that f (0) = 0. Put S := conv(f (E)) and suppose that there exist qo E C. and Ao E E such that 2lgoAol 1 - IAol but the point zo := qof (Ao) is not in S. Then, by the Hahn-Banach theorem, there exists a C-linear functional L : C" -* C such that Re L < 1 on S and Re L(zo) > 1. Put

8.3 Uniqueness of complex geodesics

259

F := L o f: E-+ C. Then Re F< I and 1

IF(Ao)I =

21X01

IL(zo)I >

1 - I.01,

I'io1

which contradicts the Borel-Caratheodory lemma (cf. Exercise 2.3).

00

As an illustration of the above results we present characterizations of complex geodesics for the following standard balls in C", n > 2 (the more complicated case of convex complex ellipsoids will be discussed in the next section).

Example 8.3.7. The case of the unit Euclidean ball B,,. Note that the group Aut(B") acts transitively on B,, and that any boundary point a E 818" is a complex extreme point for I%,, (13,, is strictly convex).

All complex geodesics cp : E -- 3" are of the form pp(A) = Aa,

AEE

(mod Aut(E) and modAut(3")),

where a E a13,, (this means that any complex geodesic cp : E -p B,, is of the

form E 9 X-+ F(X(A)a) E G, where F E

X E Aut(E), and a E

818"). Moreover, by Proposition 8.3.3 the complex geodesics in 3,, are uniquely determined (modulo Aut(E)).

Example 8.3.8. The case of the unit polydisc E".

Aut(E") acts transitively. A point a = ( a1 ,--- a,,) E a (E") is a complex extreme point for E" if la II = ... = la,, I (i.e. iff a E (aE)"). All complex geodesics cp : E -+ E" are of the form (p()) = AO(A),

A E E (mod Aut(E) and mod Aut(E")),

where gyp: E -+ 8(E") is an arbitrary holomorphic mapping. In particular, the complex geodesics in E" are not uniquely determined. Example 8.3.9. Let L,, := (z E C" : L" (z) < 11, where L. (z)

=

[11211'` + (1IzI14 - Iz

zl2)''2]

2

z=x+iyER"+iR"=C".

The norm L,, is the maximal complex norm q : C" -a R+ such that q (x) = Ilx II R" + W. L,, is called the Lie noun in C"; the ball L,, is called for all x E ]It"

VIII. Complex geodesics. Lempert's theorem

260

the Lie ball in C". Observe that L2(zi, Z2) = max{Izi - iz21, Izi + iz21},

and consequently the mapping (8.3.3)

L2 3 (zi, z2) -+ (zi - iz2, zl + iz2) E E2 is biholomorphic.

The group Aut(L,,) acts transitively on Ln; in fact, L" is one of the classical Cartan domains. Moreover,

Auto(Ln) := IF E Aut(L,,): F(O) = 0} = (e'BA: 8 E IR, A E SO(n)) (cf. [Hua]), where SO(n) := the group of all orthogonal operators A : 1R" --> 1R" acting on C" according to the formula:

C" 3 x + iy -+ A(x) + iA(y) E C. We will prove that a point a = xo + iyo E aL" is a complex extreme point for L" if the vectors x0 and yo are R-linearly dependent in R". If n = 2, then the above statement directly follows from (8.3.3). The general case may be reduced to the case n = 2 using the following three remarks: - the result is invariant under the action of SO(n), - for any a E C" there exists A E SO(n) such that A(a) E C2 x (0},

- L" fl (C2 X (0}) = L2 X (0}.

All complex geodesics gyp: E -* Ln are of the form (cf. [Aba 2]):

(P(X) _ X C

2(x)

*1(A)

*r

2i

2

0, ... , 0)

,

xEE

(modAut(L,,) and modAut(E)),

(8.3.4)

where (v", *) E O(E, 3(E2)). In particular, the geodesics are not uniquely determined.

To prove (8.3.4) observe that the case n = 2 follows from (8.3.3). The general case may be reduced to n = 2 in the following way. Ln be a complex geodesic with 9(0) = 0. Then we have Let co : E Ln(co(A)) = IXI,

A E E.

8.3 Uniqueness of complex geodesics

261

Fix 0 < a < I and choose A E SO(n) such that A(p(a)) E C2 x (0). Put

X =(XI.....X,,):=AoV. Note that

LA(:) < Lk+r(:.. w).

:E

(8.3.5)

Cc. W E C'.

with the equality only for w = 0. In particular, (XI. X2): E -> fl..2 and L,(Xi(a). X2(a)) = L"(X(a)) = a. This shows that (XI. X2) is a complex geodesic in L2. Consequently. IAI = L2(Xi(A).X2(A)) < L.(X(A)) =1), 1.

X E E,

X,, - 0. The proof of (8.3.4) is completed.

which by (8.3.5) implies that X3

Example 8.3.10. Let Mi" := (: E C": M. (:) < I). where

f (II:II2 + _

I:.:I)1/2

[11 x 112 + 11y112 + [(1l

112

- 11y 112)2 + 4(.r ..)2] I/2] 1/2

: = x+i} E C".

The norm M" is the minimal complex norm q: C" -> l8+ such that q(:) < Ilzll. E C". and q(x) = IIxII for all .r E R": cf. [Hah-Pfl I I. Note that

M2

a + i:2) E A2 :_ ((wi. w2): Iwil + (w21 < I) (8.3.6)

is biholomorphic. The group Aut(M,,) does not act transitively, in fact.

Aut(M,,) = Auto(M,,) = (e'" A: 0 E R. A E SO(n)) (cf. [Kim 1J). Any point a E 8M[" is a complex extreme point for M,,: the case it = 2 follows from (8.3.6); the general case may be proved by the same methods as in Example 8.3.9. All complex geodesics 0: E -+ M with 0 E sp(E) are of the form v(X) = .Xa.

AEE

(modAut(E)).

262

VIII. Complex geodesics. Lempert's theorem

where a E dM". Since all boundary points are complex extreme points, these complex geodesics are uniquely determined modulo Aut(E); cf Proposition 8.3.5. Note that we have discussed only the case where 0 E V(E); Q? we do not know what is the description of all complex geodesics in M"

0

The following example will show that complex yG-geodesics need not be uniquely determined even for very regular strongly pseudoconvex domains G C C"; cf. Proposition 8.3.3.

Example 8.3.11 (cf. [Sib 2J). For any k E N there exists t(k) > 0 such that for each t > t (k) the domain 12 G = G := 1(Z I . Z2) E C2: IZi +

I_

2

1 1 + tjzI _ Z' I' < I J

admits at least k non-equivalent complex ya-geodesics for ((0, 0). (1. 0)). Observe that for any t > 0 the domain G, is strongly pseudoconvex with smooth Cl-boundary. Obviously, G, C B2. Moreover, the mapping [0. I] x G, -3 (s, (z 1, z2)) -> (sZ,. s2Z2) E G,

is a contraction of G to 0. Note that for any a E ./T the mapping G, 9 (:1.:.2) - T.4 (ZI,aZ2) E G, is an automorphism of G,.

We go to the proof of the example. Fix k E N. We will prove that for t >> 0 there exists a complex ya,-geodesic p : E G, for ((0, 0). (1.0)) of the form (p(A) = (Aa, X2b),

A E E.

with b 36 0; then the mappings T. o V. a e

are non-equivalent complex

y(;, -geodesics for Q0. 0), (I, 0)). Put Z2) := IZI I + IZ212 +t,Zr - Z;12,

(ZI. Z2) E C2

(u, is a strictly plurisubharmonic COQ-function). Let *(X) = (XXI(1). X2X2(X)). X E E, be an arbitrary complex xG,-geodesic for ((0, 0), (1.0)) (G, is taut). By the maximum principle we have u,(XI. X2) < 1 on E. In particular, u,(XI (0), X2(0)) <

8.3 Uniqueness of complex geodesics

263

1. This shows that 1

IXi(0)I < maxi Iz1I: 372: u r (7 1, z2) < I} _: a,. XG,((0,0):(1,0)) =

On the other hand, if (a, b) is such that u, (a, b) = a,, then (Aa, Ab2) E G,, k E E. Consequently, for any t > 0 if u,(a, b) = a,, then the mapping W(A) _ (Aa,

'X2

b),

X E E,

is a complex xG,-geodesic for ((0, 0), (1, 0)). Moreover, in view of the definition of a,, the mapping G, E) (Z1, Z2)

z1,

\\

b Zj a2

is a holomorphic retraction of G, onto (p (E) (in fact, r(z1, z2) = rp(z1/a)). Hence, rp is a complex yG,-geodesic for ((0. 0). (1, 0)); cf. Proposition 8.1.5.

It remains to prove that for t >> 0, if u,(a, b) = a,, then b 0 0. It suffices to take t > 0 so big that max(Izl l: u,(zl, z, )} > max{Izl l u,(Zl, 0) = 1}

(note that t(k) may be given by an explicit formula). We end up this section with an example related to Proposition 8.1.5.

Example 8.3.12 (cf. [Lem 2]). Let G := ( (ZI, Z2) E C2: (1 + Iz112)(l + Iz212) < 25}.

and a := (l, 1) E G. Then there is no holomorphic retraction r : G -> S with 0, a E S and dim S = 1 (S as a holomorphic retract would be a complex submanifold of G). In particular, there are no complex CG- or yG -geodesics for (0, a). For suppose that r : G S is such a retraction. Note that G is balanced and strongly pseudoconvex. Moreover, G is not convex at the point b := a/h(a) (2, 2) E 8G (h denotes the Minkowski function of G). In particular, the point b is extreme (cf. the remark before the proof of Proposition 8.3.5). Hence the mapping E a A 4 A.b E G is the only (modulo Aut(E)) complex kG-geodesic for (0, a); cf. Proposition 8.3.5(a). One can easily prove that the mapping r o (PO is also a complex kG-geodesic for (0, a). Hence r(;Lb) = X(;L) b, A E E, for a suitable

automorphism X E Aut(E). One can see that X = idE. Thus r o (po = (po. In particular, rpo(E) = S (recall that S is one dimensional). Since cpo' o r : G -* E.

VIII. Complex geodesics. Lempert's theorem

263

we conclude that cG (0, a) = kG (0, a) = h (a). This means that G is convex at b, cf. Proposition 2.2.1; a contradiction.

8.4 Geodesics in convex complex ellipsoids For p = (p, .... , p") with p'..... p, > 0, n > 2. define the complex ellipsoid (cf. Example 6.1.6):

£(p) := ((z,. ...czn):

izi12''' < I).

i=I

Note that (p) C En is a balanced n-circled pseudoconvex domain and that B. = ((1... . M. Remark 8.4.1. (a) (p) is convex if p'..... p,, > 1/2.

(b) (p) is geometrically strictly convex if and only if p,..... p" > 1/2 and

#(j:pi=1/2)
(c) 86(p) is C"'-smooth and strongly pseudoconvex at all points : belonging to

n (C.)". (d) If p, , ... , p,, > 1/2. then (p) is strongly convex at all points

belonging to

n (C.)". Recall that a bounded domain D C R' is called strongly convex if there exist

a neighborhood U of a D and a C2-function r: U - R such that U f1 D = jr < 0). U fl 8D = (r = 0}. grad r 0 on U. and fr(x;t) > 0. X E U, 4 E (RI)., where e_r

?{r(x;$) :=

k= 18x 8xk

(e) 86(p) is CI-smooth if p,,..., p,, > 1/2. (1) 8£(p) is C2-smooth iff p,...., p, > I. (g) For p,, ... , p" > 1 the following conditions are equivalent:

(i) (p) is strongly convex; (ii) (p) is strongly pseudoconvex;

(iii) E(p) = B,, (i.e. p, _ ... = pn = I ). (h) If p,, .... p,, > 1/2. then any boundary point of (p) is a complex extreme

point for (p).

8.4 Geodesics in convex complex ellipsoids

265

From now on we will assume that £(p) is convex, i.e. that p l, ... , p" > 1/2 (cf. Remark 8.4.1(a)). Our aim is to characterize all complex geodesics cp: E -+ £(p). Observe that if ,p = (, ... , (p,,) : E -), £(p) is a complex geodesic with rp" __ 0. then the mapping 0 : (cot... . (p,,-i): E -> £(P), P := (Pi.... is a "lower dimensional" complex geodesic. For we have c(A")) <

CE(A', A") = CE(p)((P(A),

c£(p)(W(A'),

0(A")) <

cE(k', A").

A', A" E E.

Hence it suffices to describe only those complex geodesics V : E -+ £(p) for which

Sp, $0,

j = 1,....n.

(8.4.1)

Moreover, after a suitable permutation of variables, we may assume that for some

0<s
(pi, ... , cps have zeros in E and cos+i.... , Sp are without zeros in E. Proposition 8.4.2 (cf. [Jar-Pfl-Zei]). A mapping V = (VI, (8.4.1) and (8.4.2) is a complex geodesic in £(p) iff

-a ( i-- x) Up), ai--

.

..,

p,,): E -* C" with

j = 1,....s

yep,

aj

where

(8.4.4a)at,...,a,, EC,. (8.4.4b) ao, .... as E E, a,+I, ... , all E E,

(8.4.4c) ao = E= Ia1l2p'aj, (8.4.4d) I + IaoI2 = E"=i Iaj I2 ' (1 + lai I'-), (8.4.4e) the case s = 0, an = aI

(8.4.2)

= a,, is excluded,

(8.4.4f) the branches of the powers are such that 1 I'pJ = 1, j = 1, ... , n. Moreover, the complex geodesics in £(p) are unique modulo Aut(E).

(8.4.3)

Vlll. Complex geodesics. Lempert's theorem

266

Remark. (a) In the case p, = ... = p,, > 1 /2 the problem of characterizing the complex geodesics in E(p) has been studied in [Pol I I and (Din-Timl. (b) The case p, _ ... = p = 1 /2 has been investigated in IGen 31 and [Din-Tim].

(c) The case it = 2, p, = 1. p2 > 1 /2 has been discussed in IBFKKMPI; cf. Example 8.4.7. Note that condition (e) says that 0 # const. Condition (f) is of technical character, since we prefer to have a one-to-one correspondence between rp and the parameters

aj, aj, j = I..... lt. Corollary 8.4.3. Let cp: E E(p) be a complex geodesic. (a) I f p, .... , p E (; . 1), then (p extends holomorphically to a neighborhood of E. (b) If t := max(p,, ... , p ] > 1, then tP extends to an (I / t)-Hiilder continuous mapping on k (tp E 01'(E)). (c) If u := max(p j : pi # 11 < I. then tP extends to a Ct -napping on E whose first order partial derivatives are (I /u - I)-Holder continuous (tp E C1.1/U-1 (E)).

Remark 8.4.4. Proposition 8.4.2 gives a tool for finding effective formulas for r := xe(l,)(zo; Xo). Since the case where a,, j = Xo. j = 0 (for some I < jo < n) may be reduced to the lower dimensional one, we can assume that for each j E (I , ... , n j either z4). j # 0 or X,,,j 36 0. This condition assures that the complex geodesic tp: E -i £(p) with (p(0) = a), r(p'(0) = X0 satisfies (8.4.1) (note that by Proposition 8.4.2, (p is uniquely determined). Thus, to calculate r it is enough to find a permutation a of (I .... , n), a number 0 < s < n, and a j. a j, j = I.... , n, with (8.4.4) such that:

-ajaj.

j = 1....s

aj(I - 1aj12 Xo.a(i) ='Pi(0) =

u

n (do - 6j) -

(ao-Q'j))

j = 1,...,s j = s + I .... , n

There are situations where the above problem has an explicit solution (cf. Examples 8.4.7, 8.4.8 and Exercise 8.18), and consequently r = x(zo; X(j) may be described by an effective formula.

Proof of Proposition 8.4.2. First, observe that if zo E (a£(p)) fl (C.)", then the unit outer normal vector v(zo) _ (vi (a))..... to a£(p) at a, is uniquely determined and

8.4 Geodesics in convex complex ellipsoids

where p(;o) > 0. If V = (tp), ....

267

E -> C" is a bounded holomorphic

mapping with (8.4.1), then by the identity principle (p`(A) E (C.)" for almost all A E M. Combining these facts and using Remark 8.2.3. Corollary 8.2.8. Lemma 8.3.1. and Remark 8.4.1(h), we get the following criterion.

Corollary 8.4.5. Let V = V,,): E -> C" be a non-constant bounded holomorphic mapping with (8.4.1). Then (p is a complex geodesic in E(p) if m

E IV 12p' = I a.e. on dE

(8.4.5)

i=)

and there exist functions h E HI (E. C") and p : d E -- R,O such that

a.e. on M. j = I..... n.

h* = PPS

(8.4.6)

II

The proof of Proposition 8.4.2 will be divided into four steps.

Step I ". Any mapping of the form (8.4.3) with (8.4.4) is a complex geodesic in

E(p) Suppose that tP is given by (8.4.3) with (8.4.4). To verify that tP is a complex geodesic, we apply Corollary 8.4.5(F-). Obviously. op is continuous on t and non-constant. If A E dE, then by (8.4.4c,d) we get:

1-a,A12

of

ItP.i(A)12p, =

/_)

1a,12", I 1

J=)

E;

-auAl

Iajl2p,(I + Ietjl2) -2Re(;tr_=) Iaj12t''ai) I + Ia(i122 - 2 Re(lao)

=I.

which gives (8.4.5). Define (I -Q,A )2

j = I..... s

ull,

(n-u )(1-a i.)

p(A) := I I - aoAI-,

j =.r+ I..... it

.

AE E.

A E dE

(the branches of the powers are the same as in (8.4.3)). One can easily prove that

hj E H''0(E). j = 1.....n. Direct calculations give (8.4.6). The proof of Step 1" is completed.

D

268

VIII. Complex geodesics. Lempert's theorem

Step 2". Any complex geodesic ip: E -> E((po, ..., po)) with (8.4.1) and (8.4.2) is of the form (8.4.3) with (8.4.4).

Let pi = . . . = p" = po and let gyp: E -+ E(p) be a complex geodesic with (8.4.1) and (8.4.2). We are going to apply Corollary 8.4.5(-+). Let h and p be as in this corollary. In view of (8.4.6) we have

ihi E R>o a.e. on 8E,

j = I_- n.

(8.4.7)

Now, we need an auxiliary lemma.

Lemma 8.4.6 (cf. (Gen 3]). Let f E HI (E) be such that f*(A) E R>o for almost all A E 8E.

(8.4.8)

Then there exist r > 0 and a E E such that

f (A) = r(A - a)(I - a)), A E E.

Proof of Lemma 8.4.6. Put bo := f (0), b, := f'(0), P(A) := bo + b1A + &X2, A E C. In view of (8.4.8), the Cauchy integral formula (cf. Appendix H 3) gives b, > 0. In particular, P(A)

= b1 + 2 Re(boA) E R,

A E 8E.

(8.4.9)

A

Let

(x)-P(x

g(A)

x

0.

A E E.

1=0

Observe that g E H1(E) and, by (8.4.8) and (8.4.9), g*(A) E R for almost all A E 8E. Hence, by the identity principle, g - 0. Thus f = P in E. Consequently, by (8.4.8) and (8.4.9), b1 + 2 Re(boX) > 0 for almost all A E 8E, which implies that b1 ? 21bo1. We will consider two cases.

1)bo=0. Weputr:=bl,a:=0. 0. Then the polynomial P has two roots At = , (-b1 ± bf - 41bo12) # -bo/71+, a := A+ the 0 with A+11_ = 1. One can assume that A+ E E. With r proof of Lemma 8.4.6 is completed. 0 2) bo

269

8.4 Geodesics in convex complex ellipsoids

In view of the lemma, condition (8.4.8) implies that there exist ro..... r > 0.

ao..... a E E such that:

A E E. j = l.....tt.

vj(X)Iti()L) = ri(A-ai)(l -aiX).

(8.4.1 0)

V (X) . h(a) = ro(X - ao)(I - doll).

X E E.

(8.4.11)

Replacing It by - h (and p by 1 p). we can always assume that ro = I. Note that by (8.4.2) we have

a1.....a, E E.

(8.4.12)

Moreover. (8.4.10) and (8.4.11) give a

,1

/=l

i=1

aoriai and I+1aol-ri(I+Iai12).

(8.4.13)

On the other hand, conditions (8.4.5).(8.4.6). and (8.4.11) imply that for almost all

AEi3E P(A)Po = P(,k)PoE Isv;(A)I2A' _ tp'(1).h'(A) = II - aoxl'.

Hence by (8.4.6) we conclude that

I/tj(A)I = II -for almost all X E

E.

j = 1,...,n.

In particular,

the functions h1..... h are bounded.

(8.4.14)

and by (8.4.10) we have r" ISpj (A)II I - aoXI n = ri"' I 1 - diAI n) I

for almost all A E 8E,

j = I..... n. (8.4.15)

Note that the functions (I - ai X )11TH, j = I ..... is, are outer. Consequently, by (8.4.15), the decomposition theorem for H I -functions (cf. Appendix H 6) shows:

Vj(A)(I -aoA)1'I" =a,B1(A)Si(k)(I -X E E.

(8.4.16)

270

VIII. Complex geodesics. Lempert's theorem

where

IajI=r

,

j=1,...,n, j = 1, ... , s j = s + I,...,n

Bi (A)

S;(A) : = exp C-jn

eeie ;e

(8.4.17)

(cf. (8.4.2), (8.4.10), (8.4.12)) , (8.4.18)

±.kdaj(0))

aj is a singular non-negative Borel measure, j = 1, ... , n, and the branches of the powers are chosen such that I 11PO = 1.

It remains to prove that aj = 0, j = 1, ... , n. If we suppose for a moment that all the measures vanish, then: - conditions (8.4.16) and (8.4.18) imply (8.4.3); - conditions (8.4.13) and (8.4.17) imply (8.4.4c,d);

- ao E E, otherwise s = 0 and ao = a,

a E 8E which would imply

that rp = const.

We come back to the proof of aj = 0, j = 1, ... , n. First, observe that by (8.4.10), (8.4.14), and (8.4.16) there exists e > 0 such that

AEE, j=l,...,n.

1?ER-ajIII-ajtl2-!PoI1-aoX

ISiW

On the other hand (cf. Appendix H 6), Sf (A) = 0

for a1-almost all A E 8E,

j = I..... n.

Combining the two conditions above and using the fact that for any 0 E R, b > 0 the function

EBA-+IA-110explb

1 - IAIz IA - 1 Iz

is unbounded, we easily conclude that a1 = 0 for all j = 1, ... , n. The proof of Step 2° is finished. \\\`

0

Step 3°. Any complex geodesic gyp: E -+ £(p) with (8.4.1) and (8.4.2) is of the form (8.4.3) with (8.4.4). Note that Steps 1° and 3° give the proof of the first part of Proposition 8.4.2. Fix E-+£(p)bean arbitrary complex geodesic with (8.4.1) and (8.4.2). Let h, p correspond to V as in

8.4 Geodesics in convex complex ellipsoids

271

Corollary 8.4.5. Write

'p3=Bii/ij,

j=I.....n.

where Bi is the Blaschke product for (p, and f1 is nowhere vanishing (define

Bi :- I for j =s+ 1.....n). Put: Oi := :- B i *i

!i

P!

i :=Pi- hi Vi -

(8.4.19)

One can easily prove that by (8.4.5) we have n

E i=1

)I

E

I

a.e. on

E.

i=I

and that. by (8.4.6).

a.e. on

E.

j = l... . t.

Moreover, hi E H I (E): here it is important that pi < Po. j = 1..... it. Thus. by Corollary 8.4.5. the mapping 0 is a complex geodesic in E((po,, .... p())) with (8.4.1) and (8.4.2). Consequently. by Step 2", the geodesic (p is of form (8.4.3) with (8.4.4). Finally, using relation (8.4.19), we conclude that the same is true for the geodesic V.

0

The proof of Step 3" is finished.

We move to the last part of the proof of Proposition 8.4.2. Step 4". Proof of the uniqueness of complex geodesics.

The case where e(p) is geometrically strictly convex (cf. Remark 8.4.1(b)) follows directly from Proposition 8.3.3. In the general case we proceed as follows. By Proposition 8.3.2 it suffices to prove the uniqueness of the x; (p) -geodesics. Let gyp. ' : E - E (p) be two complex geodesics with

'p(0) = *(0)

and

cp'(0) = *'(0).

(8.4.20)

So far. Proposition 8.4.2 shows that: SPi - 0 if jfii - 0. Thus, without loss of generality. we may assume that 'p and >/i satisfy (8.4.1). Moreover, we assume that

'p fulfills condition (8.4.2). Put lu := 1j: >Gi has a zero in E). Now observe that X := 1(V + >i) is again a complex geodesic in E(p) (E(p) is convex, cf. the proof

VIII. Complex geodesics. Lempert's theorem

272

of Proposition 8.3.3). In particular,

x(A), p(A), *(A) E ae(p) for all I E aE. Therefore, 0. ), E aE,

argco (A) = argifj (A) = argxl()L) if w,(),)*,(A)

(8.4.21)

I(pjI=I*jI=IXjI

onaE if p1> 1/2.

Consequently, if pj > 1/2, then cps = *f on aE and so cps - i/i, on E. It remains to discuss j with pj = 1/2. Fix such a j. First note that by (8.4.21) we have Sod *j = *jcOj

on

8E.

(8.4.22)

There are four cases.

(a) 1< j< s and j E 10. Then

a

(I - a;)(1 - a;X)

and , (A) = bl

(1 - 0OA)2

(X - '0,)0 - 6ik) (I #0),)2

(b) 1< j< s and j 0 l0. Then (I - 01j) (1 - aj.X) a ! (I - aoA)2

Vf

and

*l (A) _ bI

l - f3j,

2

(I - Pox )

(c) s + I < j < n and j E I. This case is symmetric to (b).

(d)s+1<j
1 -a11 2 =aj (1 -aOA)

and

1/1(1) = b j

p )'

1-#0'/

(I

Ad.(a). Using (8.2.22) one gets

ajbj

ajbj

( 1 - aOA)2(A - fl0)2

(1 - #OA)2(A -

ao)2

So we obtain a0 = HBO. Moreover, because of (8.4.20) (cf. Remark 8.4.4), we get

orjaj = fibj

and

aj(1 + IajI2) = b;(1 + Ifi,12).

273

8.4 Geodesics in convex complex ellipsoids

which directly implies that a j = /3j and a j = bj. Ad.(b). By (8.4.22) we have

hja, (h - /3j 2 _ (I -aoA)2 l So we conclude that a

ajbj

(1 \l -A,x/

fijA\2

and therefore

bj((j(w-/3j)2-aibj(l -flA)2. Hence we have If.,I = I. Again using (8.4.20) we get

-ajar = b1

and a,(l + Ja., j2) _ -2b,/ j.

which implies that Jaj I = I : a contradiction. Ad.(c). The argument is the same as (b). Ad.(d). Directly from (8.4.20). using the forms of (pj and *j, we conclude that

aj = bj. Then. in view of (8.4.22). we obtain a = f, and next, using once again

0

(8.4.20). we prove that a) = 13j: the details are left to the reader.

0

The proof of Proposition 8.4.2 is finished.

Example 8.4.7 (cf. (BFKKMPD). A formula for

Fix in > 1/2, put f := E((I. m)), and fix (a, b) E E, (X, Y) E (C2).. We are going to find an effective formula for r := rp((a. b); (X. Y)). We already know that if a = b = 0. then r = x;((0.0): (X. Y)) _ (!e(X. Y). where q; denotes the Minkowski function of t. Observe that the number a l/q; (X. Y) is the only positive solution of the equation H2iXl2

+,,2nI1y12n =

1.

(8.4.23)

Assume that (a. b) # (0.0). One can easily prove that for any 8 E R the mapping

(:I.:2)

:, -a p;N(1 -IaI2)t:2

VIII. Complex geodesics. Lempert's theorem

274

is an automorphism of E. Thus, without loss of generality, we may assume that

a=0and0
IYI

and therefore we can also assume that X 36 0. Let V = (VI, (p2) : E -* E be a complex geodesic with (p(0) _ (0. b),

rcp'(0) = (X, Y).

Note that V satisfies (8.4.1) and (8.4.2) (with s = I ors = 2). Hence by Proposition 8.4.2 either (P(A)

=I

a,

a'

I - 6121k

- aoA

if s = I

1 - a0A

or

I

J

a,

A

a2I

1

- a2 -&Z Jl

- a2AC\' II

-

if s = 2.

where a,, a2, a, = 0, and a2 satisfy (8.4.4); cf. Remark 8.4.4.

We will prove that:

(a) if s = I, then mIYt
b); (X. Y)) _

T=

(8.4.24)

- b2"

(8.4.25)

(b) ifs = 2, then mIY1 > 61X1,

r = xt((O,b);(X. Y)) =

ma(I -t)IYI

b(1 -a2)(m(l -t)+t) '

(8.4.26)

(8.4.27)

8.4 Geodesics in convex complex ellipsoids

275

where 2m2v

t

1+2m(m-1)v+ blXI

v

( Inly

a:=

(8.4.28)

1+4m(m-1)v

)

(8.4.29)

b(1 - t) =).

(8.4.30)

Proof of (a). Using the equations from Remark 8.4.4 and conditions (8.4.4), one can easily prove that s = 1 iff there exists 0 < a < I such that 1IYI

= ba(1 -b2"'),

r

in

IIXl2+b2m(l

+a2b4m =

1

+a2),

which is equivalent to (8.4.24) and (8.4.25).

Proof of (b). This case is more complicated. First observe that 0 < v < 1 (cf. (8.4.26) and (8.4.29)) and that the number t (cf. (8.4.28)) is a solution of the equation

(in - 1)2vt2 - (I + 2m(m - 1)v)t +ni2v = 0.

(8.4.31)

Moreover, 0 < t < 1. As in (a), using Remark 8.4.4 and (8.4.4), we conclude that

s = 2 iff there exists 0 < a < l such that a2Je-l > On, 1

IYl =

T

b4"'

1+

a-2

I

b

1m(1 -a2) +a2 _

am

b`2m a2"'-`'

,

b2"'

1

a`). T` lXI` + a2m (1 +

Combining these conditions we get:

v((m - 1)a''" -

ma2m-2 + b2'n )2

- (a4m-2 - b2n'a2n' -b2ma2' 2 +b

4n,)

= 0,

ma2'n-'IYI

T

bI (m - 1)a2," -

mat,"-2

+ hem

I

(8.4.32)

Vill. Complex geodesics. Lempert's theorem

276

In view of (8.4.31). the first of the above equations may be written in the form (a2'"

-

tae'"-2

- (I - t)b2")

((ttt - 1)2 va2m -

m2v

(I - v)b2i'

t

I -t

Now, one can easily prove that the equation

a2m-ta`''"-`-(I-t)b'2"'=0

(8.4.33)

has a solution 0 < a < I with a2'"-1 > b2"'. Note that, by (8.4.23), the equation (8.4.33) is equivalent to (8.4.30). Finally, in view of (8.4.33), formula (8.4.32) gives (8.4.27).

Example 8.4.8. A formula for

Let A2 := E((;, ; )). Fix b = (b,, b2) E 02 n (R>o)2, X = (X,, X2) E (1R2).. Using the same methods as in Example 8.4.7, one can prove the following formulas. The details are left to the reader. Define

d := I - b1 - b2, co := I + hi + h2, c, := I + b1 - b2, c, := I - h, + b2, L, (X) := (I -h2)X, +b1X2, L2(X):=b2X1 + (I -b,)X2, A := dc, + i2db2, C

-2b1c2 +idc2.

E

-2d

b

B := 2db1 + idc2, D -dc1 +i2b2e,.

,+ i 1-2db,hl , F

:

=-

I2dbl

+i2d h,,

Cp,1,) :_ (z E C.: ArgA < Argz < ArgB or Arg(-A) < Argz < Arg(-B) , (z r= C.: ArgC < Argz < Arg D or Arg(-C) < Argz < Arg(-D)), Cli;,l (z E C.: Arg(- D) < Arg z <_ Arg A or Arg D < Arg z Co) (,ur or Argz <- Arg(-A)). {z E C.: Arg(B) < Argz -< ArgC Cp;r Cper

or Arg(-B) < Argz < Arg(-C)). (z E C.: Arg(- F) < Argz < Arg A or ArgF < Arg z

or Argz -< Arg(-A)). (z E C.: ArgB < Argz < Arg E or Arg(-B) < Arg;, < Arg(-E)). Cpar Q,i :_ (z E C.: Arg(E) < Argz < ArgF or Arg(-E) < Argz -< Arg(-F)).

8.4 Geodesics in convex complex ellipsoids

277

There are two cases: (a)

+

!

< I (cf. Fig. 8.1): then:

) (1X11+IX2I)

ifXEC(1) lin

(IXII + IX2I)

if X E Cn'

2Jn:e, (ILI(X)I+

ILl(X)12+ti:deiIX212

ifXEC"

,dh,,._ (IL2(X)I +

IL2(X)12 +

if X E C

Fig. 8.1: the indicatrix of x_%, (b; )

(b) lb- -I +

> I (cf. Fig. 8.1): then:

X E CI'

Jr., O X I

(ILI(X)I + VILI(X)12 + tidc,IX2I2).

X E C?

(IL2(X)I +

X E C"

xJ: (b; X) _ 2db,.

:i

IL2(X)I2 +; de2IX112

j,-h, (b2I Ll (X ) 12 + bI I L2(X ) 122

- bIb2I X I + X2122).

X E Celi

VIII. Complex geodesics. Lempert's theorem

278

Fig. 8.2: the indicatrix of xo, (b;

)

8.5 Biholomorphisms of complex ellipsoids Fix p = (pi( p i ,

q = (qi, ....

E (R>o)", n > 2, and consider the

complex ellipsoids E(p) and E(q); cf. §8.4. It is natural to ask: when are the ellipsoids E(p) and E(q) biholomorphic ?

Theorem 8.5.1. The ellipsoids E(p), E(q) are biholomorphic iff p = q up to a permutation.

Remark. (a) It was N. Kritikos [Kri] who already in 1927 studied the group of automorphisms of 6(1/2, 1/2) by exploiting complex geodesics (he himself used the notion metrische Ebene); cf. Corollary 8.5.5. (b) The biholomorphic equivalence problem for all complex ellipsoids was studied first by I. Naruki (cf. [Naru]) and then by M. Ise (cf. [Ise]); the equivalence problem for general bounded Reinhardt domains containing the origin is due to T. Sunada (cf. [Sun]). (c) Similar investigations for proper holomorphic mappings were done by M. Landucci (cf. [Landu]) and S. M. Webster (cf. [Web]). (d) More generally, we have the following result. If two bounded balanced pseudoconvex domains G 1, G2 C C" are biholomorphic, then there exists a biholomorphic mapping F : G i -> G2 with F(0) = 0. Then, by the results of Chapter III, it easily follows that G,, G2 are bilinearly equivalent. This result is due to W. Kaup & H. Upmeier (cf. [Kau-Upm]) and also

8.5 Biholomorphisms of complex ellipsoids

279

to W. Kaup & J.-P. Vigue (cf. [Kau-Vig]). Their proofs are based on Lie theory and so they are beyond the scope of this book. Recently. there is another proof given by K.-T. Kim (cf. [Kim 2]); he exploits the fact that maximal compact subgroups of a connected Lie group are conjugate.

Our proof of Theorem 8.5.1 is more consistent with complex analysis. It is based on the fact that any biholomorphic mapping between bounded complete Reinhardt domains extends to a biholomorphism of neighborhoods of the closures (cf. Theorem 6.1.10). First observe that without loss of generality we may assume that P1 = ... = X. = 1, PA+1..... P»

(8.5.1)

1,

ql = ... = q, = 1, g141,....q,,

(8.5.2)

0
(8.5.3)

(In the case k = I = n we simply have £(p) = E(q) = i».) Lemma 8.5.2. Assume that p satisfies (8.5.1) and let V E Aut(Bk), a' 9k+ 1, ... , 9» E R. Define

i

_ (W(z'), e

(z',a'

za+1

112

TP77-1

ifl

- IIa'II2

(1 - (;'. a1))2

a'))2)

C(1

1

(0),

Then 4 E Aut(£(p)).

Proof. The cases k = 0 or k = n are obvious. Assume 1 < k < n - 1. Clearly' is holomorphic and injective in a neighborhood of £(p). Observe that by Corollary 2.2.4 we have (I-Ila'112)(1-11:7'112)

IIW(z)Il=cet(0.W(z))=ca;(a,z) Hence for z =

.:.k+1, .

. . , z..» ,,

)

1-

I1-(,,.a)i 2

E E(P) we obtain (IIa'.

,

,

I4)j (z)12n; = 1 1

1

which finishes the proof.

k

a1') I2

(1 -

Iz,12p, ),

0

Vlll. Complex geodesics. Lempert's theorem

280

Lemma 8.5.3. Let p. q E (R,o )" satisfy (8.5. I) - (8.5.3) and let L : C" -+ C" be a C-linear isomorphism such that L(((q)) = 6(p). Then p = q up to a permutation a of (k + 1. . . . , n) (in particular. k = 1) and the mapping L is of the form:

L(wl,..., w,,) = L(w'. Wk+1..... wn) = (U(w'), e104+iuv0(k+I).....eiH"wa(")) where U : C"-k - C" is a unitary i s o m o r p h i s m a n d 0j+I ,

.... 0" E R.

Proof. Recall (cf. Remark 8.4.1) that 8E(q) is strongly pseudoconvex at a point

a E a6(q) if

,w")EC":wt+I .... wn=0). Moreover, aE(q) is strongly pseudoconvex at a if aE(p) is strongly pseudoconvex at L(a). Hence

L(HI)=H2

((z l...

Now, since L is an isomorphism, we easily conclude that k = I and that there exists a permutation a of (k + 1.... , n) such that L"(j)((wj = 0)) = (0).We can assume that a = id. Thus

L(w'. wk+1..... wn) = (U(w') + M(wk+I, ... , wn), Ck+iwk+I..... (,,W.)where U : Ck - Ck and M : C"_k --b- Ck are C-linear and ck+1.... , c" E C.. The case k = 0 is elementary. Assume k > 1. It is clear that U is an isomorphism and that U(Bk) = 13k (take wk+1 = = w" = 0). Hence U is unitary. It remains to prove that M - 0. Since L(9E(q)) = aE(p), we get to") E 86(q).

IIU(w')+M(wk+l..... wn)112+ > ICjwjl2p, = 1, (w', This implies that n

_ E IwjILl,+IIM(wk+1,...,wn)112+ E Icjwjl2p, =0 j=k+l

j=k+I

for all (wk+l,... , w,,) with

Iw

j=k+I

12q,

< 1.

8.6 Schwarz Lemma - the case of equality

wj-1 = wj+1 =

Taking wk+,

281

= w" = 0, we have

-xqi+IIM(0,...,0, 1,0,....O)II2x+ICjI2P/x"/ = 0, 0<x < l,j =k+1,...,n. j

Hence we get p j = q, I c j I = 1, and M (0, ... , 0, 1, 0, ... , 0) = 0 for j =

k+l,...,n.

Proof of Theorem 8.5.1. Recall that if 4: 6(q) -* 6(p) is a biholomorphic mapping with -t(0) = 0, then L := 4'(0) is a C-linear isomorphism with L(E(q)) = 6(p). Thus, in view of Lemmas 8.5.2 and 8.5.3, it remains to prove the following fact.

Lemma 8.5.4. Assume that c: E(q) -* E(p) is a biholomorphic mapping with 4)(0) _ (0, ... , 0, bk+i,... , b"). If p, q satisfy (8.5.1) - (8.5.3), then bk+I b" = 0.

Proof. Fix v with l + l < v < n and let H,, := E(q) fl {z E C': zv = 0). Because of the fact that 4 extends biholomorphically to 6(q), we have 1k+i ... 4)" = 0 on Hv; cf. the proof of Lemma 8.5.3. Then 4)(H,,) C E(p) fl {w E C": W j(v) = 0}

with I + 1 < j(v) < n. In particular, (Djtvl(0) = bj() = 0. Since v = b" = 0. bijective, we have bk+i = Corollary 8.5.5. If p j

j(v) is

I for all j = I..... n, then any 4 E Aut(E(p)) has the

origin as a fixed point.

Observe that the main tool in the proof of Theorem 8.5.1 was Theorem 6.1.10 on the extendability of biholomorphic mappings. Q It seems to be interesting to find a proof of Theorem 8.5.1 (at least in the convex case) using only the complex geodesics from Proposition 8.4.2.

0

8.6 Schwarz Lemma - the case of equality Let q be a complex norm on C". By

Gq = (Z E C": q(z) < 1) we denote the corresponding unit ball. Then any holomorphic map F: Gq -> Gq, F(0) = 0, satisfies q(F(z)) <_ q(z) for all z E Gq; cf. Theorem 2.1.1. Moreover, equality in a neighborhood of 0 implies that F is a linear automorphism of Gq.

VIII. Complex geodesics. Lempert's theorem

282

Using the uniqueness of complex geodesics, J.-P. Vigu6 showed the following generalization of the last statement above (cf. [Vig 5]). Theorem 8.6.1. Let Gq be as above and suppose that every boundary point of Gq is a complex extreme point of Gq. Then any holomorphic map F: Gq + Gq with F (0) = 0 and q (F (z)) = q (z) for all z E U, where U is a nonempty open subset of Gq, is the restriction of a linear automorphism of C".

Proof. Fix a point z' E U, z' $ 0. Then cp(A) := Xz'/q(z'), A E E, is a complex geodesic through 0 and z'. Then F o rp is the unique complex geodesic through 0 and F(z') which implies that

,kF(z')/q(F(z')),

Fo

A E E.

(8.6.1)

Now, we write F as 00

F(z) = E Pi(z), i=1

where Pi is an n-tuple of homogeneous polynomials of degree j. So. for A E E we obtain 00

P;(z'/q(z')))J.

P1(Az'/q(z')) _

F(G(A)) _ j=1

j=1

Comparison with (8.6.1) for j > 2 yields

0 = Pj (z'/q(z')),

and therefore

Pj (z') = 0.

By the identity theorem it follows that

F(z)=P,(z)

ZEGq,

for all

i.e. F is a restriction of the linear map Pi. It remains to show that P, is an automorphism of C". Otherwise, there exists a point w' E (C"). with P,(w') = 0. Now observe that the interior of

{Z E C": q(PI(z)) =q(z)) contains a point z° with q(z°) = 1. Since z° is a complex extreme point of Gq, there is a sequence of complex numbers A, -p 0 with 00

I

and

q(P,(z°+,Xw')) =q(z°+Avw'), v E N.

8.6 Schwarz Lemma - the case of equality

283

Combining these pieces of information we arrive at

I =q(z°)=q(PI (z°))=q(PI(z°+A,w'))> 1; 0

a contradiction. Hence Ker P, = ]0}.

Under the assumption of Theorem 8.6.1 the map F is not necessarily a linear automorphism of Gq as the following example shows. Example 8.6.2 (cf. [Vig 5]). Fix Of E (0, 1) and define on (C2).: O(z) = 0((z1 . z2)) := arccos(Izi I/ III).

Moreover, choose a decreasing CI-function x : [0, r/2] -> R+ such that the following two properties hold:

x (t)

-

qo(z) :=

1

.

a

if t is near 0 if t is near n/2 '

(x(8(z))I zi 122 + Iz2I2)l/2 Q

ifz0 ifz=0

is a norm on C2 with a strictly convex unit ball Gq. Then the map F(z) := (z2, az,) belongs to O(Gq. Gq) and F(0) = 0. Moreover, if z E G. and z is near (0, 1), then qo(F(z)) = qo(z). Thus all the assumptions of Theorem 8.6.1 are fulfilled. But I det F'(0) I = a < 1. So Cartan's theorem on holomorphic mappings implies that F is not a linear automorphism of Gq; cf. §8.7. In the following result we sharpen the assumptions in Theorem 8.6.1 to be able to conclude that F is a linear automorphism of Gq. Theorem 8.6.3 (cf. [Vig 51). Let q, Gq, U, and F be as in Theorem 8.6.1. Moreover,

suppose that there exists a E U with F(a) = a. Then F is a linear automorphism of Gq.

Proof. According to Theorem 8.6.1, the map F is a restriction of a linear automorphism of C" which we denote again by F.

VIII. Complex geodesics. Lempert's theorem

284

Because of the linearity of F we may assume that q(a) = I and q(F(z)) = q(z) whenever q(z -a) < r and 0 < r < 1 is suitably chosen. Moreover, we recall that

q(F(z)) <- q(z) for all z E C. Therefore, if q(z - a) < r, then q(F(z) - a) < r. Now, we consider a subsequence of (F))jEN that converges uniformly on com-

pact sets to a linear map F. Obviously, t(O) = 0 and F(a) = a.

Suppose that there is a w' E (O"). with F(w') = 0. Since a is a complex extreme point, there exists a complex number X yl- 0 with

q(a + Aw') > 1

and

q(Aw') < r.

By the construction this yields

q(F1(a + Aw')) = q(a + Aw') >

j E N.

(4.8.2)

On the other hand, we have F(a + Aw') = F(a) = a which, because of (4.8.2), implies that

I < q(a + Aw') = q(F(a)) = q(a) = 1;

0

a contradiction.

In the next theorem we assume that aGq is a real-analytic manifold. Then it is well known that every boundary point of 8Gq is a complex extreme point of Gq so that we are automatically in the situation that is discussed in Theorem 8.6.1. Nevertheless, we prefer to give a simple direct argument.

Theorem 8.6.4. Let q be a complex norm on O" and let F E O(Gq, Gq) with F(0) = 0. Moreover, assume that q(F(z)) = q(z) for all z E U, where U is a nonempty open subset of Gq. If 8G,7 is a real-analytic manifold, then F is a linear automorphism of Gq.

Proof. Fix z' E U with z' 0 0. Then E3A

Az'/q(z') E Gq

is a complex geodesic through 0 and z'. So F o cp is again a complex geodesic in Gq. In particular, we get q(z') = YE(O;q(z')) = YG9(O; F'(O)z') = q(F'(0)z').

Now consider the map

aGq 9 z

F'(0) E C.

(8.6.3)

8.7 Criteria for biholomorphicity

285

Because of (8.6.3), there is an open set V in aGq. V 0 0. with D(V) C aGq. Then the identity theorem for real-analytic functions implies that go(8Gq) C 8Gq. i.e. q(F'(O)z) = q(z) for all z E C". Hence F'(0) is a linear automorphism of C".

If we analyse the map W := F'(0)-' o F we can see that 'I1(0) = 0, 'Y E O(Gq, G9), and 'Ir'(0) = id. Hence Cartan's uniqueness theorem implies that F =_ F'(0), i.e. F is a linear automorphism of Gq.

Corollary U.S. Let F: B"

B,, be a holomorphic map such that F(0) = 0 and IIF(z)II = Ilzll for all z E U, where U is a nonempty open subset of L. Then F is a linear automorphism of B.

8.7 Criteria for biholomorphicity Let G, , G2 be domains in C" and let F : G, -> G2 be a holomorphic mapping. We say that F is a y-isometrv (resp. x-isometrv) at a point a E G, if

Sc,(F(a);F'(a)X) =Sc,(a;X),

X E C",

(8.7.1)

where S = y (resp. S = x). What is the relation between the S-isometricity of F at a point and the global biholomorphicity of F ? (Recall that any biholomorphic mapping is a S-isometry at any point.) In the case where G, = G2 = G is a bounded domain in C" and F(a) = a the full answer to our question is given by the following well-known theorem due to H. Cartan.

Cartan Theorem (cf. [Kra 1(). Let G C C" be a bounded domain, let a E G, and let F: G -* G be a holomorphic mapping with F(a) = a. Then the following conditions are equivalent: (i) if )LO is an eigenvalue of F'(a), then IAoI = 1;

(ii) I det F'(a)I = 1; (iii) F is a y-isometry at a; (iv) F is a x-isometry at a; (v) F is biholomorphic, i.e. F E Aut(G).

Remark 8.7.1. (a) (cf. [Vig 31) Let C2:

Izil + Iz21 +elz,z21 < 1) (e > 0),

G,

((z,, z2) E

G2

((:i,z2) E C2: IziI+ IZ21 < I}.

286

VIII. Complex geodesics. Lempert's theorem

Then (cf. Proposition 2.2.1)

YG, ((0, 0); (X,, X2)) = YG,((0. 0); (X,, X2)) = IXi I + IX21,

(X,. X2) E C2.

In particular F := id : G, -* G2 is a y-isometry at (0, 0) which is not biholomorphic. (b) Let G2 C C be any taut domain that is not simply connected (e.g. an annulus).

Let F: E -* G2 be an extremal disc for (F(0), F'(0)), i.e. xG,(F(0);F'(0)) = 1. Then F: E - G2 is a x-isometry at 0 which is not biholomorphic. On the other hand, we have the following positive results.

Proposition 8.7.2. (a) (cf. [Vig 31) Suppose that G, C C" is a convex taut domain G2 be a y-isometry at a and that G2 C C" is an arbitrary domain. Let F: G, point a E G,. Then F is biholomorphic. (b) (cf. [Gra 2J) Suppose that G, C C" is a taut domain and that G2 C C" is a strictly convex bounded domain. Let F: G, -> G2 be a x-isometry at a point a E G,. Then F is biholomorphic.

Proof. (a) In view of Lempert's Theorem (Theorem 8.2.1), yG, = xc;, and CG, = kG, . Moreover, G, is c-finitely compact and yG, (a; X) > 0 for all X 0 0; cf. Chapter VII. Hence condition (8.7.1) (with S = y) implies that F'(a) is an isomoris biholomorphic for an open neighborhood phism, and therefore F : U U,, C G, of a. Since top G = top CG,, we may assume that U,, = 8,.,;, (a, ro) for some ro > 0. First we will prove that

cG,(F(a). F(;.)) = cG,(a, z).

z E G,.

(8.7.2)

Take z< E G, \ 1a) and let Sp : E -* G, be a complex geodesic for (a, zo) with

(p(0) = a, p(a) = zo for some 0 < or < I (since G, is convex, the notion of complex geodesics is well-defined). By (8.7.1) and Proposition 8.1.3 we get YG,(F(a); F'(a)(p'(0)) = YG,(V(0);cP'(0)) = 1.

This means that the mapping Fop is a y(;,-geodesic for (F(a). F'(a)V'(0)). Using again Proposition 8.1.3 we obtain

c'a:(F(a). F(zo)) =cG,(FoSp(0). Fo(p(a)) = p(0. a) = cc;, W0). 00 = c'G, (a. zo),

8.7 Criteria for biholomorphicity

287

which finishes the proof of (8.7.2). Since G I is c-finitely compact, condition (8.7.2)

implies that F is a proper mapping. In particular, F is surjective and F: B,.°, (a, ro) --;- B,.(;. (F(a), ro) is biholomorphic.

(8.7.3)

Since F is proper, there exist a proper analytic set S C G2 and a number N E N such that

F: GI\F-I(S)->G2\S is an N-fold covering (see also the proof of Proposition 10.6.7). By (8.7.3) we have N = 1, and therefore F: G, -+ G2 is biholomorphic. (b) Since G 1 is taut, xG, (a: X) > 0 for all X 0 0. Hence, as in (a), condition (8.7.1) (with 8 = x) implies that F: U --), is biholomorphic for an open neighborhood UQ of a. Let F := (gyp E O(E. G 1) : V(0) = a and xG, (a; rp'(0)) = 11,

M := U t2(E).

We will prove that

1° for any92EF: CG, (p(A'), 92(A" )) = cc,, (F o 92(A'). F o (p(A")) = n(A',

E E; (8.7.4)

2° M is closed in GI;

3° F: M - G2 is injective; 4° F(M) = G2; 5° (FIM)-I : G2 -> M is continuous; in particular, M is open. Conditions 2°-5° clearly imply that F : G, -+ G2 is biholomorphic. Ad.l° Fix (p c F. Since yG. = xG_ (G2 is convex), condition (8.7.1) gives

yG:(F(a);F'(a)(P'(0)) = xG, (a; (p'(0)) = 1. Hence by Proposition 8.1.3 we have

p(X'. I") = cG,(F o

Fo

cG

w(A"))

n(

"),

which gives (8.7.4).

Ad.2° Let M a z, z0 E GI, where (,. E F, A,, E E, v > I. Since GI is taut, we may assume that Ip,, 4 0) E O(E, GI). Obviously. 9po(0) = a. Since (p,,(0) -+ rpo'(0), the continuity of xG, (cf. Proposition 3.5.13(b)) shows that 'po E F. The sequence (CG, (a, z,,));"I is bounded. Hence by (8.7.4) the sequence (p(0. is bounded, and consequently we may assume that X. -+ A.o E E. Finally. Zo = Vo(Ao) E M.

288

VIII. Complex geodesics. Lempert's theorem

Ad.3" In view of (8.7.4), the mapping F is injective on rp(E) for each Sp E F.

Now let z, = Vj (k f) E M \ 1a), 0j E F. k j E E, j = 1, 2. Assume that 02(E) and suppose that F(zi) = F(z2) =: wo. By (8.7.4) 54 z2 and 4pi(E) the mappings F o Vj, j = 1, 2 are complex geodesics for (F(a), wo). Since G2 is strictly convex, F o = F o q? mod Aut(E) (cf. Proposition 8.3.2), i.e. 1

there exists 0 E R such that F o (pi (k) = F o W2 (e'01), I E E. Consequently, (p, (k) = q' (e'"k) for k near 0 (recall that F is biholomorphic near a) and finally, by the identity principle, 401(k) = V2(e'").). X E E. In particular, cpi (E) = (E), which gives a contradiction.

Ad.4" Let wo E G2 \ { F(a) } and let *: E -+ G2 be a complex geodesic for

(F(a). wo) with *(0) = F(a). By Proposition 8.1.3,' is a yG.-geodesic for (F(a), *'(0)). Put Xo :_ (F'(a))-1 *'(0) and let V E F be a wG,-geodesic for (a, Xo). Then F o V and i/i are yG,-geodesic for (F(a), c'(0)) (cf. the proof of 1"). Consequently, by Proposition 8.3.2, Mfr = F oV mod Aut(E). In particular, wo E F(cp(E)) C F(M). Ad.5° Let G2 9 w, = F(z,,) --> wo = F(zo) E G2. We have to prove that

z,, - zo. It suffices to show that if z,, - zo E (C U (co))", then zn = zo or, equivalently (in view of continuity and injectivity of FIM), that zo E M. Suppose that z,, = rp,.(k,,), where gyp,, E F. X,. E E, v > 1. Tautness of G, permits us to assume that (+,, 4 tpo E F (cf. the proof of 2"). The sequence (cG_(F(a), w,,))'°_1

is bounded. Hence by (8.7.4) we may assume that A. --- ko e E. Finally. zo =

0

tpo(ko) E M.

Notes The main result presented in this chapter is the Lempert Theorem (Theorem 8.2.1).

The proof of this theorem is based on Lemma 8.2.2; a similar result based on a generalized winding number formula (without proof) has been stated in [Aba 3]. In the original version (used by Lempert in [Lem 2,31) this lemma has been formulated and proved under the additional assumptions that (p, h E C(E) and that (8.2.2) holds for all k E aE. We found the proof without these additional conditions, and therefore the whole proof of Theorem 8.2.1 could be essentially simplified. The remaining part of the proof of Lempert's Theorem follows the ideas taken from [Roy-Won] (notice that the original proof presented in [Roy-Won] has some gaps). We completed and simplified the proof from [Roy-Won) in the series of lemmas: Lemma 8.2.4 - Lemma 8.2.7 (see also [Aba 3]). Our method of proof allowed us to skip the discussion on the regularity on E of complex geodesics (this is the central problem in Lempert's proof; cf. (Lem 1,31). We will come back to

this problem in Miscellanea (cf. Theorem C.2) and Exercises 8.4 - 8.12. On the

Notes

289

other hand, Lempert's method is more general and it works also in the strongly linearly convex case; cf. Miscellanea Theorem C.4.

290

VIII. Complex geodesics. Lempert's theorem Exercises

8.1. Let d* := m(k) (resp. If := g) and let S := y(k) (resp. S := A). Prove that for any domain G C C" and for any point ko E E the following statements are equivalent (cf. Proposition 8.1.3): (i) 3k0 " E E \ {ko}: m(ko, kg); (ii) `dk E E: dc(V(ko), SP(A)) = m(ko, k); (iii) SG(Q(ko);V'(ko)) = Yr(ko; 0-

8.2. Let k E N and let G C C" be a taut domain. Prove that the following conditions are equivalent (cf. Proposition 8.1.5): (k)

m(k) =- k c' and Yc = XG; 0) MG

(ii) e = kG; (iii) for any zo. Za' E G, m (k)

# zo', there exist rp E O(E, G) and f E O(G, E)

such that zo = V(0), Zp E v(E), ordZO f > k and f (V(k)) = kk, k E E. 8.3. Let k E N and let G = Gh C C" be a balanced domain of holomorphy. Prove

that for any a E G with h(a) > 0 the following conditions are equivalent (cf. Proposition 8.1.2(b)): (i) the pair (0, a) admits a complex mG )-geodesic; (ii) the pair (0, a) admits a complex Ya )-geodesic;

(iii) MG (0, a) = h(a), cf. Exercise 4.3.

8.4. Let G C C" be a convex domain. Prove that there exists a sequence G / G such that each G, is a bounded strongly convex domain with smooth real-analytic boundary.

8.5 (cf. [Lem 1,31). Suppose that G C C" is a convex domain such that V complex geodesic c p : E - G 3 MO > 0, 3µ E (0, 1 ] : dist((p(k), 8G) < Mo(1- Ikl)". I E E,

2Co>0, 3aE(0,1]:XG(Z;X)>Co

(dist(IIz, XBG))a'

ZEG,XEC".

(1)

(2)

Prove that any complex geodesic cp: E G extends as an (aµ)-H61dercontinuous mapping to E. (Note that any bounded convex domain satisfies (1); cf. Exercise 8.9.)

Hint. Use the Hardy-Littlewood theorem; cf. Appendix H 9.

Exercises

291

8.6 (cf. IDin-Timl). Let G C C" be a bounded balanced convex domain. Prove that G satisfies (l) with µ = 1. 8.7 (cf. ILem 1,31. [Roy-Won]). Let G C C" be a bounded convex domain such that 3r > 0: V;,,) E i) G: 3a)) E G: II:() - at) 11 = ro and 8(ao. ro) C G.

(3)

Prove that G satisfies (1) with p = I. Hint. Fix a complex geodesic cp : E G. It suffices to show that there exists a constant M, > 0 such that cG(cp(0). V(A)) < M1 - ; logdist(cp(A). aG). A E E.

Let K := (: E G: dist(z. aG) ? ro). Put M, := max(cG((Q(0). -):

E K).

Then for cp(A) E K we have cG(cp(0). rp(A)) < M3 - ; logdist((p()L), aG) with

M3 := M: + ; logdiamG. If V(A) K, then by (3) we get CG (0)), VW) :5 M4 - ; logdist(ip(A). aG) with M4 := M, + ; log(2ro). 8.8. Let G C C" be a strongly convex domain. Prove that G satisfies (3) and that 3R,) > 0: V:,() E aG: 3ao E C": Ila) - aoll = R,) and B(ao. Ro) D G.

(4)

8.9 (cf. [Merl). Prove that any bounded convex domain G C C" satisfies (1).

Hint. (cf. Exercise 8.7.) Fix a complex geodesic )p: E -> G. Let ao := 00). It suffices to show that there exist constants M, > 0 and 0 < p < 1 such that CG(ao. V(A)) < M, -

I

logdist((p(A). aG).

AEE

(then we can take M = exp(2M,p)). Let ro > 0 be such that B(ao.4ro) C G. Define K := (;, E G: dist(:. aG) > ro) and M2 := max(cG(ao, z): E K). Let Ao E E. If W(Ao) E K. then for any 0 < p < I we have eG(ao. o(AA)

M3(p) -

I

µ

Iogdist(V(Ao), aG).

where M3 (u) := M, + ,<< log diam G. We will prove that there is a p = µ() E (0. I J (independent of (o(A1,)) such that the above inequality is also true if V(A0) ( K

(then one can take M, := M3(po)). Suppose that V(A0) ff K and let ;() E aG he such that the point V(AO) belongs to the segment (ao. )). Define A = A(ao. r,). -(,) := conv(B(ao. 2ro) U (:.o)).

VIII. Complex geodesics. Lempert's theorem

292

Let bo := 2(ao+zo). Then dist(bo, 8G) > dist(bo, 8A) = ro. In particular, bo E K. Observe that q (Ao) belongs to the segment (bo, zo). For 0 < µ < 1 define

zo + 2(I + A)N(ao - zo),

1(A)

A E E,

where (1 +,k)," := exp(µ Log(1 +,k)) and Log denotes the principal branch of the logarithm. Observe that *(0) = bo and that *(-tN) = cp(ho), where C2IIV(Ao) - zoll l M rN :

JJ

Ilao - toll

1

Moreover, p (0, -t,) < iN log diam G - i,. log II So (Ao) - zo II . Consequently, if p(E) C G, then cc(ao, (P 010)) < cc(ao, bo) + cc(bo, (P 010)) < M2 + kc(bo, (P (Ao))

< M2 + P(0, -tN) < M3(µ) -

2I logdist(rp(Ao), aG). µ

It remains to observe that if

µ := min{ 1 1092 3, 2

?Jr arctan

l

-I

IdiamG 2ro

then *1(E) C A(ao, ro, zo) C G for arbitrary Zo E aG. 8.10. Let G C C" be a convex domain. Define

aEG,XEC", LIG(E) := sup{OG(a;X): a E G, dist(a, aG) < s, IIXII = 1),

e > 0.

Prove that

XG(a;X)>

20G(diIst(a,

aG)) '

aEG,X EC".

In particular, if there exist A > 0 and 0 < a < I such that

0(e) < As", then G satisfies (2).

E > 0,

(5)

Exercises

293

flint. Use Lemma 8.3.6 or the following argument (cf. (Din-Thn]). Fix a tz G

and X E C" with II X II = 1. Let ,p : E -> G be any holomorphic mapping such that tp(0) = a and t,p (0) = X for some t > 0. Put t/r(A) := 7T fo

(I +cos0)cp(e'AA) d9. A E E. Then i,r(A) = a + '-,X. Since n(1 +

cos9) dO is a probability measure on [0, 27r], the function t1' maps E into G.

8.11. Prove that if G C C" is a convex domain satisfying condition (4), then condition (5) is fulfilled with a = 1/2. 8.12 (cf. [Lem 1,3], [Mer]. [Roy-Won]). Let G C C" be a bounded convex domain satisfying (5). Prove that for any complex geodesic cp : E -+ G there exists µ E (0, 1] such that (p extends to an (aµ)-Holder-continuous functions on E. In particular,

- if G is strongly convex, then one can take a _ 1/2, t = 1, - if G is balanced, then one can take A = 1. Hint. Use Exercises 8.5 - 8.11.

8.13. Let E := E(p) be a complex convex ellipsoid; cf. §8.4. Prove that

A (E) a A(p)ea(P),

e > 0,

(6)

where A(p) > 0 is a constant depending only on p and

I min(pi...., p,,, I }

a(p) := 2 max(p1., p"}

(See Exercise 8.10 for the definition of the function D£.) In particular, in view of Exercise 8.10,

xe(z;X) > C(P)

IlXII

(dist(z, ae))a(P)'

ZEE, X E C".

Hint. The proof will be based on the ideas taken from [Glo]. To get (6) it suffices

to show that for any z E E, X E C" with II X II = 1, 0 < r < 1 such that z +rE X C E, the following inequality holds: r2''T

po .= min(pi,... , p,,, 1), po := max(pl,.... E

C

M(p)(1 - u(z))"o, where

u(Z) _ _i Izil'P',

_

VIII. Complex geodesics. Lempert's theorem

294

Observe that this implication is a consequence of the following relation:

Vs> 2:3B(s)>0:V8E(0,11, V(t,t)EExC: tf

17) < 4S, then

In12s
mint lnl2s, InI2) < B(s)S

ifs> 1 if ; < s < 1

'

(7)

where

<43. To prove (7),fxs 1/2,0<S<1,i;'EE:, andrl#0with Then 4S > ' 17) > Inl2'As(I4/17I), where A3(r) := (r2 + r"12- + 1)s + (r2 2r2:, r > 0. Consider two cases: (a) s > 1: then r-/2- + 1)s lim ,. A.,(r) > 0; (b) 1/2 < s < 1: then 48 >

-

-

I

1? 12$,

11712 1.

8.14 (cf. [Gen 2]). Let G = Gq be a convex balanced domain (q is a seminorm).

Suppose that a E G with q(a) > 0 is such that the point b := a/q(a) is not a complex extreme point for G (cf. Proposition 8.3.5(b)). Prove that there exist complex cG-geodesics (resp. yG-geodesics) for (0, a) that cannot be continuously extended to E.

Hint. If b+E-i C G for some E (C")k, then take V(A) := A(b+f (A)i; ), k E E. where f is a suitable function from O(E, E).

8.15. Let G C C" be a balanced pseudoconvex domain such that G is strongly pseudoconvex at a point b E 8G. Then G is not biholomorphic to E". Hint. Use Corollary 3.5.7(a) and the remark before the proof of Proposition 8.3.5.

8.16. Let E(p), E(q) be convex complex ellipsoids, p = (pt. ... , p"), p, ... , p

> 1/2, q=(ql,...,q"), q1,---,q">1/2.Let (p=((p,...,c p): E->E(p)bea complex geodesic. Write (p, = B j - *j, where Bj is the Blaschke product for cpj (we put Bj :- 1 if 0 0 c j (E) or (pj - 0), j = 1, ... , n; cf. Step 3° in the proof Prove of Proposition 8.4.2. Put ipj := Bj3(rp;lgj, j = 1, ..., n, that gyp: E -* E(q) is a complex geodesic. Hint. Use Proposition 8.4.2.

8.17. Let E(p) be a convex complex ellipsoid. For b E E(p) n (C.)" denote by ZE(p) (b) the set of all X E (C"). such that the pair (b, X) admits a complex

295

Exercises

Ye(n)-geodesic V = (pt, ... , (p,,): E + E(p) with 0 f tpi (E). j = 1.... n. Prove that

Z8, (b) = {X = (Xi,..., X.) E (C").: IIXkb - bkXII? IXkI, k =

l,...,n},

b= (b1,...,b") EBnn(C.)". Hint. If gyp: E -+ IB,, is a complex geodesic, then v(E) = B" n L. where L is a one-dimensional affine subspace of C".

8.18. Let E(p) be a convex complex ellipsoid, p = (pl...., p"), Pi, ... , P, > 1/2. Fix b = (bl,... , b") E E(p) n (C.)" and put b* := bc' (the power is ar(bi , ... , b;). Note that b* E B,,. For bitrarily fixed), j = 1_., n. Let b*

X = (X1....,X,,) E C" write X .= pi(bj/b1)X1 j = 1__n, X* _ (X*,,..., X ). Prove that (cf. Exercise 8.17):

;

l,...,Xn)E(C")*:

E

(a)

"

EIpkbk,-lXkbp'

-brpJbJj-lXf12

>

PiIbk12p,-21XkI2, k

= 1,...,n).

j=1

(b)

Ye(p)(b;X) = Ye.(b*>X*) pilb.12f''_2bjXj

1Lsl=1

pj2lb,Izp'-2IX;I2 lbil2p'

+

(I

-

Hint. Use Exercises 8.16 and 8.17.

I

Ei=1

12

lbil2r')"

t X E Sf(n)(b)

Chapter IX

Product-property

The product-property we are going to discuss in this chapter may be intuitively formulated as follows. Suppose we are given a contractible family of functions d (resp. a contractible family of pseudometrics S). We are interested in characterizing those domains G1, G2 for which dG, XG, (resp. SG, .G,) is equal to the maximum of the corresponding functions (resp. pseudometrics) on G, and G2. More precisely, let d = (dG )GEQSO be a contractible (or an m-contractible) family of functions. We say that d has the product-property on G 1 x G2 at (z , 4), (z , zZ )

(GjE00,Z,zjEGj,J=1,2)if ,

dG,XG,((zi z2)

(zI

,

z2)) = max(dG,(z,

,

dG2(z2,

z2I,)}.

(9.1)

If (9.1) is satisfied for all zj, z, E G j, j = 1, 2, then we say that d has the product-property on G 1 x G2. If d has the product-property on G, x G2 for all G, , G2 E 00, then we simply say that d has the product-property. We already know that k, k*, k, and k* have the product-property (cf. Proposition

3.7.1 and Theorem 3.7.2.) and that c(') has the product property on G x E for arbitrary G (Example 2.5.8(2°)). Notice that the product-properties for d and d* (cf. Remark 4.1.3) are equivalent, and therefore to verify the product-property we can always pass from d to d` or vice-versa. In the case where S = (SG)GEma is a contractible family of pseudometrics, the

product-property of S on G, x G2 at (z°, z2), (X°, X2) (G; E 6o, Gj C C"", z° E Gj, X° E C"', j = 1, 2) has the following form Sc, XG2 ((z° zi); (X?, Xz)) = max(8G, (zO; X°), SG, (Z02; X2)}.

(9.2)

Similarly as above we introduce the notion of the product-property of j on G 1 x G2, etc. Recall that x and x have the product-property; cf. Proposition 3.7.1.

297

IX. Product-property

Note that the product-properties may also be studied for more general objects, e.g. for Dd; see Proposition 9.3(b). It is clear that the product-properties are very useful in all cases where we want to determine dG, xc2 (or Sc, )Ic2) given formulas for dc, and dc2 (resp. for Sc, and Sc2) separately. This problem frequently appears when we construct examples. For some special families d or S (like m(k) or y(k)) one can also study more general "product-properties"; see Exercise 9.2. We would like to point out right now that, in general, for given G, and G2 there exists a large set of (z',, z2), (zi , zZ) (resp. (z°, z2), (X°, XZ)) at which the productproperty is fulfilled for any It (resp. $); see Remark 9.1(b)(c). Nevertheless, the fact that for given G,, G2, and d (or 8) the product-property holds ford (or S) on the whole G, x G2 is rather very exceptional; see Examples 9.2 and 9.9.

Remark 9.1. (a) Since the natural projections G, x G2

Gj, j = 1, 2, are

contractions, the inequalities ">" in (9.1) and (9.2) are always true. (b) Suppose that for fixed (z'I, z2), (Z'1', z2) E G, x G2 there exists a holomorphic

mapping F: G, -+ G2 with F(z',) = z2 and F(z',) = z2. Then, using the map G, a z, -+ (z,, F(zi)) E G, x G2, we easily conclude that in (9.1) the inequality "<" is also satisfied. Consequently, in this case the product-property holds for any d at (z', z2), (Z'1', z2) E G, x G2. It is clear that such a mapping F exists if z2 = z2 . In particular, (9.1) is fulfilled

if z', =z"'orz2=z?. If G2 is taut, then one can easily construct such a mapping F for all points for cf. Exercise 5.2. For let f E O(G 1, E), (p E O(E, G2) be such that f (z',) = 0, f (zi) = v, := (z',, z') and (p(0) = z2, p(a2) = z2, where o2 := ka2 (z2, z2") (cf. Proposition 3.2.4). Then we can take which we have kc2 (z2, z2) -< cc, (zi ,

F := w(a; f) (c) The discussion in (b) can be easily repeated in the case of pseudometrics. If

F: G, -+ G2 is holomorphic and F(z01) = zi, F'(z°)X° = X2, then (9.2) is fulfilled for any S. In particular, the product-property is satisfied if X° = 0 or XZ = 0. If G2 is taut, then such an F exists in all cases where xc2(z2;XZ) < Yc, (z°; X°).

(d) If dc, XG, is a pseudodistance, then (a) implies max(dc, (zi z'), ), dc2(z2, Z2))


dG,(zl+zl)+dG2(z2,z2)

Similarly, if Sc,xc2((z°, z°); ) is a seminorm, then in view of (a) we get max(SG, (z°; X°), Sc2(zi; Xi)} Sc. xc2((z0. z°);

(X°, Xi))

Sc, (z?; X°) + Sc2 (z2°; X0).

IX. Product-property

298

(e) If G; is biholomorphic to a convex domain, j = 1, 2, then in view of the Lempert Theorem the product-property is fulfilled on the whole G, x G2 for any d and S.

Example 9.2. If cGo 0 kGo (e.g. G° := P = the annulus), then there exists a contractible family of functions d such that d does not have the product-property on G° x E; cf. Remark 9.1(e). We will now prove a slightly more general result. Let d(o) _ (dG))GEoo ° and P) _ be contractible families of functions. Put d(`) G

:= I - t d(°) ) G + td(l) G (

P) := (d(')) G

0 < t < I.

dot It is clear that d() is also a contractible family. Suppose that d(o Go G for some G° E 0o. Then di'i does not have the product-property on Go x E for any

0
For let z . zo' E Go be such that M° := d j(zo, zo') dca (zo, za') _: M,. Assume that Mo < M, . Fix 0 < t < 1 and let Or(t) E (0, 1) be such that p(0,Or (t)) E (Mo. M,) and

t<

Mo

M, - M°

Then using Remark 9.1(a) we get dcoXE((zo 0), (zo a(t ))) > p(O, a(t )) = max{dGo (z0, zo), de`)(0, a(t))}.

Notice that similar examples can be easily constructed (using the same idea) on the level of m-contractible families of functions, (m-) contractible families of pseudodistances, and contractible families of pseudometrics; the details are left to the reader. Note that one could start the above construction with the annulus

Go = P and d(l),

d(o), dM E {M(k), g, k`},

d(O)

S(°), S(') E {y(k), A, x},

j(o) 0 S(').

Now, we would like to select basic operations under which the product-properties are invariant.

Proposition 9.3. Let d = (dG)GEeo be a contractible family of pseudodistances such that d has the product-property on G, x G2 (G,, G2 E so). Then (cf. § 4.3) (a) d' has the product-property on G, x G2;

IX. Product-property

299

(b) Dd has the product-property on G1 x G2. Let S = (SG)GEG0 be a contractible family of pseudometrics such that S has the product-property on GI x G2. Then (c) 8 has the product-property on G 1 x G2; (d) f S has the product-property on G 1 x G2 provided that SG, and 3G. are upper semicontinuous.

In particular,

[the product-property for x] - [the product-property for x and k], [the product-property for c]

[the product-property for c' and y].

Proof (a) Let us first make a remark on the Ldp;-length. Fix G E 00 and let a: [0, 1] -+ G be an arbitrary continuous curve with I := Lda(a) < +oo (cf. §4.3). Put Oo(u) := LdG(al (o.Ul),

0 < u < 1.

Then *o: [0, 1] -> (0, 1] is a continuous and increasing function (which, in general,

need not be strictly increasing). Define *,,(u) := ro(u) + cu, 0 < u < 1, E > 0. [0, 1] --o- [0,1 + E] is a strictly increasing bijection. Define It is clear that

(l+s))),00.Then < (1 + e)(t" - 1'),

LdG(aSl

0 < t' < t" < I.

Now, we go to the main proof of (a). Fix i j, z' E Gj, j = 1.2, and e > 0. Let aj : [0, 1 ] -+ G j be a II II-rectifiable curve such that a(O) = zj, a(l) = z", and

Lj,I(aj)-d, (zj,zj) <e, j = 1,2. In view of the above remark, we can assume that Ld,..I (ay I[".r")) <

(I!

+ e)(t" - t').

0 < t' < 1" < 1,

where l j := LdG, (a j), j = 1, 2. Put &(t) := (al (t), a2(t)), 0 < t < I. Obviously, < IN = 1 we a : [0, 11 - G, x G2 is 11-rectifiable. Moreover, for 0 = to < II

ix. Product-property

300

have: N

1: dc,xc2(a(ti-I), a(ti)) i=j N

E max{dc, (al (t1-1), al (ti)), dc2(a2(ti--l ), a2(tj))) i=1

N

E max{(1i + --)(ti - ti-,), (12 + s)(ti - ti_01 i=1 5 max{11,12)+6

< max{dc, (z;,

dG2(z', z2)1 + 2e.

Consequently,

d'c, xc2 ((z', z'2) ( z", z2))

- Ld

G,xG2

(i) < max{dc (zi, zi'), dc2(zz,

Letting s -+ 0 we get the required result. (b) Use the definition of Dd. (c) Use standard functional analysis argument. (d) Use the same method as in the proof of Theorem 3.7.2.

z2))+2e.

0

Remark 9.4. By Lemma 4.2.3 and Proposition 4.2.11(b) we get: (a) if g has the product-property on Gi X G2, then so has A; (b) if m(k) has the product-property on Gl X G2, then so has y(k). Now, after all the above general remarks, we would like to discuss more concrete situations. Namely, we are going to investigate the product-property for m(k) (in particular, for c(*)) and for g. We will prove the following three results: - c(*) has the product-property (cf. Theorem 9.5); - g has the product-property in the class of domains of holomorphy (cf. Theorem 9.8); - for each k > 2 the family m(k) does not have the product-property even for very elementary domains (cf. Example 9.9).

Theorem 9.5. The pseudodistance c(*) has the product-property on G 1 x G2 for arbitrary domains G 1, G2. In particular, c' and y have the product-property.

Proof. We will show that the product-property for c is a consequence of the product-property for k !

IX. Product-property

301

First, recall that the product-property holds on convex domains; cf. Remark 9.1(e). In particular, we have the following result.

(*) Let N E N and let W: CN X CN

C be given by the formula %P(t, n) =

t,..., N),>)=()7i,...,r1N)ECN.Let W,VCC"be absolutely convex bounded domains. Then

I'I'(,11)1

U), c'y(rjo, n)}

IIWIIwxv

for all (go, r!o). (t, n) E W x V with

rlo) = 0.

Remark. Condition (*) is a consequence of the Lempert Theorem (and of the product-property for k). 0 We do not know how to prove (*) directly. Note that the domains W and V are very regular; they simply are balls in suitable norms in CN. Moreover, the function P is also very elementary.

Now, we will prove the product-property by means of property (*).

Fix arbitrary domains G j C C"/, j = 1, 2, and let z', z. E G j, j = 1, 2. We have to prove that for any F E 0(G1 x G2, E) with F(z',, z2) = 0 the following inequality is true:

IF(zi,zi)I < max{cG,(z,,z,),cG'(z2,z2 )}. 11

Let us fix such an F. Let (G j.,), , be a sequence of relatively compact subdomains of G j such that zj, Z j" E G j, / G j, j = 1, 2. In view of Proposition 2.5.1(a), it suffices to show that

IF(z,,z2)I <max{cG,

v> 1.

(9.3)

Fix v=voENandletDj:=Gj,,v, j=1.2. It is well known (cf. Theorem 6.1.13 and Appendix HF 14) that the function F may be approximated locally uniformly in G, x G2 by functions of the form Ns

(zi, Z2)

L.a(zi)gs.u(z2),

(z1. z2) E G, x G2,

(9.4)

where f,,µ E 0(G, ), g,,,2 E 0(G2), s > 1, it = 1, ... , Ns. Observe that the F.,(zj', z2), s > 1, also have the form (9.4) and that F,=:4F. functions k Hence, without loss of generality, we may assume that F, (z',, z2) = 0. s > I. Let m3 := max{1, IIF.IIn,xD,) and FS := F,/m s > 1. Note that m, - 1, and therefore F5=F. Consequently, we may assume that F,(D, x D2) C E, s > 1.

IX. Product-property

302

To verify (9.3) (with v = vo,). it is enough to prove that

IFr(z,,z2)I < Fix s = se E N and let N

z,).

, c.n,(z2,

2 )).

s>1.

fr, := ff,, r gr, := g.v,, r µ = 1,

,

N. Set

f:=(f,... .fN):G,->CNandg:=(91,.. ,gN):G2-,CN.Put K := { = RI-- W W E CN: IE,,I < IIfpIIn,, u = I... . N. g(z2))I < I, Z2 E D2).

It is clear that K is an absolutely convex compact subset of CI with f (D,) C K. Let

L:={n=(r7,,...,gN)ECN:1171,1
Then again L is an absolutely convex compact subset of CN, and moreover. g(D2) C L. Let (W,,)"'=, (resp. (V )" ) be a sequence of absolutely convex bounded domains in CN such that CC W" and W. \ K (resp. V and V. \ L). Set M. := Il W II w. x v a E N. By (*) and by the contractibility of the mappings f : D, --> W", g : D2 --> V we have IF,(zi.z2)I = 141 (f (z', ), f (z1)), c'v, ((g (z'2). g(z2 ))) M max (cn, (: i , .. ), cL, (Z2, z2) ).

Letting a - +oo we get the required result.

0

We will apply Theorem 9.5 to present another proof of Theorem 2.6.1 (cf. [Pet]). Observe that K. Peters worked with the Kobayashi-Buseman metric instead of the Caratheodory-Reiffen metric.

Theorem 9.6. Let Gj, G , C C"' be bounded domains. j = 1, 2. Let rp = (VI, SP2) : G, x G2 --)' G, x d2 be a biholomorphic mapping. Assume that there is an open non-etnptyy subset U of G, x G2 such that for every (z. W) E U we have (p, (z,

w)(C,,, x (0)) = C"

Then (p is of the form rp(z, w) = (g, (z), g2(w)). Z E G,, w E G2, where gi : Gi G1, j = 1, 2, are biholomorphic mappings.

IX. Product-property

303

-

I.1. Theorem [1 immediately leaclc to u In the case when i.;` _ lJJ for if new proof of Theorem 2.6.1. The proof of Theorem 9.6 is based on the following general lemma from linear algebra.

Lemma 9.7. Let V, V and W, W be n- and m- dimensional normed real vector spaces, respectively. Assume that V x W and V x W are endowed with the corresponding maximum norms. If (p = (v I, (P2) : V x W isometrv with Cpl (V x 10}) = V, then we have

V x W is a linear surjective

(P2(V x {0}) = (0}, (P2({0} X W) = W, 0({0) X W) = {0} and

IIV2(0, w)II = IIwII, II(PI(v.0)11 = IIvll, V E V, w E W.

In particular, f : V V given by f (v) := (pl (v, 0), v E V, and g : W -+ W given by g(w) = (p2(0, w), w E W, are linear surjective isometries satisfying w(v, w) _ (f (v), g(w)) on V x W. We postpone the proof of Lemma 9.7 and continue the proof of Theorem 9.6.

Proof of Theorem 9.6. Without loss of generality, we may assume that there are

points a E G 1, b E G2, and a number r > 0 such that U = B(a, r) x B(b, r). Then fixing (z, w) in U we see that (P' (Z' to) : C"' X C", - C"' X Cm

is a C-linear isomorphism. Now, we endow the "left" C"', C"', and C"' X C"z with the norms yG, (z; ), yG, (w; ), and yG, X G2 ((z, w); ), respectively, and the "right" C"', C"=, and C"i x C"2 with the norms yd, ((PI (z, w); ), yd,((P,(z, w); )and respectively. Then (p'(z.w) becomes a linear surjective isometry. By virtue of Theorem 9.5, Lemma 9.7 applies and we obtain

9' (z, w)(C"' x 10}) = (0} and (pi(z, w)((0} x C"2) = (0). Thus we have aV2

ay

(z,w)=0 forI<j
J

aw,(z, w) =0 for 1
IX. Product-property

304

Since (z, w) E U was arbitrary, it follows on U that V2 (z, w) = V2(a, w) and rp, (z, w) = (pl (z, b).

The identity theorem then implies that for every (z, w) E G1 x G2

Q2(z, w) _ q' o i0(w), where ia(w) := (a, w), vi (z, w) = pt o jb(Z), where jb(Z) := (z, b). Finally, it is clear that V2 o iQ : G2

G2 and W l o jb : G i -+ G I are biholomorphic

0

mappings which completes the proof.

Proof of Lemma 9.7. First, we introduce the following temporary notation for a normed real vector space (T, II II):

BT:=(xET:IIxII <1),

BT:=(xET:IIxIII), ST:={xET:Ilxlll). Moreover, if L is an arbitrary linear subspace of T we write

NT(L):={x EST:IIx - yll> 1 for allyEL). Then the following inclusions are easy to obtain:

B;, x Nµ,(92(V x (0))) C Np.*((p(V x (0))) C (B, x NC,(9p2(V x (0)))) U (N (0 1 (V x (0))) x Bg,) U (SV x Sf.). Since cpi (V x (0)) = V, it is clear that N f, ((pj (V x (0))) = res. Let us assume that there is a point (v, w) in Ny,tw((p(V x {0})) that does not belong to the closed set Bp x NW ((p2 (V x (0})). Then we find an open subset U of

Nj,((p(V x {0})) with (n, w) E U and U C Sp x S. By hypothesis, 0 is a linear isometry from V x W onto V x W. Therefore, U := cp-1(U) is an open subset of Nvxw(V x (0))) = By x Sw. Thus, V I u is a topological map from U onto U, although the dimensions of Sp x Sw and 13v x Sw are different; a contradiction. We have therefore obtained x NW(

(V x (0))) = NVXW((p(V x {0}))

= co(Nv,,w(V x (0))) = v(Bv x Sw).

IX. Product-property

305

Defineg: W-> Wbyg(uy):=tp2(0.uw)foru'EW.Then g(Sw) C tp,(Bt, x Stiv) = N,(ipi(V x (0))) C S11,.

(9.5)

i.e. g is a linear map preserving the norms, and therefore g is injective. Because of dim W = dim W it follows that g is a surjective linear isometry from W onto W. So, from (9.5) we deduce that N11.((p,(V x (0))) = S. and so tp2(V x {0))) = (0). On the other hand. the fact that tp2(10) x W) = g(W) = W implies, with the same

argument as before, that f : V - V defined by f(v) := tp,(v,0) is a surjective

0

linear isometry and p, ({0} x W) = (0). Now, we go to the product-property for the complex Green function.

Theorem 9.8. If G,. G, are domains of holomorphy, then g has the productproperty on G, x G,. Consequent!'. A has the prodrtc t-property in the class of all domains of holomorphy (cf. Remark 9.4(a)).

In particular, if G,..... G" C C1. then

z ).(

...gc (=n.;")}.

z;. z E Gi. j = I..... it.

A(;,x..xu ((:,.....zn):(X,..... Xn)) = max{AG,(:,;X,)..... AG,,(z,,:Xn))

:;EG,.Xj EC.j=1.....n.

Remark.

.'

trary domains

We do not know whether g (or A) has the product-property on arbi.' : cf. Example 4.2.8.

Proof. The proof will be based on the methods of the Monge-Ampere operator (cf. Appendix MA). Recall that any domain of holomorphy may be exhausted by an increasing sequence of hounded hyperconvex domains. Hence, in view of Proposition 4.2.7(a). it suffices to prove the product-property on products G :_

G, x G, of bounded hyperconvex domains G, C C",, j = I.2. Fix a = (a,. a2) and b = (h,, h,) E G. Let it := n, + it,. We want to show that 9G (a. h) = max{g(;, (a,. b, ). g(;: (a,, b2)).

IX. Product-property

306

Recall that the inequality ">" is always true (cf. Remark 9.1(a)). By Remark 9,1(b)

we can assume that aj 0 bj, j = 1, 2. Fix s > 0 and define

u_(z) :_ (I + e) loggG(a, z), / Z2)), u+(z) := logmax(gG1(al, zt), 8G2(a2,

Z = (Zl, Z2) E G.

It suffices to show that u_(b) < u+(b) (then, letting a - 0 we get the required inequality). We are going to apply the domination principle of E. Bedford and B. A. Taylor (cf. Appendix MA 1). For r > 0 define G,. := G \ B(a, r). We will prove that for sufficiently small r the functions u_ and u+ satisfy all the necessary assumptions of this principle on Gr. It is clear that u_, u+ E PSH(G,) f1LO0(Gr); cf. the proof of Proposition 4.2.7(e). Moreover,.by Proposition 4.2.7(d), for any

EBGweobtain lim u_(z) = Flimb-U+ (z) = 0.

Z-Oz

u+(z) - u_(z} > log(max{

11 z, -al11

11z2-a211

diamG1 ' diamG2

}) - (1 + s) log

11z-all R

>logM - slogllz -all R

where M > 0 is a constant independent of z. Consequently, there exists ro = ro(s) E (0, R) such that for any 0 < r < ro and for any E aB(a, r) we have lim

f (u+(z) - u-(z)) > 0.

Thus, if r is sufficiently small, then

lim lnf (u+(z) - u_(z)) >- 0, E dGr. By Proposition 4.2.7(e), (dd`u_)" = 0 in G\(a). We will prove that (ddcu+)" = 0 in G \ (a). We need the following result. Proposition (cf. [Zer]). Let ft, be an open subset of C") and let a1 E PSH(SZ j) l L°O(SI,loc) be such that (dd''u j)"i = 0 on l,, j = 1, 2. Define u(Z1, Z2) := max{ul (zi), u2(z2)},

Then (ddcu)"'+"2 = 0 on Q.

(Zl, Z2) E n := SZl X Sl2.

IX. Product-property

307

In view of the above proposition (and Proposition 4.2.7(e)), we have

(ddru+)" = 0 in (G, \ (a11) x (G, \ (a21). It remains to show that (ddnu+)" = 0 in an open neighborhood of the set

((G1 \ jai)) x {a,)) U ((at) x (G2 \ {a2})).

Fix, for instance, a point :0 = (a,. zo) with :° 0 a,. Then a+(;.) = loggG,(a,.:2) for z = (:,.:2) in an open neighborhood of and therefore (by Proposition 4.2.7(e)) (cld` u+ )" = 0 in a neighborhood of

Example 9.9. For any k > 2 the family y«I does not have the product-property

on P x E. where P = P(R) is an annulus with sufficiently big radius R > 1. Consequently, for any k > 2 the family m(A' does not have the product-property on P x E (cf. Remark 9.4(b)). For we fix k > 2 and we will prove that there exists R(k) > I such that for any R > R(k) we have (k)

YPxe((u. 0):

(A) (A) (1. Y)) > max{yP (a: I ). yy (0;Y)(9.6)

where P := P(R). a = a(R.k) := RIM. Y = Y(R.k) := yP )(a;1). Obviously. the right hand side of (9.6) is equal to Y. By Proposition 5.5(b) we have

Yi (a: 1) = [Ra

f

(a.

a)

-a)]

n(aa.

_: *+(R. k).

ForR>2define h(,%.fl := ailri(A)

+aAhL(1).

where

a,

_

_

2

2 + R2%(A+1)'

aA

R21(4+ 1)

2 + R2/(A+1)'

!t, (k) := Ri- f (a.A) f C R . -A, I hA(X):= RA[f(a,A)lAf

CR.-n).

.

XE P.

IX. Product-property

308

Observe that ord(Q.o) h > k and a, Ih, I + ak Ihk I= 1 on 8 P. This implies that

YPXe((a,0);(I, Y)) > (a1 h'I Jr(a,a) a

(a)IYk-I

+ «kI(hk)(k)(a)I)Ilk

f (.-a) Yk_I +Uk I 2

(Yr(a,a))kf (R, a

2

*-(R.k). Then direct calculations show that lim

which gives (9.6).

*-(R,k) *+ (R, k)

_

A_1 (2-'+ k

2

Ilk

> 1.

-a

]Ilk

Exercises

309

Exercises

9.1. Let GA := ((zI.z2) E C2: Iz(:;--I < 11, k E N. k > 3. Using Proposition 2.7.5. prove that tn(k) and y(A) do not have the product-property on GA x E.

9.2 (Generalized product-properties for m(A) and y(A)). We say that m(l) has the generalized product-property on G( x G2 if for any : E Gj. j = I, 2, (A)

,

(z(.Z2n))

= max{[(mg,(zl, zi))t(InG,

l =0.....k}, (I)

where m(0) :- I. Similarly, we say that y(k) has the generalized product-property on G ( x G, if. whenever Z j E Gi C O"' 3 X1, j = 1.2, then

(Xi. X2))

YG

= max([(Y6,)(z(;X1))'(Yc, where y(o)

t)(z2;X,))A-t1(fk: / =0.....k}.

(2)

1.

(a) Prove that (2) is a consequence of (1).

(b) Prove that in (I) and (2) the inequalities ">" are always satisfied. (c) Observe that for k = I.2 the generalized product-properties coincide with the standard ones. (d) Note that the standard product-property implies the generalized one.

(e) Let Gt C O"- be an ni-circled domain with 0 E G;, j = 1, 2. Assume that G( and G2 satisfy the cone condition. Prove (using Proposition 2.7.5) that the generalized product-properties are fulfilled for m('() (resp. for y(A)) if (z;, z;) _

(0.0) (resp. if (z,, z2) = (0.0)). (f) Prove that in the situation described in Example 9.9 the generalized product property does not hold. Hint to (/). If R >> I. then max([(y()(a; I))tYA-tJ(1A: 1 = 0, .... k) = Y. (g)

Try to find the correct form of the product-property for m(A) and y(A).

.'

Chapter X

Comparison on strongly pseudoconvex domains

From Chapter VIII we know that the equalities yo = xG and co - kg hold for any convex domain G. Moreover, on the unit ball R. the Bergman metric and distance coincide (up to a constant) with ya and cg, respectively. Since it is well known that strongly pseudoconvex domains share a lot of properties with B, one may expect that all the above objects behave very similarly on strongly pseudoconvex domains, at least near the boundary. The discussion of this problem is exactly the content of this chapter.

10.1 Strongly pseudoconvex domains Let G be a bounded domain in C. We recall that G is strongly pseudoconvex if there exist a neighborhood U of 8G and a C22-function r : U -+ IR satisfying

(i)

(ii) (iii)

(iv')

G ft U = (z E U: r(z) < 0}, (C" \ G) fl U = (z E U : r(z) > 0), dr(z) # 0 for every z E U, (,Cr)(z;X) > 0 for all z E 8G, X E (C"), n

with j=I

8r 8zj

(10.1.1a)

(10.1.Ib)

(I0.1.Ic)

(z)Xj = 0.

Under these assumptions it is well known that then the signed boundary distance gives a new C2-defining function for aG; cf. [Gor], [Kra 1), [Kra-Par]. Therefore, one can choose new U and r such that in addition to (i), (ii), (iii) the following

10.1 Strongly pscudoconvex domains

311

conditions are also satisfied (iv)

(Gr)(z;X) > 0 for all

(v)

II

(10.1.1d)

gradr(z)II = 1 . Z E aG , where

gr ad r ( z )

(vi)

z E U, X E (C"),t;

_

(( ..... ar

z)

ar .

(z)

(10.

;

1

.

1e)

for any z E G n U there exists exactly one point

rr(z) E aG such that dist(z. aG) = Ilz - n(z)II.

(10.1.1f)

We say that a strongly pseudoconvex domain is given by a pair (U. r) if (i) - (vi) are fulfilled. Most of the theorems in this chapter are based on the existence of good solutions of a-equations. For the convenience of the reader we formulate the result we need in such a form that is used in the sequel. Of course, the proof of that deep result

is beyond the scope of this book and so it is omitted. We refer the reader, for example, to the books [Hen-Lei] or (Ran 2], where other references may also be found.

Theorem 10.1.1. Any strongly pseudoconvex domain G C C" admits a positive constant C such that if a = =i ajdzj is a a-closed (0, 1) form of class C'° on G with IIa II G := E _ j 1101i II G < oo, then there exists a C' -function f on G with and Ilflfc CIfahIG

of = a

We know that any strongly pseudoconvex domain is locally biholomorphic to a strictly convex one; cf. [Ran 21. Thus it is clear that for any boundary point E aG one can find a local peak function f. i.e. f E 0(V n G) for V = V(%') sufficiently small with f 1 and If (z) I < 1 whenever z E G n v \ In fact, even more is true as the following result shows.

Theorem 10.1.2. Let G be a strongly pseudoconvex domain in C". Then for any sufficiently small positive ni we can find: 0 < r72 < 17 1. constants d,, d2, and a domain G D G such that there exist functions E CM(G), i; E aG, that satisfy the following inequalities:

(i)

I

and Ih(z;l;)I < 1. Z E G \ (.}.

(ii)

11 -

(iii)

lh(z;2;)I -< d2 < 1,

di Ilz - 2;11,

z Ed n

(10.1.2a) rl2),

(10.1.2b)

z E G. 11 z -'11 > nf (10.1.2c)

X. Comparison on strongly pseudoconvex domains

312

Proof. Because of the strong pseudoconvexity of G we can find a neighborhood U = U(aG) and a C2-function r on U such that the properties (10.1.1) are fulfilled. For a point i; E aG put z

s,j=I

i=1

is called the Levi polynomial of r at . We choose a sufficiently small s, > 0 such that U' := Si) CC U. Then the assumptions on r yield (Cr) (z; X) > C, II X 112, z E U', X E C", where C, < I is a suitable positive constant. With C2 := C1/2 the Taylor formula then leads to the following inequality

r(z) = r(t;) - 2Re P(z;C) +

>-2

z - ) + o(IIz - II2) if

where s2 is independent of . Thus on

llz - III < s2<si, EaG,

E2) we have

2Re P(z;i;) >- C211z - Ilz - r(z).

(10.1.3)

Now fix any positive number qi < E2 and choose a CO°-function X: R --> [0, 1]

with ,(t)= 1 fort q,. Putting we obtain a C°°-function on C" x C". Let (P(z;i;)

(1 -

:=

E aG, Z E C".

If 11z - II < q, /2, then Q (z; ) = P (z; ). In particular, q(.;) is holomorphic on

the ball B (i; , q, /2). Moreover, (10.1.3) and C2 < 1 yield the following inequality

2ReV (z;)

if

qi/2 and r(z) < C2rli/8. (10.1.4)

We choose
taken independent of .

Summarizing, V(;) is a C'-function on C" which does not vanish on G \ B(i;, q,/2) and which is holomorphic on B(%, q,/2). Therefore, 8(1/(p(;)) defines a 8-closed form a(;C) = Ei= a;(;)dz; of class C°O on G. Because of

a;(z;) =

J0

if zEGnB(i;,q,/2)

1 - az" (z;)/('2(z;)

if z E G \ B(, qi/2)

10.1 Strongly pseudoconvex domains

313

c) II d < C3, where the constant C3 is independent

and (10.1.4), it follows that Ila j

of . Hence by Theorem 10.1.1 there exist C°°-functions v(.; ) on d with 6v(.; C) II c < C4, where C4 does not depend on . where {z E G: Put on d \ C4 0). Then belongs to 0(G \ Z(t )) and because of (10.1.3) and (10.1.4) we have R ef > 0 on (G \ B g, /2)) U (G \ { }) =: G. Therefore, with the function is holomorphic on G and its values on G \ () lie inside the unit disc. If Z E G fl B q i /2), z ¢ Z( ),

a(.;) and II v

then

P(z;0

g(z;0 = 1 -

C4)

remains bounded near

it extends holomorphically through

Z(i;) fl G fl B(i;, q,/2), and therefore

E 0(G). Observe that there is a

But since

positive C5 such that

E aG, Z E (j.

IP(z;0I < C511Z -0,

Thus, if 0 < 112 < min{q,/2, 1/(4C4C5)}, then it follows that

Ig(z;0I< 1 -

C5 2CC5114C511zI 11Z -

ZEG, 11Z-y11<172,

- Il

where C6 := 2C5. Now, we choose C7 in such a way that lea - 11 < C,IAI for I,LI < C6,12. Then we obtain 11 -

C6C7IIz -III =: di Ilz -III,

In particular, this shows that h (Z ; ) = 1. What remains is the upper estimate for Ih G, Ilz-III>-q,.Then we get 4. Take

Re g(z; ) = Ilz -

1121 + Ilz

-

Z E G, IIz -III <'12

)1 away from the boundary point

I12(C4 - Re v(z; i ))

C4)12 II - Ilz gi /(l + (diam G)2 2C4)2 =: C8.

Hence we have Ih (z; ) I < e-c" =: d2 < 1.

0

314

X. Comparison on strongly pseudoconvex domains

Before presenting a result on approximation of bounded holomorphic functions we recall the following stability property of strongly pseudoconvex domains. If K is an arbitrary compact subset of the boundary of a strongly pseudoconvex domain, then there are arbitrarily small strongly pseudoconvex enlargements d with K C G

and aG = ad away from K. Observe that for the case K = aG this property has been just exploited in the proof of Theorem 10.1.2. In the general form this property will be used in the proof of the following result on approximation which has been found by I. Graham (cf. [Gra I]). Theorem 10.1.3. Let G be a strongly pseudoconvex domain in C. Then for sufficiently small R the set G n B(2;, R) is connected whenever E aG and there exists p = p(R) < R such that the following property holds. Given s > 0, there exists a number L = L(e, R) > 0 such that for any E 3G, f E H°O(G n R)), and W E G n p) there is an f E H°°(G) satisfying:

D" f(w) = D" f (w),

(i)

a E (Z+)", lal 5 1,

(ii)

IIIIIG 5 LIIf IIGnB(C.R),

(tu)

Ill - f II GnB(C.p) < E.

Proof. According to Theorem 10.1.2 we choose the data 112 < r1,, d,, d2 < 1, G D G, and E 0(G), E 3G, so that G n B(i;, R) with R := 2111 is connected for all i; E 3G. Introducing i;) + 3)/4, we can also require that

1/2,ZE0,1; E3G. Fix d3 E (d2, 1) and then take 0 < 11 < 112 such that

1,) C G and

E 3G. With p := min(11, 11i/5) there is a finite number of boundary points p). Then we choose N such that aG C UI , strongly pseudoconvex domains G; D G by modifying G near j so that d1,

IIz - III < 11,

G n B(t;, 2p) CC G3 C G and G \

4p) = Gj\ B(i;;, 4p).

Now let i;o E 3G, f E HO'-(G n B(to, R)), and W E G n

p). Then we find

jO with i o E B(i;j0, p). For simplicity we take i; fo = i;i.

We denote by X a cut-off function with 0 5 X 5 1, x - 1 on 6111/5), and X = 0 outside Then put a (aX)f on G n B(iO, 2171) and a := 0 on G \ Observe that a = 0 on G n and on G n (C" \ 9111/5)). Hence, by trivial extension, a can be thought of as a 5-closed (0, 1)-form on G, of class C°°. Instead of solving the equation au = a we will deal with the equations avk = k E N, where the right hand side is again a 5-closed (0, 1)-form of class

315

10.1 Strongly pseudoconvex domains

C00 on G1. Applying Theorem 10.1.1 gives a solution vk E CI(GI) with IIvkIIG, <

C depends only on X and so only on q j. Then the functions and are holomorphic on G and on G1 n B(1o, rl), respectively, with

fk := Xf -

C1C(d2/d3)kllf IIGnR(;o.R)

Now, if z Ed n

p), then IIz - c1 II

IIz - coil +

II

< 2p, i.e.

p) C GnB(i;1,2p) CC G1. Hence there is an L1 < 1, independent of w E G n B(i;o, p), such that the Cauchy inequalities imply that Lc(d)k al>k(w)

aZ> (w)

d311111

Let e < 1. We fix the exponent k so large that

cnB(xo.R)

< E with 8

,C (d3) k

s[2(1 + n(n + diam G))(1 + 11f IIGnB(C(,.R))]-1. So we obtain

max(Ilfk - f IIG,nB(Co.q),

I afk (w)

_ az (w)

1

} < Ellf IIGnB(;o.R)-

J

At the next step we define on G a new holomorphic function f by f (W)

f(z)

lk(Z) +

\

n

- fk(w) +

(8z; (w) f

A (w) I (z' - w')'

/

We note that i(w) = f (w) and .L(w) = 1(w). Moreover, since p < r),

it

follows that Il i - f llGnB({o.p)

IIIIIG

s,

IIfkhIG+IIIIIGnB(xo.R)

With L := max(2 + 2kCj Cd2 : 1 < j < N) the proof is complete.

0

X. Comparison on strongly pseudoconvex domains

316

10.2 The boundary behavior of the Caratheodory and the Kobayashi distances Looking at the explicit formula of the Carathe odory distance for the unit ball one easily sees that c-B. (0, ) behaves like -(log 8B"))/2 near the boundary. It will turn out that the same boundary behavior remains true in any strongly pseudoconvex domain (cf. [Aba 1], [Fad 2], [Vor]).

Theorem 10.2.1. Let K be a compact subset of a strongly pseudoconvex domain G C C". Then there exists a constant C > 0 such that

cG(zo, z) >- -(1/2) logdist(z, 8G) - C,

Zo E K, Z E G.

Proof. Because of C2-smoothness of 8G we can find a positive so such that for any point z E G with dist(z, 8G) < so there is exactly one point i; (z) E 8G satisfying

z = (z) - dist(z, 8G) v(i;'(z))

(10.2.1)

Here v(t; (z)) denotes the unit outer normal to 8G at the point 2; (z). Now, we choose positive real numbers 112 < qi, d1, d2, and the function h according to Theorem 10.1.2 with 2q, < so and q, < dist(K, 8G). Fix Zo E K. Now let z E G with dist(z, 8G) < 113 := min{q2, (1 - d2)/di }. Then, with the aid of the point i; (z) E 8G, we get the following estimate: cc(zo, z)

> Ih(z;4(z)) h (zo;

(z)) I

ll II + Ih(z;(z)) - II

>

11

> I - d2 - Ih(z;C(z)) - I

- 1-d2+Ih(z;l;(z))-lI >

-

1-d2-d, llz-i;(z)II >0.

Using (10.2.1) it follows that cG(zo, z) >

- 21

log

(1 - d2)

d,Ilz-i;(z)II

= 2 log (I d' d2) - 2 logdist(z, 8G).

10.2 The boundary behavior of the Carathdodory and the Kobayashi distances

317

On the other hand. if z E G with dirt(;., aG) ? q3, then cG(;.O,:) ? 0 > (- 1/2) log dist(z. aG) - C,

with a suitable Ci > 0. Analyzing the proof of Theorem 10.2.1 immediately gives the following local version of Theorem 10.2.1.

Theorem 10.2.1'. Let G be as in Theorem 10.2.1 and let e > 0. Then there exist positive numbers p2 < p, < £, and C such that for any E aG the following inequality is true:

CG(z, m) ? -(I/2) logdist(z. aG) - C.

E G n B(Z. p2), w E G \ B(1. pi).

Remark 10.2.2. (a) The proof of Theorem 10.2.1 shows a strong relation between the lower estimate of the Caratheodory distance and the existence of good peak functions.

(b) Moreover, if G is as above and if

are different boundary points of G. then even the following inequality has been claimed in (Fad 21, 1Vor1:

cG(:'.:") > -(1/2) logdist(:', 8G) - (1/2) logdist(:". aG) - C. whenever .7'. Z" E G.:' is near 4', and :" is near i; ". We will not use this result. Later on we will show a similar inequality for the Kobayashi distance which is much easier to obtain. As we have already mentioned, the lower estimate for the Caratheodory distance depends on deep results of complex analysis whereas the following upper estimate relies only on the smoothness of the boundary.

Proposition 10.2.3. For a bounded domain G in C" with smooth C2-boundary and a compact subset K of G there is a constant C such that

k(;(zo. z) < -(1/2) logdist(z. aG) + C.

zo E K.: E G.

Proof. If we choose e,> > 0 sufficiently small. then any point z E G such that dist(z, aG) < e() lies inside the ball 8(z', ei)) C G with := i; (z) i; (z) E )G. and moreover 2 = 4(z) - dist(z. aG)v(i; (z)). Applying the triangle

318

X. Comparison on strongly pseudoconvex domains

inequality we obtain kG(z, zo) < kG(z, z') + kG(z', zo) < kB(t'so)(z, z') + C, I _ -log so+Ilz - z'II +C, 2

I

I

2

2

--logdist(z,aG)+log(2so)+C,,

so - Ilz - z'II

where C, := sup(kG(w, w): w E K and W E G with dist(w, 8G) > so). The remaining case, namely z E G with dist(z, 8G) > so, can be handled as in the proof of Theorem 10.2.1. Combining the previous two theorems yields the following result.

Theorem 10.2.4. (cf. [Aba I]). For a strongly pseudoconvex domain G C C" and

zoEGwe have lim

cG(ZO, Z)

_

kG(zo, z)

zl8G - log dist(z, 8G)

- log dist(z, 8G)

I

2

where the limits are locally uniform in the first variable zo.

Moreover, we get a precise description of the boundary behavior of complex geodesics in strongly pseudoconvex domains (cf. Exercise 8.9). Corollary 10.2.5. Let gyp: E --> G be a complex CG-geodesic in a strongly pseudoconvex domain G. Then there exist k,, k2 > 0 (depending only on cp(0)) such that k, dist(cp(k.), 8G) < 1 - IAI < k2 dist(ep(A), aG),

k E E.

Proof For a compact set K C G with ep(0) E K, using Theorem 10.2.1 and Proposition 10.2.3, we obtain

- C, - (1/2)logdist(ep(.k), aG) < cG(V(o), V(X)) I

= P(0, X) = 2

log 1 + IXI < C2

1-IXI-

- 12 logdist(cp(A), aG)

with appropriate constants C,, C2 which depend only on K. Thus it follows that exp(-2C, )/ dist(ep(A), aG) <

1 + I'XI

< exp(2C2)/ dist(ep(A), 8G),

1 - IAI -

10.2 The boundary behavior of the Carathdodory and the Kobayashi distances

319

and so exp(-2C2) dist(ip(A), aG) < I - IXI < 2 exp(2C,) dist(O(X), 8G).

Corollary 10.2.5 will be used at the end of this section to show that complex cG-geodesics in strongly pseudoconvex domains are continuous up to the boundary.

Now, we turn to the lower estimate of the Kobayashi distance between two points near two different boundary points; cf. Remark 10.2.2(b). Proposition 10.2.6. Let t;', i;" be two different boundary points of a strongly pseudoconvex domain G C C". Then, for a suitable constant C, we have

kG(z', z") >- - (1/2) log dist(z', aG) - (1/2)log dist(z", aG) - C,

whenever z', z" E G. z' is near ' and z" is near ". Proof. Theorem 10.2.1' implies that there are disjoint neighborhoods U' = UV') and U" = such that for suitable smaller V' = CC U' and V" = CC U" and for an appropriate constant C the following inequalities are true:

kG(z', G \ U') > -(1/2) log disc (z', aG) - C,

kG(z", G \ U") ? -(1/2) logdist (z", aG) - C,

Z' E V',

(10.2.2a)

Z" E V".

(10.2.2b)

Now fix z' E V' and Z" E V" and choose e > 0. Then we can find a C'-curve a : [0, 1] --> G, a(O) = z', a (l) = z", such that

kG(z',z")+e >

f

'

XG(a(t);a'(t))dt.

(10.2.3)

0

With 0 < t, < t2 < 1 such that a([0, ti)) C U', a(ti) E RU', and a((t2, I]) C U", a(t2) E aU", conditions (10.2.3) and (10.2.2) imply that kG (z', z") + e > kG (Z', a(t I)) + kG(a(t2). Z")

> -(1/2) logdist(z', aG) - (1/2) logdist(z", 8G) - 2C. Since s is arbitrary, the proposition is verified.

Remark 10.2.7. A version of Proposition 10.2.6 with less restrictive conditions may be found in a paper of F. Forstneri6 and J.-P. Rosay (cf. [For-Ros]). They

X. Comparison on strongly pseudoconvex domains

320

only assume that G is a bounded domain whose boundary is C2 and strongly pseudoconvex in neighborhoods of the points ', c". In case where two points, whose k-distance is measured, converge to the same

boundary point the following upper estimate is also due to F. Forstneri6 and J.-P. Rosay. They dealt with bounded domains with C'+`-boundary. Here we will assume that the boundary is of class C2.

Proposition 10.2.8. Let G be a bounded domain in C" with smooth C2-boundary and let o E 8G. Then there exist a neighborhood U = U and a constant C > 0 such that 2

kG(zi, z2) <-

-2

logdist(zj, 8G) + 2 E log(dist(zj, 8G) + Ilzi - zzlI) + C j=1

J=1

forzi,z2EGnu. Proof. Since the boundary of G is of class C2, there exists R K 1 such that

(i) (ii)

1/8,

z-

E G and dist(z -

whenever z E G fl u, U := B(go, R), (iii)

E 8G n B(i;'o, 2R);

8G) > 38/4, 8R) and 8 < 2R; E 8G n

B(i; -4Rv(C), 4R) C G U

e 8G;

denotes, as before, the unit outer normal to 8G at . Now fix two points zi, z2 E G n u and choose the uniquely determined points

here

E 8G with Ilzj - jll = dist(zj. 8G). Recall that zj = j - Ilz; - ;Nv( j) Moreover, we obtain that

IIkj - doll S IRRj - zj 11 + Ilzj - roll :5 211zj - i;'ohI < 2R,

lie in G and dist(wj, 8G) ? (3/4)Ilzi -z211 because of (ii) above. So the triangle inequality

i.e. j E 8G n B(4a, 2R). Therefore, the points w j := zj - Ilzi -

10.2 The boundary behavior of the Caratheodory and the Kobayashi distances

321

leads to the following upper estimate 2

kG(zi, z2) < kG(wi, w2) +EkG(wi, zJ) i=1

< kG(wi, w2) + EkB( y.p,)(wi, zi) i=1 1logy'+Ilzi-will

=kG(wI,w2)+

pJ - Ilz/ - tUi ll

j=1 2

< kG(wi, W2) +

log

2 i=1

2

dist(zj,

aG)

where p. := Ilwj - i; jll < 4R. What remains is to estimate the term kG(wl, w2). First observe that we have l1wi

- w2II < (5/4) Ilzi -z211; here we use (i) above. Then we consider the analytic

C defined by rp(A) := w1 +,k(w2 - WI), A E C. If Al I< 3/5 (resp. IA - 11 < 3/5), we obtain

curve gyp: C

(.k) - will < (3/5)(5/4)Ilzi - 2211 < dist(wi, 8G) Ilsv(A) - w211 = I1- XI IIwi - w211 < (3/4)Ilzi - 2211 < dist(w2, aG)), ll

(resp.

i.e. if D := B(0, 3/5) U B(1, 3/5) C C, then VIo is a holomorphic map into G. So we finally conclude that kG(zI, z2) 2

< kn(0, 1)+log2-

2i=1

2

logdist(zj, aG) +

2i=1

log(dist(zf, 8G)+Ilzi -z211)-

O

Now, we intend to apply Theorem 10.2.1 and Proposition 10.2.8 to prove a theorem on the boundary behavior of proper holomorphic mappings between strongly pseudoconvex domains. For a more general treatment and an extensive bibliography the reader should consult the book of K. Diederich and I. Lieb [Die-Lie]; see also [Fors].

322

X. Comparison on strongly pseudoconvex domains

For the convenience of the reader we collect those properties of proper holomorphic mappings we will need. Details can be found in Rudin's book [Rud 2], pages 300-305.

Let F : G, -+ G2 be proper and holomorphic (G j , j = 1, 2, are domains

in C"). Put M := {z E GI: detF'(z) = 0). Then M

G1, F is an open

mapping, and the set of critical values F(M) of F is a proper analytic subset of G2. Moreover, FIG,\F-,(F(M)) : G, \ F-1 (F(M)) -- G2 \ F(M) is an unbranched proper holomorphic covering of finite order. After this short summary we can formulate the result we want to discuss; also cf. Exercise 10.5.

Theorem 10.2.9. Let G, and G2 be two strongly pseudoconvex domains in C". Then any proper holomorphic map F : G, --* G2 extends to a continuous map from G1 into G2.

Proof. We assume that G, is given by a pair (U1, r,) satisfying conditions (10.1.1).

Now let U' be an open set with 8G C U1 CC U1. Put K2 := F(G, \ Ui) and define

r2(w):=sup{r,(z):zEGandF(z)=w),

WEG2\K2.

Since F is proper and holomorphic, it turns out that r2 is negative and continuous on G2 \ K2 and psh on (G2 \ K2) \ F(M). So, r2 is a psh function on G2 \ K2 W-

Appendix PSH 21) with r2 (w) -) 0. w-OGz Since G2 has a C2-boundary, there exist positive numbers Eo, do, and Co with the following properties: for any point w E G2 with dist(w, aG2) < eo one can find points wo E G2 and t(w) E 8G2 satisfying (1)

B(wo, eo) C G2 \ K2,

(ii) (iii)

11w - is (w) 11 = dist(w, 8G2),

(iv) (v)

B(t(w) - 2(wo -- t(w)), So) _: B. C G2 \ K2, r2 < -Co on B..

B(wo, so) n aG2 = {g(w)

Now, we are going to prove a fact which is known as the Hopf Lemma. Fix w, wo, and t (w) as above. Then the function u : E - R defined as J r20 (WO +

I Y-1

1

ifA=1

10.2 The boundary behavior of the Caratheodory and the Kobayashi distances

323

is continuous and its restriction to E is subharmonic. If h : E -+ (-oo, 01 is the solution of the Dirichlet problem with u as boundary values, then Harnack's inequality implies that

u(A) < h(A) < h(0) I - I)_I < h(0)(1 - Al), IA E E. 1 + IAI

In particular, we obtain

r2(w) = u

(e0 - dist(w, aG2))

h(0)

EO

EO

dist(w, aG2).

Because of (iv) and (v), the number C, := h(0) = in J0 u(eie)d9 is negative, i.e.

r2(w) < C, dist(w, aG2)/e0 = -C2 dist(w, 8G2)

with

C2 :_ -C, /EO > 0.

Hence for Z E G, sufficiently near aG, we obtain C2dist(F(z), 8G2) < Ir2(F(z))I < Jr, (z)I -< C3 dist(z, aG), where C3 denotes a suitable positive constant. With a correctly chosen positive C we finally get the following inequality:

dist(F(z),aG2)
(10.2.4)

Z E G1.

The last step is to prove that F extends continuously to G1. Here Propositions 10.2.6 and 10.2.8 will do the main job. Obviously, it suffices to prove that for C G, with lim,,_a Z,, = ZO E 8G1 the image sequence any sequence (F(zV))VEN is also convergent. Observe that proper.

8G2)

V_ 00

0 since F is

Now, we argue via contradiction. Suppose that there are two different accumulation points w('), w(2) of i. e. there exist two subsequences F(zvj)) (z(i))VEN C (zv)VEN with = w(i) E aG2, j = 1, 2. Then Propositions 10.2.6 and 10.2.8 imply the following chain of inequalities:

-

2 logdist(F(z(i)), 8G2) - C1 < ko2(F(z,(,')), F(42))) < kc,(z('), z(2)) i=1 2

2

<- E 2logdist(z,(,'),aG,)+E j=1

j=1

where C1, C2 are suitable positive constants.

X. Comparison on strongly pscudoconvex domains

324

With the aid of (10.2.4) we can easily continue this estimate, and so we end up with

-C3 <

2 log(dist(z,(i), 8G,) + Ilz;' l _Z(2)11)+C2,

V E N,

0

which is obviously impossible.

As a simple consequence we get the following weak version of a deep result of C. Fefferman (cf. [Fef}, see also [Vor]). Corollary 10.2.10. Any biholotnorphic mapping between two strongly pseudoconrex domains in C" extends to a homeomorphism between their closures.

As promised after Corollary 10.2.5, we conclude this section with a general regularity result for complex CG-geodesics in strongly pseudoconvex domains.

Theorem 10.2.11. Let G be a strongly pseudoconvex domain in C". Then any complex CG -geodesic cp: E -+ G is 1/2-Holder continuous, and therefore it extends continuously to E.

Before we start the proof we establish the following lemma on good exhaustion of strongly pseudoconvex domains (cf. [For-Ste]). Lemma 10.2.12. Any strongly pseudoconvex domain G C C" admits a strictly psh C` function p : V -+ ]l1; defined on an open set V D G such that (i)

G = (Z E V : p(z) < 0),

(ii)

dp(z) g-` O for Z E G.

Proof. Assume that G is given by a pair (U, r) satisfying (10.1.1). Now choose

an open set U' ccUwith Moreover, let XI: R -)- IR be an increasing convex C° -function with the following properties:

X, (t) = 3.5E if t <4e. Then the function X, o r can be regarded as a C2-function on V := G U U' (by setting X, o r = 3.5E on G \ U'), which is psh and, in addition, strictly psh on

{zEU':r(z)>3e}.

10.2 The boundary behavior of the Caratheodory and the Kobayashi distances

325

Finally, we choose a C' -function X2 on C" with

X2(:)= 1 if:EG\U'orif: EGfU'andr(z) <2e, or if:EGfU'andr(:)>e. X2(:)=0if:¢G Obviously, for a suitably chosen positive constant a the function r,(:) cX2(:)II:II' + X, c r(:) fulfills all the requirements of Lemma 10.2.12.

0

Proof' of Theorem 10.2. / 1. According to Corollary 10.2.5, it suffices to show that

there is a positive C such that

xG(:: X)?CIIXIII dist(:,aG).

:EG. XEC".

(10.2.5)

Namely, from (10.2.5) and Corollary 10.2.5 it follows that Cillw,(

l

> xF(J.: I)

(I - IAI)

and therefore

II'v'(k)II <

C,(1

11X1)"2.

), E E.

Hence the Hardy-Littlewood theorem (cf. Appendix H 9) implies the claim of Theorem 10.2.11. To verify the inequality (10.2.5) we may assume that r: V -+ R is the function from Lemma 10.2.12. Then there exists a positive at such that

(1r)(::X)>a,11X112.

:EG.X EV".

We mention that it suffices to prove (10.2.5) for points : E G with sufficiently small boundary distance such that -r(:) < a2 dist(:, 1G) with a2 > 0. Now fix such a.:() E G and an Xo E (C"). and let E 0(E, G) with >'(0) = ;4) and a*'(0) = X,) for a > 0. Let ,) E aG with -:()II = dist(:o,. aG). Then r(:) := r(:) - a311z i;()II2 is a psh function on G. where a::= a,/2. Therefore. we obtain

(:l)) = r

-

V/ (0)

2ir ./)

II*(e) - ))II'd9.

326

X. Comparison on strongly pseudoconvex domains

Applying the Cauchy integral formula it follows that "

II'(0) II =

1

I2,ri j=1 I

I

1.i 0.) - 6'j dx A-

1

1

2n 1/2

(2rr (1 + a2/a3)'12

dist(zo. aG)..

0

Hence we end up with (10.2.5).

Remark 10.2.13. The main argument in the proof of Theorem 10.2.11 uses the fact that G admits a good strictly psh exhaustion function. Such functions also exist for any C°'-smooth bounded pseudoconvex domain whose boundary is B-regular

(for a precise definition see [Sib 5]). To be more concrete. for any 0 < r) < I there exists a defining function r on U = U(G) of 8G such that the function z E G, is a negative psh exhaustion function of G with rl (z) :(Cr) (z; X) > C II X 112, z E G, X E C", where C > 0. Modifying the argument in the proof of Theorem 10.2.11 we find that any complex cG-geodesics cp: E -> G is >)/2-H61der continuous on G. In particular, this result holds if the boundary of aG is real-analytic.

10.3 Localization In this section we study how to estimate our metrics on strongly pseudoconvex domains dealing only with local information. Both the results we will present were found by I. Graham (cf. [Gra I]).

Theorem 10.3.1. Let G be a strongly pseudoconvex domain in C". Then for a sufficiently small R the set G fl B(%', R) is connected and we have lim

Z_ zeGn8(x.R)

whenever (C,I)

YGnR(x.R)(z;X) = 1

YG(z;X)

E 8G, X E (C").. The convergence is uniform in i; E 8G and X e

10.3 Localization

327

Proof. According to Theorem 10.1.3 we can choose R, sufficiently small, and

p = p(R) < R. We recall that for z E C n R(t. R), g e aC, the inequality YG(z; )

YGnB((.R)(Z, ) always holds.

Now fix E > 0 and put s' := E/(2 + E). For this s' we choose L = L(e', R) via Theorem 10.1.3. Moreover, for E a G let It (.; ) denote the peak function of Theorem 10.1.2 with the data 172 < 111 < p, d2, d1, and G. With a suitable k E N we find that dZ L < I and

I -E'

if IIz - III <- 1) (E)

112

After these preparations we fix zo E G fl B(i;,11(e)) and X E (C").. Then we R), E) with find an f E O(G fl

f(zo) = 0

YGnB({.R)(ZO,X) =

and

of

" I(zo)Xjl = lf'(zo)Xi. azj I=1

J

According to what we said in Theorem 10.1.3, there exists f E HO0(G) with the following properties: f (zo) = 0, YGnB((.R)(zo; X) = I f'(zo)X I , IIf IIG < L. and Ilf -f IIGnB((.p) < E'.

Putting f :=

f gives a holomorphic function with (1 -E')YGnB((.R)(zo;X).

f(zo) =0. If'(zo)XI = Moreover, we get

IIf

d2 4L < 1

and

IIIIIGnB((.,i,) 5 I + E'.

Thus, if zo E B(i;, 11(E)), then

a(f/(1 +E'))

1 +E'

YGnB((.R)(zo;X)

- I - e'I

j=1

3Z jI

< (1 +E)YG(zo:X).

The analogous result for the Kobayashi-Royden metric is also due to 1. Graham (cf. [Gra I]) and will be a consequence of Proposition 7.2.9 and Theorem 10.1.2.

328

X. Comparison on strongly pseudoconvex domains

Theorem 10.3.2. Let G be a strongly pseudoconvex domain and let U = U(lo) be a neighborhood of a boundary point i;o E 8G such that G n u is connected. Then XGnU (Z; X)

lim

Z--zf

:ecnU

XG(z;X)

=I

.

where the convergence is uniform in X E (C")..

Proof First, recall from Proposition 7.2.9 that

xanU(z;X)
h for i;o. With r := inf(Il - h(w)I: w E G \ U) > 0 we see that V(s) := (z E G: 11 - h(z)I < s) C U if 0 < s < r. Obviously, V(s) 3 G n V, for an appropriate V, = V,(4). Following the calculation of the proof of Theorem 10.2.1, we obtain for z e G n V, and W E G \ U that kG(z, ur) > cG(Z. w)

(1/2) log(r/s)--o oo.

Hence for any s > 0 we have

x(;nU(z;X) < (1 +e)xG(z;X)

if Z E Gn v, s << 1.

0

Observe that the claim of Theorem 10.3.2 remains true if G is only assumed to be a bounded domain that admits a global peak function for 4o E G. As a consequence of the previous two results it turns out that the KobayashiRoyden and the Carathdodory-Reiffen metrics have the same asymptotic boundary behavior on strongly pseudoconvex domains. Theorem 10.3.3. For any strongly pseudoconvex domain G we have lim

xG (Z: X)

= I uniformly for X E (V).-

- -+aG y0 (z; X)

Proof. Suppose the contrary. Then there exist Co > 0 and sequences z,. = 4o E 8G and (X-,),.EN C (C"). with

x0(Z,,;X,.) ? (I +fo)ye(:,.;X,.).

C G. (*)

329

10.3 Localization

Because of the strong pseudoconvexity, Lempert's Theorem gives a neighborhood U = U (.O) such that ycnu = xcnu Therefore, by Theorems 10.3.1 and 10.3.2, it follows that lim

XG(zy; X,,)

V-+OC YG(zy,Xy)

= lim

XG(Z,.; X,,)

Ycnu(zv; XV)

-1

''-'x' xcnu(zV,XV) Ya(Z";XV)

which contradicts (*).

So far we have discussed localization theorems for the Caratheodory-Reiffen and the Kobayashi-Royden metric, respectively. Similar results are also true for the corresponding distances. At the moment we can only prove a theorem for the Kobayashi distance. The result for the Caratheodory distance will follow from a more general comparison result.

Theorem 10.3.4 (cf. [Ven 1]). Let U =

be a neighborhood of a boundary

point o of a strongly pseudoconvex domain G C C". Suppose that u fl G is connected. Then for any sequences lim,,_,' WV = o, z,, 54 w,,, we have lim

V--.OC

(WV),.EN C G fl u with limy

.z_

kcnu(zy, w,,) = 1

kc(z,,, w,)

The proof of this theorem relies on the following lemma. Lemma 10.3.5. Let G and i;'o be as in Theorem 10.3.4. Moreover, assume that there are sequences (z,,),.EN, (wv)vEN (t,,),,EN of points z,,, w,.. t,, in G, respectively, and a positive constant C with

lim z, = lim w,, = o and kG(Z,,, t,.) + kG(t,,. w,,) < kc(z,,, wV) + C, v r= N. V- OC

-

Then lim,...,,C t,, = o.

Proof. Suppose the contrary. Then there is a subsequence of again by (ty) with lim,,.. t,, = to jk 0. Case 1'. If to E G, then Theorem 10.2.1 implies that

which we denote

- (1/2) logdist(zV. aG) - (1/2) logdist(wV, aG) - C1 kc(zy, to) + kG(w, to) < kc(zy, w,,) + 2kc(ty, to) + C. which contradicts the upper estimate in Proposition 10.2.8.

330

X. Comparison on strongly pseudoconvex domains

Case 2°. If to E 8G, then by Proposition 10.2.6 we have

- (1/2) logdist(z,,. 8G) - (1/2) logdist(w,, 8G) - logdist(t1,, 8G) - C, < kG (z,, 0 + kG (t,,,

kG (z,,, WO + C,

which again does not fit with Proposition 10.2.8.

Proof of Theorem 10.3.4. Applying the localization result of Theorem 10.3.2, we find for given e > 0 a neighborhood V = V C U such that xGnU (Z; X) < ( I + e)xG (Z; X),

z E V f1 G, X

Moreover, there are C' -curves a,,: [0, 11 --> G connecting

C".

and w, such that

i

10

xG(a, (t);a;,(t))dt < (I +ev)kG(z,,, w,) with 0 < e,, < min{e, l/kG(z,,, w,,)].

Now, we claim that if v >> 1, then a ([0, 1]) C V. For otherwise there is a sequence (ri )JEN C (0, 1) with a,,, (Ti) _: ti 0 V. On the other hand, we obtain i

kG(z,,,, ti) + kG (ti, w,,;) < Z NG(ar,(t);a'V,(t))dt < kG(z,1, w,,;) + 1.

Hence by Lemma 10.3.5 it follows that limi.,, ti = o, i.e. ti E V for large j; a contradiction. Therefore, we conclude that for v >> 1 we have kcnu(Zv,WV) -< J KGnu(a,, (t); a,, (t)) dt 0

f1

< (I +e)J xG(a,(t);a;,(t))dt < (I +e)2kG(zv, o

0 Finally, we formulate a localization result for the Bergman metric, more precise

than the one of Theorem 6.3.5 (cf. [Die 1], [Hbr 1]). It can easily be obtained by modifying the 8-problem in the proof of Theorem 6.3.5 with the aid of an appropriate peak function (cf. Th 10.1.2); details are left to the reader.

10.4 Boundary behavior of the metrics

331

Theorem 10.3.6. Let G be a strongly pseudoconvex domain and let R > 0 be such that fur any o E 0 G the intersection G n B(to, R) is connected. Then for every e > 0 there exists S = SF E (0, R) such that MG(Z; X)

(i)

KG(Z;z) < KGnB(A,.R)(z;X) < (I +e)KG(Z;z).

Gt)

(iii)

MGne(z0.R)(z; X) < (I + e)MG(z; X )

(1

<18G(Z;X) <

(1 +e))BGff(r0R)(z;X)

whenever Z E GnB(to. 8), X E C". Moreover, if R is sufficiently small, then S = SF can be chosen independently of the boundary point o.

10.4 Boundary behavior of the Caratheodory-Reiffen and the Kobayashi-Royden metrics When one tries to understand the boundary growth of various metrics on a strongly

pseudoconvex domain, the following observation becomes important: strongly pseudoconvex domains locally (up to biholomorphisms) look like strictly convex domains, and therefore, up to localization, metrics should "coincide" near the boundary. The precise formulation of this asymptotical equality will be the main goal of this section. It was I. Graham who studied the boundary behavior of xG and yG on strongly pseudoconvex domains (cf. (Gra 11, also (Hen-Chi]). He obtained an asymptotic estimate of the length of normal and tangential vectors separately. In [Ala], G. Aladro obtained an estimate for the length of a general vector without specifying the asymptotic constant. He needed the boundary to be of class C". The theorem we are going to present here is in the spirit of the one recently found by D. Ma (cf. [Ma 1,2]), which is more precise than Graham's and Aladro's results. N. Sibony (cf. [Sib 41) also established estimates of the Kobayashi-Royden metric on domains which carry a "good" psh function. Moreover, estimates of these metrics near the boundary of pseudoconvex domains of finite type in C2 were found by D. Catlin

[Cat]. More recently, J. McNeal [McN] extended this investigation to smoothly bounded pseudoconvex domains G of finite type in C" under the additional hypothesis that G is "decoupled near Zo E 8G", i.e. up to a biholomorphic change of coordinates w = w(z) near zo, u'(zo) = 0, the domain G is locally given as {w E C": r(w) = 2 Re wt + E'=2 ri (wj) < 01 with rj smooth. subharmonic but not harmonic, and such that rj (0) = dri (0) = 0.

X. Comparison on strongly pseudoconvex domains

332

To prepare the proof of the main theorem we have to introduce the notion of an analytic ellipsoid. Let n

H(z, w) := E aijzitjlj,

z = (ZI, ... , Zn), w = (w), ... , wn) E

i.j=l

be a positive definite Hermitian form. Then the domain

E=E(H):={z=(ZI,..., Z,,)EC": 'I(z) :=-2Rez1+H(z,z) <0) (10.4.1)

is called an analytic ellipsoid. We point out that the Levi form of a strictly psh function can be taken as an example of such an H. Since H is given by a positive definite Hermitian matrix A = (ai j) I <;, j -,,, there

exists a unitary (n - 1) x (n - 1)-matrix S such that if we put

S:=

1

0

0S

'

jv, := S' A S has the following properties:

then the matrix B = (bi j) I

bjj>0 ifj>2, bij=0

if

i,j>2, i &j.

Setting

T

.

_

1

0

-bl2/b22

1

...

0

0 ,

-bin/b.,. 0

1

we find that C = (c; j) I O, j > 1. Therefore, R

z, - Zl+Ecjjzjzj <0) J=1

_ (Z E C": CIIIZ1 - 1/CIII2+cjjzjij < 1/c11

.

J=2

Using the transformation

CII,...,Znj' Cnn +(1/cll,0,...,0)

333

10.4 Boundary behavior of the metrics

we get n

(z E C":

1z112 < 1/c1i).

i=l Combining all these maps we have the following explicit biholomorphic mapping

F:£-li,, with F(z) := ,/cii4,-l (((S . T)-I zc)c) Using F and the formula for xgq, the next lemma becomes an easy exercise.

Lemma 10.4.1. Let E = £(H) and (D£ be as in (10.4.1). Then

xe(z:X)=Ye(z:X)=

2) 1/2

H(X,X)+ H(X, z) - X -4'E(z)

In the sequel, analytic ellipsoids will serve as local comparison domains in th process of estimating metrics on strongly pseudoconvex domains. The main result is the following (cf. [Ma 1,21) Theorem 10.4.2. Let G be a strongly pseudoconvex domain in C" that is given by (U, r) (cf. (10.1.1)). Then for every E E (0, 1) there exists S = S(E) > 0 such that the following properties hold: (i) for every z E G with dist(z, aG) < S there is a unique 7r (.7) E 8G with 11z - II = dist(z, aG), (ii) for every such z E G and even, X E C" the following inequalities are true

(1 -E)

(Gr)(r(z);X(,)) 2dist(z, 8G)

i/2 II

+ 4dist(z.3G)2)

< (1 +E)

-

yc(z;X) < xa(z;X) 1/2

(Gr)(7r(z);X(,}) + 2dist(z, aG)

4dist(z, aG)2)

'

where X is split into its normal and tangential components X(,,) and X(,) at the X(,)) = 0. and'=,(8r/8zi)(tr(z))(X(,))i point7r(z), i.e. X = X(,,)+X(,),

=0. The following technical lemma will be used in the proof.

Lemma 10.4.3. Let E = E(H) be as in (10.4.1) with H(z, z) - F;'j_, aij zizJ- >a II z I12. Z E C", where 0 < a < 1. For 0 < X < a let £a denote the analytic

X. Comparison on strongly pseudoconvex domains

334

ellipsoid

2<0). Ex _":

Let a E C and assume that la 1 < M and jai/ 1 < M, 1 < i, j < n, for fixed M > 1. Finally, fort > O put z(t) := (t - ate, 0, ... , 0) E C". Then

z(t)EEe CEj i f O<x 0 there exists a positive constant po < mina, pi), depending only on a, M, and e, such that 2

/2

(1-s)(H(2t'X)+1412)' <xea(z(t);X)

<xe (z(t);X)<(1+e)(H(ZtX)+1412`

i/z ,

whenever 0< x< po, 0< t< po, and X E C". Proof. First, a simple calculation shows that for 0 < t < p, the following inequality is true:

-2t(1 + SM3t) < `De (z(t)) <

hex

(z(t)) < -2t(l - 5M3t) ;

in particular, z(t) E Ek . For the remaining inequality we will only prove, for example, the lower estimate.

So, if 0 < t < pi, Lemma 10.4.1 leads to

(xe-(z(:);x))

-k (z (0)

I -x/a (1+5M3t)2

H(X, z(t)) - AX,t(1 - at) - X,

H(X,X)-XIIX112

2

H(X, X)

(

2t

X1I2

+

-

+ 412

C 4t2

2

-4)£z (z(t))

,z

(IIXII't +

2

t+IlXll Ix,lr})

where the positive constant C can be chosen in such a way that it depends only on M. If IX1 I < II X 11 ft, then it follows that 2

`xe"

(z(t)'X)

1 - x/a (I +5M3t)2

H(X, X)

1X112 412

+

2t

(1 -3C2a)

10.4 Boundary behavior of the metrics

335

II1II f we obtain

wbereac. for 11, I

1 -A/a (xe-(z(t);x)) > - (1 +5M-t) 2

(H(X, X)2

+

2t

4t2

(1 - 3Cf))

Therefore, we have the following lower estimate XC' (z (I); X)

>

-

a

X12

3C 2a

I

t

1/2

1 - 3C ft 1/2

1 + 5M 3t

H (X, X 21

1/2

X2 4t2 J

whenever 0 < f < a/(3C). Now, it is clear how to choose the constant po in Lemma 10.4.3.

O

Proof of Theorem 10.4.2. First, we choose a S, < I sufficiently small to have the following situation: (a) (8G)s, := {z E C": dist(z, 8G) < S, I CC U; (b) if z E (8G)8,, then dist(z, 8G) = 11z - 7r(z)II, where zr(z) E 8G is uniquely determined;

(c) (Cr)(z;X) > alIX112 and a < Ildr(z)II

<-

I/a for a suitable a E (0, I)

whenever Z E (8G)s,, X E C".

Moreover, for any y > 0 we can find S(y) < min{S,,a2) such that ID'3r(z) DAr(lr(z))I < y for all z E (8G)a(y) and for all multi-indices f E (Z+)2n, If1 < 2. Here we use the fact that r is a C2-function.

Now fix e > 0 and let o be an arbitrary point of G. We point out that the further construction will depend on ip. Nevertheless, we will omit the index i;o to keep the notation simple. On the other hand, all the constants in the estimates will be chosen independently of the specific o. For o we take a unitary matrix A = Ax,, transforming the vector into

the vector (-I,0,...,0), i.e. (-1, 0, .... 0) = (A

Thus T: C" C", T(z) := biholomorphically maps G onto the strongly pseudoconvex domain G := T(G) which is given by the pair U := T(U) and i := r o T-'. Note that gradr(0) = (-1, 0...., 0) and that (86)a, CC U. Therefore, we obtain the following Taylor expansion of r on B(0, S 1): n

F(z) = -2Re(z, -

CijZiZ;) + (Cr')(0;z) +,e(z)IIZII2.

(10.4.2)

2

where for any y > 0 we have I fl (z) I < C, y provided that II z II < 6(y) < B1. Here the constant C, > I can be chosen independently of o. Moreover, we have

X. Comparison on strongly pseudoconvex domains

336

? alla112. C,, = cj;. and Ic;jI < C2. C2 is again

(Gr)(0;a) =

independent of o. Let F : C" -+ C" be the map defined by 1

F(zl..... z,,) := (zl

-

7

?I

E CIjzizj, z2...

.

,,)

(10.4.3)

!-j=I

We will show that for every A with 0 < A < at and for all sufficiently small (depending on A) positive S < SI/2 the following inclusions are true:

6A nB(0,S)C F(GnB(0.25))cEc.

(10.4.4)

where ) := {wEC": -2Rewl+(11)(0;w)±AIIw112 <0). Moreover, for 0 < t < S/2 we claim that

F(t, 0,

..., 0) _ (t - 2c,It2, 0.....0) E e n B(0. S).

(10.4.5)

Observe that (10.4.5) is an immediate consequence of Lemma 10.4.3. (Note that the coefficients a; j of (&) (0; z) and c; j are uniformly bounded by a constant M > 1 which is independent of to.) To prove (10.4.4) we choose a positive Ro < SI, independent of to. such that

IIF(z') - z' - F(z") + z"II < 11 Z' - z"II/2,

z', z" E B(0, Ro).

Therefore, F := Fl

(10.4.6)

biholomorphically maps B(0, Ro) onto an open set V containing B(0, Ro/2): compare a proof of the inverse mapping theorem. Suppose now that 0 < A < a is fixed and take a point w E Ej n B(0. Ro/2). Put z := F-I (w) E B(0, Ro). Then, since IIzIl < 211w11 (cf. (10.4.6)) and since (V)(0; z) - (Gr')(0: F(z)) = 0(11Z 11-1). we obtain

i(z) _ -2 RewI + (F)(0;z)+p(z)IIzII2 < (Gr`)(0;z) - (f-r`)(0; F(z)) + (4i6(F-I(w)) - A)11w112 < (C3IIw11 +

provided that

4i6(F-I(w))

- A)IIw112 < 0

8 < Ro/2, i.e. Z Ed n 8(0.25).

10.4 Boundary behavior of the metrics

337

The second inclusion of (10.4.4) will follow similarly. Let Z E G n B (0, Ro) and

put w := F(z). Then -2 Re w, +(&)(0;w) -,111w112 < i (z) + (Gr')(0; w) - (&)(0'Z) - XIIw112 - $(z)IIzI12 (C311zII -18(z) - A/4)IIz112 < 0

provided that Ilzll < 23 and S sufficiently small. Hence the inclusions (10.4.4) are proved. So Ea and Ea can locally serve as an inner and an outer comparison domain for G, respectively. Now, we are able to apply Lemma 10.4.3 with H(z,z) _ (0; z). We fix

X = po < a and we choose a positive S < min(Ro/2, po) such that (10.4.4) and (10.4.5) hold and that for all 0 < t < S the point z(t) := (t, 0, ... , 0) belongs to G with dist(z(t), ad) = t. Of course, S can be chosen independently of o. Then, using Proposition 7.2.9, for X E C" and 0 < t < S/2 we obtain the following upper estimate (note that F(z(t)) E E, n B(0, S)): xd(z(t); X) < xg+nB(o.a)(F(z(t)); F'(z(t))X)

< cothke+(F(z(t)),£)+, \ B(0, S)) xE+(F(z(t)); F'(z(t))X) I

B(0.8))((C)(0;F'(z(t))X)

< (I +s)cothke+(F(z(t)),Ex \

+ IFi(z(t))X12)l/2. 4t2 (10.4.7)

2t

On the other hand, if we assume that S is sufficiently small, then we can apply Theorem 10.3.1. So we get Yc(z(t);X) > (I - s)Y6nB(0.8)(z(t);X) provided that t is sufficiently small, say t < SF < 6/2. Then (10.4.4) and Lemma 10.4.3 give

Yc(z(t); X)

(1 - s)YE, (F(z(t)); F'(z(t))X) I F1(z(t)) X 12)' /2 > (1 - s)2 ( (U)(0; F'(z(t))X) + 2t

4t2

(10.4.8)

To replace F'(z(t))X and F' (z(t))X by X(,) := (0, X2, ... , X") and X(") (X,, 0, ... , 0), respectively, we make the following observation:

X. Comparison on strongly pseudoconvex domains

338

there are positive constants C4, C5 such that the following two inequalities are true

I(1)(0;F'(z(t))X) - (CT)(0;X(,))I

<- C4(IXu12/.,,/t- + C5(%1t-IX112

I IFi (Z(t ))X 12 - IX1121 <

11X(,)112),

+ t312IIX(,)112).

Thus, since (Li)(0;X(,)) > aIIX(t)112, we derive

(1 +O(-It-)) ((C?)(0;

X(, ))

+ 11121 = (Ci)(0; F'(z(t))X) +

IF( 4(t))XI2J

4t2

(10.4.9)

Hence (10.4.7), (10.4.8), and (10.4.9) imply that

ya(z(t);X) > (1 -E)3

((/)(0; X( ))

X-I

12\'n

2

and

xc(z(t);X) < (1 +E)2 ((Gr)(0'X(t))

14112\ ,2

+

B(0,a))

if 0 (C)(0;z) +AIIz112 > aIIzII2, z E E,+. As in the proof of Theorem 1.0.2.1 we obtain

kC (F(z(t)), £x \ B(0, 8)) > cc, (F(z(t)), £x \ B(0, 8)) >

1 - d2

1

10

2

g duIIz(t)II

1 to 2

I -- d2

g d1(1

with d2 = d2(S) < 1, d, > 0, and t sufficiently small. Hence we have verified that

xo(z(t);X) < (1 +E)3

((Li)(0;X(,)) + 11211/2 2

+ M)t

10.4 Boundary behavior of the metrics

339

provided that 0 < t < Sf < Sf < 1/23. So the proof is complete for any : E G with dirt(:, aG) < SF. Remark 10.4.4. Under the additional hypothesis that G has a C'-boundary D. Ma (cf. [Ma 2.3)) obtained stronger results in which he even specified the order of the asymptotical convergence. Moreover, in the case of the Kobayashi-Royden metric he has the following precise result. Let G be a bounded domain in C", not necessarily pseudoconvex. Assume that M is a relatively open subset of )G, M is a Cz strongly pseudoconvex hypersurface. and G is on the pseudoconvex side of M,). Let M be a compact subset of Mt). For S > 0 let QA := (: E C": dist(:, M) < S) and Gb := G n Qb. Moreover. assume that for a positive S we have a) 8G n Qb,, is relatively compact in b) there is a strictly psh function tq E with Gb, := (: E Qa,,: (p(:) < 0) and II grad w(:) II = I whenever : E Q,%, n M. Let S < 30 be a positive number such that for each : E GA there is a unique point

;r(:) E M n Qb with dist(:. dG) = 117r(z) - :II and such that M, := 2r(GA) cc Mo, n Q,,,.

Then there exists a positive C = C(G. M. S) such that for each : E GA the Kobayashi-Royden metric satisfies the following estimate:

exp(-C

u (: ))

((CQ)(n'(:): X(,)) IIX"II 11/ 2u(:) + 4u'(-)/J /-,( < exP(CV(

))

"c (z : X)

((C)(3r(:);X(s)) + 2u(z)

IIX"II'`

4u2(-.)1

where u(:) := dist(:. dG) and X = X(,)

E C" as in Theorem 10.4.2. There is an example (cf. Exercise 10.1) which shows that in the above estimate the factors exp(±C u(:)) cannot be improved to exp(±C(u(:))(WW')+, ) It should be mentioned that the proof of this precise result cannot be used as an approach to find the optimal estimate for the Caratheodory-Reiffen metric.

From Theorem 10.4.2 we immediately derive Graham's formulation of the asymptotic behavior.

Corollary 10.4.5. Let G and r he as in Theorem 10.4.2 and fix i;() E i)G. Then

lim SG(:;X)dist(:. )G) = z

4,

X E C".

where X(,,) denotes the normal component of X at () and where 8(; belongs to a holnmorphically contractible f,mih. of pseudometrics S. If X is a

344

X. Comparison on strongly pseudoconvex domains

complex tangent vector to dG at i;n, then lim 8G(z; X)2 dist(z. aG) = (1/2)(Cr)(Co; X).

We point out that the proof of Theorem 10.4.2 is based on the holomorphic contractibility. Nevertheless, it turns out that the estimates there remain true if we substitute yG and xG by the Bergman metric PG. This result, and much more information about the boundary behavior of derivatives of the Bergman kernel, were obtained by K. Diederich (cf. [Die 1,2]). Theorem 10.4.6. The inequalities of Theorem 10.4.2 remain true if yG and xG there are replaced by the Bergman metric divided by In -+ 1. Before we go into the necessary modifications of the proof of Theorem 10.4.2 we have to present the formula of the Bergman kernel for analytic ellipsoids.

Lemma 10.4.7. Let E = E(H) be as in (10.4.1) with H(z, z) = E"i=l aiizi,i Then the Bergman kernel of.6 is given by

n! det aii i-2

ZEE.

tit-here 4) (z) = -2Rez; + H(z. z). Proof Use a biholomorphic map from £ to B,,; cf. the proof of Lemma 10.4.1.

Proof of Theorem 10.4.6. Here we will use the same notations as in the proof of Theorem 10.4.2. We will discuss only those steps of the proof which are different now.

Fix e > 0 and choose w = po and S,, < 8 as in the proof of Theorem 10.4.2. Moreover, we may assume that S is chosen so small that the estimate of Theorem 10.3.6 holds uniformly with respect to the boundary points. Now let zo E G with dist(z(), aG) =: t < SF < 8, co E aG with l o - zoll = t, and Y E C". As above we put z(t) := (t, 0, ... , 0) and G := T (G). Then we begin the upper estimate of MG (see Remark 6.2.7 for the definition): MG(zo;Y) = Mc(z(t);X) < MGns(o.'_a)(z(t):X) = Mt'(Gne(o,2a))(F(z(t)):F'(z(t))X) IdetF'(z(t))I < M+0( F(z(t));F'(z(t))X) I det F'(z(t))I,

10.4 Boundary behavior of the metrics

341

where X := A Y. Here we have used the transformation rule for MG (cf. Remark 6.2.7) and its monotonicity in G. Moreover. if 3, is sufficiently small, by Theorem 10.3.6 we obtain KG(z4). Zo) = KG(z(t). z(t)) > (I - e)Kdne(o.26)(Z(1). z*))

> (I - E)KF(cne(o.za))(F(z(t)), F(z(t ))) I del F'(z(f))12

> (I - E)Kt (F(z(t )). F(:(t))) I del F'(z(t))I2. Hence the above inequalities yield (cf. Theorem 6.2.5)

PG(zo:Y) <

Mt+ne(o.a)(F(z(t)):F'(z(t))X)

(I - e)iiz

(10.4.10)

K. (F(z(t )). F(z(t)))

To increase the numerator of (10.4.10) we apply Theorem 10.3.6 again and we find that

Mt: nn(o.a)(F(;.(t)):F'(:(t))X) <(1 +E)Me, (F(z(t));F'(z(t))X) whenever 0 < r < SF and SF sufficiently small. On the other hand, the explicit formula for the Bergman kernel gives that

Ke, (F(:(t)). F(z(t))) ? (1 - E)Kt, (F(z(t )), F(z(t ))) provided that I = po is sufficiently small. So the final inequality looks like this: IJG(zo:X) <

+Ffi. (F(z(t)),F'(z(t))X)

_I+E

-E (n+1)x1. (F(:.(t)):F'(z(t))X).

cf. Example 6.2.1. Now, we can turn to the proof of Theorem 10.4.2 and continue starting with (10.4.7). We move now to the lower estimate. First, we apply again Theorem 10.3.6 to get

MG(;,o:Y) = Md(z(t):X) > (1 -E)McfA(o.aa)(z(t);X) > (I - e)MFiGng(U.2a1)(F(z(t)): F'(z(t ))X) I del F'(z(r))I

(1 - E)Me (F(z(r)): F'(z(t))X) I det F'(z(t))I

342

X. Comparison on strongly pseudoconvex domains

if 8, is sufficiently small. Moreover, we have the following chain of inequalities for the Bergman kernel (cf. Theorem 10.3.6): KG(ZO, Zo) = Kc(z(t), z(t)) $ Kcne(o.2a)(z(t ), z(t))

= KF((jns(o.2a))(F(z(t)), F(z(t))) I det F'(z(t))12 < Ke+ne(o.s)(F(z(t)), F(z(t))) I det F'(z(t))12

< (1 + E)Ke' (F(z(t)), F(z(t))) I det F'(z(t))12 if 8f is sufficiently small. Combining the last two inequalities we can conclude that #G(ZO:Y)

(1 - E) Mc; (F(z(t)): F'(z(t))X) (1 +E)i/2 KEA (F(z(t)), F(z(t)))

The last step then is similar to the one in the upper estimate, and therefore it is left to the reader. 0

10.5 A comparison of distances Recall that Theorem 10.2.4 has taught us that the quotient cG(zo, z)/kG(ZO, z) tends to one if z approaches the boundary. Moreover, the convergence is locally uniform in the first variable zo. Some more work leads to the fact that the above convergence is even uniform in the first variable. More precisely, we have Theorem 10.5.1 (cf. [Ven I]). Let G be a strongly pseudoconvex domain in C". Then for every E > 0 there exists a compact set K = K(E) C G such that CG (Z', Z") < kG(Z', Z") < (1 + *G W, Z"), whenever z' E G and Z" E G \ K.

The proof of Theorem 10.5.1 needs a very deep result of J. E. Fornaess (cf. [For]) which we state here in a form appropriate for our purposes. Theorem 10.5.2. Let G be a strongly pseudoconvex domain in C". (a) For any to E 8G there exist a domain G' J G, a neighborhood U = C G', a convex domain D CC C" with C2-boundary, and a holomorphic mapping

10.5 A comparison of distances

343

4: G' -> C" which satisfies the following properties:

4(G) C D,

(1)

(iii)

L(U\G)CC"\b, --I (4)(U)) = U,

(iv)

4)Iu is injective.

(ii)

(b) There exist a domain G D G, a bounded strictly convex domain b C CN, and

a holomorphic map *: G - CN such that is biholomorphic onto a closed submanifold of CN, (ii)

>(i(G)Cb andt/r(G\G)CCN\D,

(iii)

*(G) intersects ab transversally.

Proof of Theorem 10.5.1. Let us suppose that the claim of Theorem 10.5.1 does not aG hold. Then we find Co > 0 and sequences (; },,Er . (z;; ),,EN C G with z;'

r-.x

such that (1+Eo)cG(z,,,z,)
In particular, we have z,, 54 z;;. We may assume that lim,,,,,;, z; =: z" E aG and lim,,-x z;, =: Z' E G exist. If Z' E G, then Theorem 10.2.1 and Proposition 10.2.3 imply that

(1+So) <

kc;< -(1/2)logdist(z;;,8G)+C,

---p 1, cG(Z;,, z'') - -(1/2) logdist(;.", 8G) - C2 ,'-.x

which gives a contradiction. If Z' E aG with z' O z" we are led via Propositions 10.2.3, 10.2.6, and Theorem 10.5.2(b) to the following chain of inequalities: ',

(1 +80)

kb(VI(zv),*(zv)) cb(*(zv), (z")) -(1/2) logdist(z,,. aG) - (1/2) logdist(z,;, aG) + C, - -(1/2) logdist(*(z',), 3D) - (1/2) log dist(*(z,'), aD) - C2 CG(Zv.z'V )

1,

Employing *(aG) C ab gives dist('/r(w,,). aD) < C3dist(w,., aG)

for v > 1,

X. Comparison on strongly pseudoconvex domains

344

where w = z;, or w = z;;. Inserting this in the upper inequality leads again to a contradiction. It remains to consider the case that z' = z". According to Theorem 10.5.2(a) we

choose G', U = U(i;o) with io := z' = z", and 4. Put V :='(U); V is an open neighborhood of b E aD and 1b: U -+ V is biholomorphic. Now choose a ball V' around with V' C V and put U' := -b-1(V'). Then U' = C U.

Moreover, D n V' is connected, and therefore G n U' is connected, too. So. for v >> I we obtain

(I + Eo) <

kG(z, Z,,) <

kG(z' ,

(z"))

4'(z)) .

cG(Z', z;;) - kcnu'(z,,

because of Theorem 10.3.4; a contradiction. Therefore, Theorem 10.5.1 is completely verified.

Corollary 10.5.3. Let G be a strongly pseudoconvex domain in C" and let U = be an open neighborhood of io E aG such that G n u is connected. Then lim z'#:

CGnu (z'. z")

= 1.

CG(Z', Z")

Proof. Suppose the contrary. Then there are sequences (Z" ),0, (z' )vEN C G n u, , with CGnu (Z;,, z;;) > (1 + EO)CG (Z;, Z,"). Hence, by Theorem 10.5.1, z;, 56 CG(Z . z;,')(1 +EO)h12 > kG(z', z;;) provided that v >> 1. Thus it follows that (1 +so) 112kG(zV', z') kGnu(,

Z,,.

which contradicts Theorem 10.3.4.

Remark 10.5.4. 10.5.3.

0

0 It would be interesting to find a direct proof of Corollary

10.6 Characterization of the unit ball by its automorphism group The aim of this section is to characterize the unit Euclidean ball in C" by its automorphism group. An even stronger result dealing with unbranched proper holo-

10.6 Characterization of the unit ball by its automorphism group

345

morphic mappings will be presented. Roughly speaking, we will show that if there are sufficiently many unbranched proper mappings between two domains in C", then both domains are biholomorphically equivalent to the unit Euclidean ball.

Theorem 10.6.1 (cf. [Lin-Won]). Let G and D be two bounded domains in C". where G has a smooth C) -boundary, and let q E G. Assume that there exists a D. j E N. with lim j-,, Fj (q) _ sequence of unbranched proper maps F1: G wo E a D. such that D has a strongly pseudoconvexr boundary near w(). Then G and D are biholoniorphically equivalent to the unit ball B.

Before we go into the proof we should mention two consequences which were already announced in the title of this section (cf. [Ham-Sak-Yas], (Ros 11. [Won 11).

Corollary 10.6.2. (a) Any' strongh pseudoconvex domain G C C" is biholomorphlc all equivalent to B" if Aut(G) is not compact. (b) A bounded domain with smooth C2-boundary is biholomorphically equivalent to B,, ijf'the group Aut(G) acts transitively on G. Proof. One has only to recall that any bounded domain with smooth C--boundary admits at least one strongly pseudoconvex boundary point. Then, setting G = D, Theorem 10.6.1 applies.

Remark 10.6.3. Because of Corollary 10.6.2. the only (up to a biholomorphism) strongly pseudoconvex domain with a non compact automorphism group is the unit Euclidean ball. In the case of bounded pseudoconvex domains of finite type there are analogous results by E. Bedford and S. Pinchuk (cf. (Bed-Pin 1.2,31). For example, a pseudoconvex domain G CC C2 of finite type, for which Aut(G) is not compact. is biholomorphically equivalent to a Thullen domain (: E C2: Iz)12 + I:2l2A < 1) with k E N.

Now, we are going to prepare the proof of Theorem 10.6.1 giving a series of various lemmas.

Lemma 10.6.4. Under the assumptions of Theorem 10.6.1 there exists a subsequence (Fj, ),.EN C (Fj with Fj, F E O(G. C"). where F(z) = u,() for every z e G.

Proof. Since D is bounded. there is a subsequence (Fj,) C (Fj) such that F E f)(G. D). In particular. we have F(q) = w,). By hypothesis. the point w)) is a strongly pseudoconvex boundary point of D. Therefore there exist a neighborhood V = V(wo) and a function f E O(V. C) Fj,

346

X. Comparison on strongly pseudoconvex domains

with f (wo) = 1, If (w)I
(10.6.1)

Remark 10.6.5. Observe that this lemma shows that all information about G is already hidden in the shape of D near wo. So, using the local convexity of D at wo, it is easy to deduce that G is a simply connected domain. Since we will not use this fact, we omit the details of the proof. The proof of the next auxiliary result could be based on a certain fixed point theorem from differential geometry which is due to E. Cartan; cf. [He[], Th. 13.5. One only has to observe that the sectional curvature of the Bergman metric of B" is non-positive. Nevertheless, here we give a proof which does not depend on the concept of curvature. Lemma 10.6.6. Any unbranched proper holomorphic mapping r : B,, --> D to a domain D C C" is biholomorphic.

Proof. Since n is proper, it is surjective. So it remains to show its injectivity. Let us suppose the contrary, i.e. the existence of an integer N ? 2 such that ##r-I (w) = N for every w E D. Now fix w* E D and let ,r-1(w*) = (zi, . . . , zN) C B". Then, obviously, N

ro:=inf(r>0: nBk..(zj,r)00} j=1

N

is a positive number. We put K := n Bk., (zj, ro). Observe that K is a non-empty j=1

convex compact subset of B".

Let g C Aut(B,,) denote the finite group of covering transformations for

ir:B"--* D. Then forgEgandzEK we get ka.(zj, g(z)) =

z) = k,,(z+(j), Z) < r0,

i.e. the set K is invariant under the action of g. So K consists of at least N points with N > 2. Let us take two of them, say w', w" E K with w' # w". Since K is convex, the segment w'w" belongs to K.

10.6 Characterization of the unit ball by its automorphism group

347

Because of the minimality of r) and the convexity of kB,(z1. ), we conclude that there is at least one index j)), say jo) = 1. such that kB.(z), w) = ro for every III E

Now, if we look at the formula for kB, = tanh-) (c*.). we immediately see (using the identity theorem for real analytic functions) that kB,(z1. iv) = r() as long as u' and in B". But this contradicts the completeness lies on the line through w' and of B,,.

By virtue of Lemma 10.6.6 it remains to establish that G is biholomorphic to B,,. The idea is to apply the following classification result which is due to C. M. Stanton, cf. [Sta 2]. Proposition 10.6.7. Let G be a bounded pseraloconvex domain in C" with smooth

C' -boundar .Moreover, assume that ya = rG = 7-11~G. Then G is biholonrorphically equivalent to 13,,.

Proof. By Theorem 7.6.5. G is complete with respect to the Bergman distance. So G is a complete Hermitian manifold whose distance is given by kG = ca = i

+)bG' Let us fix a point zo E G and put

13:= it, E C":

1

Observe that B is given by a Hermitian scalar product. so it is obviously biholomorphic to the Euclidean ball B,,. In the next step we will use some well-known facts from differential geometry.

Namely, for any vector t' E C" there exists a uniquely determined C--curve a,.: R - G with a,,(0) = z,) and a,.(0) = v which is a geodesic with respect to the Hermitian structure of G and such that ar,.(t) = a,.(rr). t, r E R. Moreover, there is a small positive eo such that the mapping

4': (tv E C": N(r) < e()) - G with 4'(v) := exp.-,,(v) := a;,(I) is a C"-diffeomorphism onto a neighborhood of zj). Now, we are able to introduce the map F): B - G we will be interested in. We r E 3,,. Note that v -- tank-'(N(v'))/N(t') 4'(tanh-1(N(v)) put Fg(v)

extends to 3,, as a C"-function. Hence F, is a Cl-map near the origin. To see that F, is also holomorphic we want to apply Forelli's theorem (see IRud 21, Theorem 4.4.5). Thus we have to verify that F is slicewise holomorphic.

For if we fix a E C" with N(a) = I. then there exists an extremal analytic disc V E O(E. G) for ra (recall that G is taut) with tp(0) = zo. tp'(0) = a. In

348

X. Comparison on strongly pseudoconvex domains

particular, cp is a complex ca-geodesic. Then, if A E E., we study the C°°-curve at over R defined by a(t) := (p(tanh(tlAI)A/lAI),

t E R.

It is clear that &(0) = zo and &'(0) = Aa. In addition, for positive r we have

kG(zo,&(r)) = p(0,tanh(rlAI)A/IAI) = nIAI = / T MG(a(r);&'(t))dt. 0

Hence & is the shortest curve. In particular, it is the geodesic (starting at zo into direction of Aa), and so we obtain tanh-'(IAI)ka

Fo(Aa) = exp

_&

(tanlc'(IAI)\

Im

77 (p (.X)

IAI

Since A is arbitrary in E we find that the map E 9 A -). Fo(Aa) is holomorphic. Hence Forelli's theorem implies that Fo is holomorphic. The considerations above have also led to the following equality

kG(zo, Fo(a)) = tanh-'(xc(zo;a)),

a E EA.

(10.6.2)

Therefore, Fo is a proper holomorphic mapping from 3 to G. So it is surjective. According to the general properties of proper holomorphic maps, we know that

if M:= (v EBB: detFF(v)=0),then F: go \Fo'(Fo(M))-+ G\Fo(M) is a proper covering map and that Fo(M) is a proper analytic subset of G. On the other hand, (10.6.2) shows that F0 maps xG(zo; )..balls into kc, (zo, )-balls. Thus, generically, the fibres of Fo near zo consist of one element only, which implies 0 that Fo is globally injective.

Remark 10.6.8. The proposition above remains true under slightly weaker assumptions; cf. [Sta 2]. In this context we should also mention the recent work by M. Abate and G. Patrizio [Aba-Pat 1]. The next steps toward the proof of Theorem 10.6.1 consist in successive establishing the assumptions of Proposition 10.6.7.

Lemma 10.6.9. Let G, D, q, wo, and F j be as in Theorem 10.6.1 and (10.6.1). Then G is a pseudoconvex domain.

349

10.6 Characterization of the unit ball by its automorphism group

Proof. Obviously, it suffices to exhaust G by a sequence of pseudoconvex domains. By assumption, the domain D is strongly pseudoconvex near wo. Therefore, we D fl V is biholomorphic to find a neighborhood V = V(w0) of wo such that b a convex domain. Now, we are going to construct sequences (Dk)kWN and (Gk)kEN of subdomains Dk of b and Gk of G, respectively, and a subsequence (Fj,)kEN of (Fj)jEN such that

Fj, (q) E Dk CC Dk+, , Dk is biholomorphic to a convex domain, U Dk = D; k=,

(10.6.3)

x q c- Gk C Gk+1, UGk = G. and Fj, IG, : Gk -+ Dk is biholomorphic. k=I

(10.6.4)

We only indicate the first step of this construction. Without loss of generality we may assume that F, (q) E D. Then we fix an arbitrary subdomain D, CC D with F, (q) E D, such that D, is biholomorphic to a convex domain. We denote by G, the connected component of F, 1(D,) that contains the point q. Obviously. G, is a relatively compact subdomain of G. Moreover, since D, is simply connected and since F, is a covering map. F, IG, is a biholomorphic map between G, and D,: in particular, G, is pseudoconvex. For V E N and for a sufficiently small positive number a we put

G` := (z E G: dist(z, 8G) > a/v} CC G, D,' :_ (w E D: dist(w, aD) > a/v) CC D.

(10.6.5)

(10.6.6)

Then we fix an index j2 > j, := 1 such that Fj_(G, U Gi) CC D. After that we choose a domain D2 CC Din such a way that D, U D*u Fj, (G, U G i) CC D2 and that D2 is biholomorphic to a convex domain. Denote by G2 the connected component of F1.1 (D2) containing q. Then G2 is a relatively compact subdomain of G with G, C G2. Moreover, FJ21(;, -). D2 is biholomorphic and G2 is pseudoconvex. The continuation of this procedure then results in sequences whose existence was postulated.

Now, we turn to the comparison between the Kobayashi-Royden and the Carathdodory-Reiffen metrics on G x C". Lemma 10.6.10. Under the assumptions of Theorem 10.6.1 we have YG = xG on G x C".

350

X. Comparison on strongly pseudoconvex domains

Proof We will use the sequences (DA)AEN, (GA)keN, and (Fj,)AEN from the proof

of Lemma 10.6.9; cf. (10.6.3) and (10.6.4). If (z. X) E G x C", then applying Lempert's Theorem we conclude that

YG(z;X)=Alien yc;A(z;X)=Alien yij,(Fj,(z);Fi,(z)X)

= lien x& (Fj,(z);FjA(z)X) = lim xc;,(z;X) A

ac

A x>

xG(z;X)?Yc;(:.;X

i.e.yc;=xGonGxC". We recall that the Bergman metric is not holomorphically contractible. Nevertheless, the following statement is true. Lemma 10.6.11. If G C C C" is the union of an increasing sequence of subdomains GA, then PG = lim&-x PGA.

Proof We already know (cf. Theorem 6.1.15) that lim&., KG, = KG. It remains to prove that limA-.,o MG, = MG; cf. Theorem 6.2.5 and Remark 6.2.7. But this convergence is a simple consequence of Montel's argument. The details are left to the reader. Finally, we are in a position to complete the proof of Theorem 10.6.1.

Proof of Theorem 10.6.1. First, we recall that G is a pseudoconvex domain with smooth Cl-boundary for which yG = xG holds. To apply Proposition 10.6.7 it

suffices to verify that xc; _ fG/ n + 1 on G x V. Now fix z E G and X E (0), Moreover, choose a strongly pseudoconvex

domain b C b in such a way that D f1 U C D. where U = U(vu) CC V is a sufficiently small neighborhood of w,) and such that b is biholomorphically equivalent to a convex domain. Here D and V are taken from the proof of Lemma 10.6.9.

We denote by G', the domains of (10.6.5) and by b,* the domains corresponding to (10.6.6) defined now with respect to b instead of D. We are going to slightly modify the construction of the sequences (GA)Ac-N, (DA )AEN, and (Fj, )AEN; cf. Lemma 10.6.9.

10.6 Characterization of the unit ball by its automorphism group

C GR CC G. Di C

Assume that we have constructed subdomains Gt C C DA CC D. and mappings (F,,)A,-t satisfying

(a)

zEG; CG,.+i.

(b)

Fj, 1i:.: G,.

I-

(c)

1

1
(;)X)

<

3v

351

< I +2v- .

3D.(FJ,(`);F,,(z)X) -

--

1 < v
where S=x. y, or #. Next. we choose an index jj_' > jA such that FJ,-, (GA U GA) CC D n U. Moreover. we take a domain DA+i CC D with DA CC DA+t in such a way that DA+i D D,' U Fj,_, (GA U GA. ). that DA+t is biholomorphic to a convex domain. and (c) becomes true for v = k + 1. The further construction goes as before. Then we have the following chain of equalities:

fa(::X) = Yc(c:X) 0 lim

= lim

X) = lim fi (Fj,(:):FF,(:)X) Ye',(z;X) A--- yn,(Fi,(z):F),(z.)X)

Iim

A

(Fi,(z):Fj',(:)X) Y,)(Fj(:):FF,(z)X) lim lim X F, ' F F F X A

A

8i)(Fj,(z);Fj,(z)X)

=

A-1 yc)(F,, (z): F1, (--) X) (.)

F

F'

X

rt+1.

where (*) is a consequence of Theorems 10.4.2 and 10.4.6. Since z and X are 0 arbitrary, we find drawing on Lemma 10.6.10 that yu = xG = fc;/ n + I.

Remark 10.6.12. According to a result due to S. Pinchuk (cf. [Pin[) it is known that any proper holomorphic mapping between strongly pseudoconvex domains in C" is unbranched. Therefore, the formulation of Theorem 10.6.1 becomes very simple if G and D are assumed to be strongly pseudoconvex domains.

352

X. Comparison on strongly pseudoconvex domains

Notes

Most of the results in this chapter are based on the existence of precise solutions of the 5-equation on strongly pseudoconvex domains; cf. for instance, [Hen-Lei] and [Ran 21 for detailed information. The results on the boundary behavior of the Carathe odory and Kobayashi distances are mainly taken from the work of M. Abate [Aba I J and F. Forstneri & J.-P. Rosay [For-Ros]. They lead to a weak form of the beautiful extension theorem of C. Fefferman [FefJ; see also [Bel-Ligi. More general results are also true for proper holomorphic mappings. The book of K. Diederich and I. Lieb [Die-Lie] may serve as a source of further information; see also [For]. The boundary behavior of the Carathe odory-Reiffen (resp. the Kobayashi-Royden) metric was studied by 1. Graham [Gra 11, G. Aladro [Ala], and recently by D. Ma [Ma 1,2,31. In Chapter 10 we tried to follow the estimates given by D. Ma. The analogous results for the Bergman metric are due to L. HOrmander [Hdr I J and K. Diederich [Die 1,21. In the case of domains that are of finite type the boundary behavior of the metrics is studied by D. Catlin [Cats; see also [Her 41. The characterization of the unit ball by its automorphism group was initiated by B. Wong [Won 1 ] and J: P. Rosay [Ros I]. The general formulation given here is due to E. B. Lin and B. Wong [Lin-Won]. Recently, much effort is done to generalize this result by substituting the unit ball by complex ellipsoids as model domains; e.g. see [Kod-Kra-Maj.

351

Exercises 10.1 (cf. [Ma 3]). Define G := {z E C2: - 2Re(z1 +:1:2/2) + Izi + ziz2/212 + Iz212 < 0).

(a) Using (zi, z2) -# (zi - I + zIZ2/2, z2) prove that G is biholomorphic to 132.

(b) For z(t) := (t,0) and X(t) := (,17, 1). where t a small positive number, calculate the Kobayashi-Royden metric as

XG(z(t);X(t))=

3t+1;1=-3t2/4 21-t2

(c) Compare this formula with the estimate stated in Remark 10.4.4.

10.2 (cf. [Kra 2]). For 3/4 < t < 1. in := 1/(2 - 2t). put G, := {z E C2: 1 < Izi 12 + Iz21' < 4).

Note that G, is a domain with smooth C2-boundary. Prove that there exists C > 1 such that for points z(3) := (-1 - S. 0) E G, and X (1, 0) (S small) the following inequalities are true:

(1/C)dist(z(S), 8G,)-' < x(;,(z(S); X) < Cdist(z(S), aG,)-'.

(*)

Compare (*) with Theorem 10.4.2.

Hint. Use the analytic disc (P E 0(E, G,), 4(A) := (-1 - S + (S'/l0)A, )'2), for the estimate from above. Observe that for any V = (VI, 402) E Q(E, G,), 4(0) = z(S), aV'(0) = X (a > 0). the function g(A) A E E, has its values in the annulus (I E C: 1 - S < IXI < 41.

10.3. Let G be any bounded domain in C2 with smooth C2-boundary. Show that there are positive numbers Eo and C such that whenever z E aG and t E (0, so), then z-tv(z) E G and xG(z-tv(z); v(z)) > Ct" 31x, where, as usual, v(z) denotes the unit outer normal of G at z.

Hint. Use the shells B(z + eov(z), R) \ B(z + eov(z). co) D G as comparison domains.

354

X. Comparison on strongly pseudoconvex domains

10.4 (cf. [Ran 1]). Let G :_ (z E C3: IzlI2 + Iz212 + Iz314 < 1)

and put X := (0, 1, 0), X* := (0, 0, 1). Prove that yo(z; X) and yc(z; X*)

disc(-, aG)-'12 dist(z, aG)-1 14 if z __> (1, 0, 0) along the inner normal to aG at

(1,0,0). 10.5 (cf. [Die-Lie]). Let F: Gi -* G2 be a proper holomorphic mapping between bounded domains with smooth C2-boundaries in C". For S = y or S = x assume that

(a) there exist C, e > 0 such that dist(F(z), aG2) < Cdist(z, aG, )s, z E GI; (b) there is q > 0 with So2(w; X) > CIIXII dist(w, aG2)-", w E G2, X E C". Prove that F extends continuously to 61.

Miscellanea

As we have already mentioned in the Preface, we collect here various topics which belong to the theory, but which are somehow outside the main stream of the book. We report (without proofs) on the following problems: A: Lie structure of Aut(G), B: holomorphic curvature,

C: fixed points of holomorphic mappings and boundary regularity of complex geodesics, D: criteria for biholomorphicity, E: boundary behavior of contractible metrics in weakly pseudoconvex domains.

A The automorphism group of bounded domains Let G be a bounded domain in C". Then the Carathe odory distance CG defines the standard topology of G. If the automorphism group Aut(G) of G is provided with the compact-open topology, then Aut(G) carries the structure of a real Lie group. This result is due to H. Cartan (cf. [Cart 11). Here we discuss a method of proof which uses invariant distances (cf. [Kob 41). We define

lso, (G) := 10: G -+ G : ¢ is a bijective CG-isometry). Then, according to a result of van Dantzig and van der Waerden (cf. [Dan-Wae]), Iso,(G) provided with the compact-open topology is a locally compact group. Moreover, Aut(G) is an effective transformation group of G, i.e.

4: Aut(G) x G

G, 4(g, z) := g(z),

g E Aut(G). Z E G

is continuous with (i) (ii)

(iii)

4'(g: 'D(g1, Z)) = 40(92 a gi, z), g f E Aut(G), z E G; 4;(idG, Z) = z. Z E G; idG is the only group element satisfying (ii).

Miscellanea

356

Obviously, c(g, ) describes a CZ-transformation of G. Therefore, we can apply the following theorem of S. Bochner and D. Montgomery (cf. [Boc-Mon], [Mon-Zip]).

Theorem A.I. Let G be a locally compact effective transformation group of a connected C'-manifold M (i.e. 4: G x M -+ M as above) and let each transformation 'F(g, ): M --> M be of class C'. Then g is a real Lie group. Summarizing we obtain

Theorem A.2. The autontorphism group Aut(G) of any bounded domain G in C" is cr real Lie group.

We only mention that the Lie-structure of Aut(G) is often used to study the biholomorphical equivalence problem for classes of domains in C".

B Holomorphic curvature A thorough study of the Ahlfors-Schwarz Lemma in Chapter I leads to the conclusion that this result can be thought of as one from differential geometry, where the notion of curvature is mainly involved. To be able to deal with metrics, which are only upper semicontinuous, we first introduce the notion of the generalized

(lower) Laplacian. Let G c C be an open set and let u: G -- R be an upper semicontinuous function. Then

/

(su)(A) := 4 lim inf ' 12I

Jo

u(A + re'N)dO - u(A)) E [-oo, 001

is called the generalized Laplacian of it. If a is of class C2. then Du coincides with the standard Laplace operator. Moreover, if a general u takes a local maximum at Xo E G, then, as usual, (Du)(Ao) < 0. By means of this generalized Laplacian we introduce the notion of holomorphic curvature (cf. [Suz 1], (Won 2]). Namely, let G be a domain in C" and let Sc : G x C" -* [0, oc) denote an upper semicontinuous pseudometric on G. Then

the holomorphic curvature of SG at (Zo; X) E G x (V). with Sa (cc; X) > 0 is defined as the number h-curv(zo; X; SG)

sup

([alogS (gp;(p'))(0)

f

-2S3(zo, X)

. r > 0, V E 0(rE, G), v(0) _ ;.c, (p'(0) = X

II t-tolomorphic curvature

357

Obviously, if SG (zo; X) > 0, then

h-curv(zo;X;SG) = h-curv(zo;X/IIXIIe'O;SG),

GE R,

i.e. the holomorphic curvature only depends on the complex direction of X. Moreover, the holomorphic curvature is a biholomorphic invariant when it is considered with respect to holomorphically contractible metrics. For the Caratheodory-Reiffen and the Kobayashi-Royden metrics the following estimates are true (cf. [Suz I], [Won 2]).

Theorem B.1. (a) For any y-hyperbolic domain G C C" we have the following inequality: h-curv(.; ; yG) < -4 on G x (C"),,. xG) > -4 on G x (C").. (b) If G denotes a x-hyperbolic domain, then As a consequence of the Lempert Theorem we obtain the following corollary.

Corollary B.2. For a bounded convex domain G in C" the holomorphic curvature with respect to yG and xG is identically equal to -4. Now, we rewrite the Ahlfors-Schwarz Lemma of Chapter I in terms of holomorphic curvature.

Theorem B.3. Let G be a domain in C" and let SG be an upper semicontinuous metric on G (recall that SG (z; ,kX) = I A I SG (z; X) and SG (z; X) > 0 if X

0). If we

assume that h-curv(.; ; SG) < -c2 < 0 on G x (C" )., then the following inequality holds: SG(Z; X) < (2/c)xG(z; X),

Z E G, X E (C")..

A proof of this theorem can be found, for example, in [Din]. Theorem B.3 gives a tool, at least theoretically, to find comparison results for general metrics and the Kobayashi-Royden metric. Since it seems to be very difficult to estimate the holomorphic curvature in general, we restrict ourselves to the case of Hermitian metrics, in particular, to the Bergman metric.

Lemma B.4. Let G C C" and let SG(z;X) = (Eij=I gij(z)XiXj]1122 be a Hermitian metric on G with gij E C2(G) (z E G, X E C"). Then the holomorphic curvature of SG at (Zo; X) E G x (V). coincides with the holomorphic sectional curvature at zo in the direction of X, i.e. n

h-curv(zo;X;SG) = -2

Rijkl(z)XiXjXkXI/SG(ZO;X), i. j.k.l =1

(B.1)

Miscellanea

358

where

Rij,u(z) : =

ag1

(z) azk a,

n

a.6=

$i (z)g°` a

z)

a$p!

au

)

with (ga1(z)(g,j (z)) being the unit matrix.

Formula (B.1) can be found in [Won 2) and [Wu 2]; for a more explicit calculation compare [Aba-Pat 21. In the case of Thullen domains (complex ellipsoids) K. Azukawa and M. Suzuki

(cf. [Azu 2], [Azu-Suz]) found upper and lower estimates for the holomorphic sectional curvature of the Bergman metric, and therefore for the holomorphic curvature.

Theorem B.S. Let De := 6(l, 1/p) = {z E C2: Izil2 +

1), p > 0, and

Do := {z E C2: Izi I < 1, Iz2I < 1). Then the following inequalities are true:

(a)for0
(1 +2p)2

< 1 .h-curvz,X;PG) < 2

(

_2(2+llp+15p2+8p3) (2+ p)(1 +3p)(4+5p)

(b) for p > 1:

2(2+llp+15p2+8p3) (2+p)(1+3p)(4+5p)

< 1 .h-curv(z;X;Pc):5 _ 2

2 2+p

Remark B.6. The above theorem shows that for all D. p > 0, the holomorphic curvature of the Bergman metric is bounded from above by a negative constant. But, in general, such an estimate does not remain true for all complex ellipsoids. For example we have limp,o supX O h-curv(O; X ; &,(p, p)) = 4, where £(p, p) _ (z E C2: Izt I2P + Iz212p < 1) (cf. [Azu 2]).

Applying Theorems B.3 and B.5 we obtain a comparison between the Bergman metric and the Kobayashi-Royden metric on Thullen domains Dp. As Remark B.6 shows, Theorem B.3 cannot be used to get comparison for general complex ellipsoids. Nevertheless, using the localization for the Bergman and the KobayashiRoyden metric, one is led to the following result (cf. [Hah-Pfl 2]).

Theorem B.7. For p > 0, q > 0 there exists a positive constant Cp,q such that !3e(p,q)(Z X) < Cp,q) E(p.q)(Z; X ). Z E C(p, q), X E C.

C Complex geodesics

359

0 We do not know whether such a comparison is true for all pseudoconvex balanced Reinhardt domain in C2. Moreover, it seems to be open whether Theorem B.7 remains true in higher dimensions. 0 Evaluating Theorem B.5 in the case p = 1, i.e. D, = B2, gives

-4/(n + 1). -4/3. In general, it is easily seen that The following result of P. Klembeck (cf. [Kle]) again illustrates the affinity of strongly pseudoconvex domains to the ball.

Theorem B.8. Let G C C" be a strongly pseudoconvex domain with C'-boundary. Then, near the boundary of G, the holomorphic curvature of the Bergman metric

approaches the constant value -4/(n + 1) of the holomorphic curvature of the ball B".

The proof of Theorem B.8 relies on the asymptotic formula for the Bergman kernel due to C. Fefferman (cf. [Fef]).

C Complex geodesics Lempert's Theorem (Theorem 8.2.1) is a powerful tool of complex analysis on convex domains. There are various applications of this result. For example, we have

Theorem C.l (cf. [Vig 4]). Let G C C" be bounded convex domain and let M be an analytic subset of G. Then the following conditions are equivalent: (i) there is a holomorphic mapping F: G -+ G such that M = {z E G: F(z) = z); (ii) there exists a holomorphic retraction r : G - M. The proof of Theorem 8.2.1 we present in the book is based on the ideas taken from [Roy-Won]. Our proof is much more simpler and elementary than the original proof by Lempert (cf. [Lem 1,2]). On the other hand, Lempert's results on regularity

of complex geodesics are deeper and his methods may be also applied to more general situations. For instance, we have Theorem C.2 (cf. [Lem I]). Let G C C" be a strongly convex domain with CAboundary, where 3 < k < w. Then (a) any complex geodesic cp: E -+ G extends to a CA-2-mapping on E; (b) if h is a mapping as in (8.2.19), then h extends to a CA-2-mapping on E; (c) if k > 6, then kG is a CA-4 -function on G x G \ diagonal.

In [Lem 3] L. Lempert generalized his results from [Lem 1,2] to the case of strongly linearly convex domains (where our methods do not work). Abounded

Miscellanea

360

domain G C C" given by a defining function r E C2 is called strongly linearly convex if n

(cr)(a; X) > I

a

p.v=i

a

2

az' (a)X,X,I,

aEaG.X=(X,,...,X")E(C")swith

azo

(a)X,=0.

Example C.3. Any strongly convex domain is obviously strongly linearly convex but not vice versa. For we put

G := [z = (zi,

..., zn) E C": IIz112 + (Re(z' ))2 < 11.

Then G is a strongly linearly convex domain (with C'O-boundary) but not convex. We do not know whether G is biholomorphic to a convex domain.

0

0

Theorem C.4 (cf. [Lem 3]). Let G C C" be a strongly linearly convex domain with Ck-boundary, where k E loo, w}. Then: (a) CG = kG = kc. YG = xc,

(b) complex cc-geodesics are uniquely determined modulo Aut(E), (c) complex yc-geodesics are uniquely determined modulo Aut(E), (d) any complex geodesic extends as a Ck-mapping to E.

Recently, new approaches to complex geodesics were proposed by J. Agler ([Agl]) and Z. Slodkowski ([Slo]). For instance, using the methods of dilatation theory Agler proved the following generalization of Theorem 8.2.1. Theorem C.5 (cf. [Agl]). Let G C C" be a bounded domain and let zo, zo be points in G, zo 0 za. Then there exist a holomorphic mapping (p: E --> cony G and points Ao, X"0 E E such that zo = v(Aa). zo = tp(Ao), and cc(4, zo) = P(k', 0I0 ")-

In the meantime C. H. Chang, M. C. Hu, and H.-P. Lee studied complex geodesics for points in the closure of a domain. Let G be a taut domain in C" such that cc = kG (and yc = xc; cf. Proposition 8.1.5). Let 4. zo' E G, zo 0 zo'. We say that a complex geodesic gyp: E -- G is a complex geodesic for (z4, zo') if (p extends to a continuous mapping on k with zo, z4' E (p(E). Similarly, if zo E aG

and X0 E (C"), then we say that a complex geodesic rp : E -+ G is a complex geodesic for (zo, Xo) if cp extends to a CI-mapping on t such that there exist Ao E aE and ao E C with zo = V(o) and aov'(Ao) = Xo. Theorem C.6 (cf. [Cha-Hu-Lee]). Let G C C" be a strongly linearly convex domain with Ck -boundary.

D Criteria for biholomorphicity

361

(a) If k > 3, then for any points zd. zo E G with z' # zo there exists exactly one (modulo Aut(E)) complex geodesic for (zo, zp).

(b) If k > 14, then for any zo E 8G and for any Xo E C" with iXo E TZ(aG), (Xc, v(zo)) > 0, where v(zo) denotes the outer normal vector to 8G at zo, there exists exactly one (modulo Aut(E)) complex geodesic for (zo, Xo).

If G is not strongly linearly convex, then the above theorem is not longer true (even if G is strictly convex).

Theorem C.7 (cf. [Cha-Lee]). Let E = £(p) be a complex ellipsoid in C" with p = (P1. ... , p") E N". Suppose that £ is not strongly convex (i.e. p 9 (1, ... , 1); c f . Remark 8.4.1). F i x z o := (1, 0, ... , 0) E M.

(a) Let 4 _ (l;i,

.

. .

, ,,) E a£ \ (zo}. Suppose that either p, = I or j E {0} U aE.

Then there exists a complex geodesic for (z , 4). (b) There exists a function M : [0, 1) -* R,o such that if z0" I), then (z . z) admits a complex geodesic. 2:1=2 I jI2p; <

(c) If p, > 1, then there exists a point z E a£ (resp.

E £) such that

E £ and zp )

admits at least two non-equivalent complex geodesics.

D Criteria for biholomorphicity Criteria for biholomorphicity (cf. §8.7) were studied by several authors. There are two main streams of problems.

I° We are given two domains G1, G2 C C" and a holomorphic mapping F: GI -+ G2 such that F is a 8-isometry at a point a E G,, where 8 E (y, x). We would like to decide under what conditions F is biholomorphic; cf. Proposition 8.7.2.

21 We are given a domain G C C" and a point a E G such that yG(a; ) _ xG(a; .). We would like to decide whether G is biholomorphic to the indicatrix 1G (a) := (X E V: xG(a; X) < 1); cf. Proposition 10.6.7. Note that IG (a) is biholomorphic to B,, if the mapping C" 9 X -> xG (a; X) E R+ is a Hermitian form. In direction 1° we mention, for instance, the following result.

Theorem D.1 (cf. [Bell). Let G C C" be a bounded c-complete domain and fix

a E G. Let q: C" --* R+ be a C-norm and let B := (z E C": q(z) < 1). Suppose that F : G -* B is a holomorphic mapping such that xB (F (a ); F' (a) X) _ xG (a; X), X E C". Then F is biholomorphic.

362

Miscellanea

Problems of direction 2" are more difficult than those of I". We would like to mention the following results (in the chronological order). Theorem D.2 (cf. [Sta 11). Let G C C" be a c-finitely compact domain. Suppose

that yG (a; ) = xG (a; ) for a point a E G and (X E C": yd; (a; X) < 1

E.

Then the domain G is biholomorphic to E".

Theorem D.3 (cf. (Sta 2]). Let G C C" be a complete hyperbolic domain. Suppose

that yG (a; ) = xG (a; ) for a point a E G and suppose that yG or XG is a CO° Hermitian metric on G x V. Then G is biholomorphic to B".

Theorem D.4 (cf. [Pat]). Let G C C" be a strongly convex domain with C'°bou ndary. Then the following conditions are equivalent: (i) G is biholomorphic to B"; (ii) there exists a E G such that the function G 9 z -->

is of class C'°; (iii) there exists a E G and its neighborhood U such that xc; is a C'° Hermitian metric on U x C". We say that R is a real ellipsoid in C" if, after a C-linear change of coordinates, 7Z may be written as

R= ((:i.....<,,) EC":

, (Iz;I'+AjRe(zj)) < I]. i=1

where 0 < Ai < 1. j = 1..... n.

Theorem D.S (cf. [Kay]). Let ?Z C C" be a real ellipsoid. Then R is biholomorphic to B,, iff xR (0; ) is a Hermitian form.

Theorem D.6 (cf. [Aba-Pat 11). Let G C C" he a taut domain. Suppose that a E G is such that:

- yG(a. ) = xG(a; ) - there exists a neighborhood U of the point a such that xG is a C" function on U X (C" \ (Q}).

- the indicatrix I (a) is strongly there exist 0 < ri < r2 and a biholomorphic mapping 4) : Bx,; (a. r,) - Bc,; (a, r2) such that 4)(a) = a. Then G is biholomorphic to XG (a).

Theorem D.7 (cf. [Aba-Pat 11). Let G C C" be a taut domain. Then G is hiholomorphic to B. if there exists a point a E G such that y(, (a; ) = xG (a; ) and xG is a C'° Hermitian metric.

E Boundary behavior of contractible metrics on weakly pseudoconvex domains

363

Theorem D.8 (cf. [Tis]). Let G C C" be a strongly pseudoconvex domain with simply connected boundary. Then the following implications hold.

(a) If aG is C°° smooth and xr is a Hermitian metric in a neighborhood of 8G. then G is biholomorphic to B,,. (b) If aG is C2 smooth and xr is a C" Hermitian metric in a neighborhood of 8G, then G is biholomorphic to B. (c) If aG is smooth real analytic and xc, is a Hermitian metric in a neighborhood of a boundary point, then G is biholomorphic to On.

E Boundary behavior of contractible metrics on weakly pseudoconvex domains In Chapter X we discussed estimates from below and above for the Bergman, Carathdodory, and Kobayashi metrics near the boundary of strongly pseudoconvex domains. For various purposes it is worthwhile to know similar estimates also on pseudoconvex domains that are not strongly pseudoconvex. In dimension two the following result is due to E. Bedford and J.-E. Fornaess.

Theorem E.1(cf. [Bed-For 2]). Let G be a bounded pseudoconvex domain in C2 with smooth real analytic boundary. Then there exist C > 0 and 0 < e < 1 such that YG(z: X) > C11 X II(dist(z. aG))-F.

z E G, X E C2.

The necessary information for this inequality is taken from the construction of peak functions (cf. [Bed-For I ]). In the case where, in addition, G is convex the following result due to M. Range is true even in arbitrary dimensions. Theorem E.2 (cf. [Ran I ]). Let G be a bounded convex domain in C" with smooth real analytic boundary. Assume that G is given by G = [z E C" : r(z) < 0 } with a defining function r. Then there are positive numbers C and e < I such that

yG(z;X) > C(

IIXII

+ IE"_, (ar/az,,)(z)XI).

dtst(z. aG)8

dist(z, aG)

Z E G, X E C".

We only mention that the exponent a in Theorem E.2 is somehow related to the order of contact certain supporting analytic hypersurfaces have with 0G. It is still open whether the estimates in Theorem E.1 remain true in higher dimensions.

0

Miscellanea

364

For arbitrary dimensions a lower estimate for the Kobayashi-Royden metric has been found by K. Diederich and J.-E. Fornaess.

Theorem E3 (cf. [Die-For 1 ]). If G is a pseudoconvex domain in C" with smooth real analytic boundary, then for suitable C > 0 and s > 0 the following inequality is true

xG(z; X) > C dist(z, aG)-E,

Z E G, X E C".

Details can be also found in [Die-Lie]. As a consequence (cf. Exercise 10.5) one can prove that any proper holomorphic mapping F : G, -+ G2 is Htllder continuous on G1, where G, CC C" is a pseudoconvex domain with smooth C2-boundary and G2 CC C" is pseudoconvex with smooth real analytic boundary. Now, we come back to C2. We assume that a bounded pseudoconvex domain

G C C2 is given by G = {z E U fl G : r(z) < 0), where U = U(aG) and r E CI(U, R) with dr(z) # 0, z E U. A boundary point Co of G is said to be of finite type m if

m=sup{ordoroV: 4PEO(E,C"),9(0)=Co,V'(0)#0]. For more information about points of finite type see [Blo-Gra] and [Koh]. For simplicity, we will assume that Co = 0 and (ar/az2)(0) 0 0. Then for z near Co we put

Li(z):= (1,

ar (z)/ar (z)),

(*)

L2(z) :_ (0, 1).

Obviously, L, (z) and L2(z) form a basis of C2. Moreover, for j. k E N we set (Gj.k)(Z) a ar

azl - (azj

ar a a (z)l FZ2 (z)) lj-1I

az2az,

ar

- (az,

ar a (z)/ az2az2 (z))

]k-'

(Gr)(z; L i (z))

Observe that the assumption on the type implies that

if j + k < m,

(Cj,k)(0) = 0 54 0

for at least one pair (jo, ko) with jo + ko = m.

By means of the U k's, for I E N and z near 0 we define C,(z) := max{I(Gj,k)(z)I : j + k = I).

E Boundary behavior of contractible metrics on weakly pseudoconvex domain,

365

If now X is an arbitrary vector in C2 and if z is near 0, we have the following unique representation of X: X =: Xl(z)Ll(z) + X2(z)L2(z). With this notion in mind we finally define M

MM(z; X) := IX2(z)IIr(z)I-I + IXI (z)I > ICr(z)/r(z)11"'. !_2

After these preparations we can describe the size of the metrics in the following "small constant-large constant" sense. These estimates have been found by D. Catlin.

Theorem E.4 (cf. [Cat]). Let G be a pseudoconver domain with C°°-bouiulary. Assume that o is a boundary point of G of finite type m (we may suppose that (*) is satisfied). Then there exist a neighborhood U = U (to) and positive constants c and C such that for all z E G n U and all X = X, (z)Ll (z) + X2(z)L2(z) E C22 we have

CM.(z;X) <SG(Z:X)
We mention that if G is a bounded convex domain with smooth C^`-boundary

in C, n arbitrary, and if o E dG is a boundary point of a suitable finite linear type, then a similar "small constant-large constant" estimate has been established by J.-H. Chen [Che]. But, in general, the boundary behavior of contractible metrics on weakly pseudoconvex domains in C", n > 2, seems to be unknown except for special cases. We conclude this section with a recent result of G. Herbort into this direction. Let G be a domain in C", n > 2, which is given by

G:=(z=(z,, z)EC"=C' xC"-' :Rez,+P(;,) <0), where P is a real-valued psh polynomial on C"-' without pluriharmonic terms. We say that G is of homogeneous finite diagonal type if:

(a) there exist mj E N, 2 < j < n, such that P(t'/2i:Z2,

,

tl/2m"

4,) = tP(z),

Z = (Z2, ... Z,+) E

C"-', t > 0;

(b) P(z) - 2s J'j=2 Iz.;I`'", is psh on C"-' for a suitable s > 0.

Miscellanea

366

To be able to formulate the final result, the following auxiliary functions are needed: r

A,.j(z) := max(I 8za

8z!

2 < j < n, 2 < 1 < 2m j;

(z)I : v, µ E N, v + it = 11, 2mj

Ar

j C1(z) := f (_r(z))t/r,

2<j
r=2

MG(z;X) := I, az (z)X/VIr(z)I +ECj(z)IX3I, !=1

Z E G, X

E C".

j=2

'

Under the additional hypothesis that any term of P contains at most two of the . , z,,, the following comparison between MM, yc , and xG is true.

variables Z2,

Theorem E.5 (cf. [Her 4,5]). Suppose that G :_ (z E C" : Re z, + P(z) < 0) is a domain of homogeneous finite diagonal type and suppose that P has the form

P(z2,... , z") = F,Pj(zj) + E Pjk(zi, zk) j=2

,

2<j
where Pj, Pjk are real-valued polynomials and where for j c k we have

a-

Pjk(zj,zk) _

ZkrZ-ks

capyBZ7

.

a+ti>_ 1, y+b> I

Then there exist positive constants c and C such that

cMG(z; X) <_ S0(z; X) < CM;(z; X),

z E G, X E C",

where Sc = yc, SC = xc, or 8c = $c A particular case of Theorem E.5 is contained in [McN].

Appendix

HF Holomorphic functions References: [Chi], [Con], [Gun-Ros], [Kra 1], [Ran 2]. Since the whole book is based on the classical Schwarz Lemma, this lemma should be given first.

HF 1 (Classical Schwarz Lemma). Any f E O(E, E) with f (0) = 0 satisfies If (A)I < IAI for all A E E and I f'(0) I < 1. Moreover, if If (Ao)I = IAoI for at least

one Ae E E. or if I f'(0) I = 1, then f is just a rotation, i.e. f (A) = e'"A, A E E, where 0 is a suitable real number.

HF 2 (Classical Schwarz-Pick Lemma). If f E O(E. E), then A' - A" 1 -X"A' I

f (A,) 1-

If {

I

1

1 - IAIz'

1- If P-111'

'

1',X" E E.

(HF.1)

A"

(HF.2)

E E.

Moreover, if there are A', A" E E. A' 34 A", (resp. A" E E) such that equality holds in (HF.1) (resp. in (HF.2)), then f is of the form

f (A) = e'"

A

with a suitable 0 E R.

Recall that the proof of HF I is based on the maximum principle and the theorem on removable singularities. Therefore, the Schwarz Lemma remains true in more general situations; for example, compare PSH 22.

HF 3 (Fatou). Let f be a bounded holomorphic function on E. Then for almost all E 8E the function f has a non-tangential limit at 2;, i.e. the limit

f`W = lim f(A) = k-C a

f(A)

Appendix

368

exists and is independent of a > 1, where

{A E E: IA -1; J < a(I - Al)). I

HF 4 (Little Picard Theorem). Except the constant function there is no entire holomorphic function f E 0(C) omitting two different complex numbers as values.

HF 5. Any proper holomorphic map f E 0(P, P), where P := {x E C: 1/R < Al I< R}. R > I. is biholomorphic.

For a proof the reader is referred to Theorem 14.22 in [Rud 11 and asked for the necessary modifications of the proof there. HF 6 (Theorem of Bohr, cf. [Boh]). Any continuous injective function on a domain G C C is holomorphic or antiholomorphic if for every zo E G the following limit exists: lim f (z) - f (zo) E (0. oo).

Observe that the assumption "injective" is important here as the following example shows: f(~(A

Sl l

if lm'X >0 if Im'l < 0

The two papers [Gra-Mor] and [Trot may serve as a source of helpful information about HF 6 and similar questions.

A holomorphic covering is a holomorphic map n : MI -+ M2 between two connected complex manifolds satisfying the following property:

any point wo E M2 has a neighborhood U = U(wo) such that ,r (U) is the union of pairwise disjoint open subsets V; of MI with rr l v, : V; -a U being biholomorphic, i E 1; Jr is called the projection map.

HF 7 (Uniformization Theorem). Let G be an arbitrary domain in C. Then there exists a holomorphic covering 7r: M -+ G, where M is either C or E. such that for any other holomorphic covering n : Al - G and for any points z' E M, z' E Al with 2r(z') = there is a unique holomorphic map f : M -+ M satisfying

ito f =stand f(z')_:'.

Recall that it : M -> G is uniquely defined up to a biholomorphic mapping commuting with its projections; rr : M

G is called the universal covering of G.

369

HF Holomorphic functions

In short, the above result says that any domain G in the plane has E or C as its universal covering.

So far we have collected the main results from the classical complex analysis of the plane which were applied in this book. Now, we discuss a few results from several complex variables. HF 8 (Hartogs' Theorem). A function f : G --> C, where G is a domain in C", is holomorphic if it is holomorphic in each variable separately, i.e. if z° E G and I <

j < n are fixed, then the function f (z°, ..., z°_1, , z°+1.... , z") is holomorphic on the open set (A E C : (z°, ... ,

Z9

zn) E G).

HF 9 (Montel). Any locally bounded family .F of holomorphic functions on a domain C .F has a subsequence (f,,, )t EN that G C C" is normal, i.e. any sequence (f,

converges locally uniformly to an f E O(G). HF 10 (Holomorphic extension). Let G be a domain in C".

(a) If K C G is a compact subset of G such that G \ K is connected, then any f E O(G \ K) extends to an f E O(G), i.e. fIG\K = .fIG\K (b) If F C G is a closed (with respect to G) subset of G with H2"-2(F) = 0, then any f E O(G \ F) has a holomorphic extension to G.

(c) If F C G is as in (b) and Hen-'(F) = 0, then every f E HOD(G \ F) is holomorphically continuable to the whole G.

In particular, we recall that any proper analytic subset of G satisfies the condition in (c). Therefore, the classical Riemann Removable Singularity Theorem is a particular case of (c). HF 11. Let G and G be domains in C" such that any bounded holomorphic function

on G extends to a holomorphic function f on G with fIG = f IGo, where G° is a fixed non-empty open subset of G fl G. Then II f III

II f 11 G

HF 12. Any distribution T (in the sense of L. Schwartz) on G C C" with aT = 0 is given by a holomorphic function f, i.e. < f, (p > = < T, V > for all test functions

tpECO(G). We denote by Lt2,(G) the set of all square integrable holomorphic functions on a domain G. HF 13 (Holomorphic extension for L2-holomorphic functions). Let A be a proper analytic subset of a domain G C C". Then any function f E Lh (G \ A) extends to a holomorphic function on G.

Appendix

370

HF 13 can be found in [Ran 2] as exercise E 3.2; a proof of it is contained, for example, in [Bell 1].

HF 14. Let G; C C"' be arbitrary domains, j = 1, 2. Then N

{Ef1gi:NEN,ffEO(G1),giEO(G2),1 <j
is a dense subset in O(G 1 x G2) with respect to the topology of uniform convergence on compact subsets.

A proof of HF 14 may be found in [Nar 11, Ch. I, §5.

PSH Subharmonic and plurisubharmonic functions References: [Kli 3], [Rad].

For an open set G C C', an upper semicontinuous function u : G [-oo, +oo) is said to be subharmonic (shortly sh) if for every relatively compact open subset

Go of G and for every function h, continuous on do and harmonic on Go, we have: if u < h on BGo, then u < h on Go. We will write: u E SH(G). Now let G C C" be open. An upper semicontinuous function u : G -- R, is called plurisubharmonic (psh) on G if for each a E G and X E C" the function

{XEC:a+XX EG}9A->u(a+AX)E[-oo,+oo) is subharmonic (as a function of one complex variable). The set of all psh functions

[0,+00) is on G will be denoted by PSH(G). We say that a function u: G logarithmically-plurisubharmonic (log-psh) on G if the function log u is psh on G.

PSH 1. The set PSH(G) is a convex cone, i.e. for any U,, U2 E PSH(G) and for any ti, t2 > 0 we have t,u, + 12U2 E PSH(G). PSH 2. The plurisubharmonicity is a local property, i.e. a function u : G -+ R-" is psh on G iff for each a E G there exists an open neighborhood U of a in G such that uIU E PSH(U).

PSH Subharmonic and plurisubharmonic functions

371

PSH 3. Let u: G -+ [-oo, +oo) be upper semicontinuous. Then u E PSH(G) iff for any a E G, X E C", and r> 0 with a + rEX C G we have

a)

u(

j21r

u(a+reX)d6.

PSH 4. If f E O(G), then If I is log-psh on G. PSH 5. Let D C C" be a domain and let it E PSH(D), u $ -co. Then u belongs to L I (D. loc ). In particular, the set u - I (-oo) has Lebesgue measure zero.

PSH 6 (Maximum Principle). Assume that D C C" is a bounded domain and let

u E PSH(D), u $ const. Then

u(z) < sup{Iimsupu(z)l, reaD D3z-r

z E D.

PSH 7. Let G; be an open set in C",, j = I.2, and let F : G i - G2 be a holomorphic mapping. Then for any u E PSH(G2) the function u o F belongs to PSH(G1).

PSH 8. Let I C [-oo, +oc) be an interval and let cp: 1 -+ R be an increasing convex function. Then for any psh function u : G -> I the function (P o u belongs to P S H (G). In particular,

- if u E PSH(G), then e" E PSH(G), - if u is log-psh on G, then u is psh on G, - if u : G - IR+ is psh, then for any a > I the function ua is psh on G. PSH 9. If u 1, U2 are log-psh on G. then u i + u2 is log-psh on G.

PSH 10. If q : C" -> I[8+ is a C-seminorm, then q is log-psl: on C".

PSH 11. Let h : C" -+ R+ be such that h(Az) = IAIh(z). X E C, Z E C". Then h is psh on C" iff h is log-psh on C".

PSH 12. If (u,)x , C PSH(G) and u,. \ uo pointx'ise on G. then uo E PSH(G). PSH 13. If (u,.),',` , C PSH(G) and u,. 4 uo, then uo E PSH(G).

Appendix

372

PSH 14. If (ua),.EA C PSH(G) is locally uniformly bounded from above, then the function Ito := (Supua)* aEA

is psh on G, where "* denotes the upper semicontinuous regularization.

PSH 15. If

C PSH(G) is locally uniformly bounded from above, then the

function

uo := (limsupu,,)* is psh on G.

PSH 16. Let it E PSH(G), ao E PSH(Go) with Go C G. Suppose that lim supuo(z) < u(l; ).

E (dGo) fl G.

Then the function

u(,)

max{u(z), uo(z)),

z E Go

u(z),

zEG\Go

is psh on G.

Let 4) E CC(C". R+) be such that

z = (.I..... z,,) E C" suupp 4, = A,.

J

441 = 1.

Put

4),(z):=--4(F), E>0. EC". If G is an open set in C", then we put

PSH 17. If U E PSH(G), then u * 4), E PSH(GF) fl C' (G,.) and u * 4), \ u pointwise on G as E \ 0. Here "*" denotes the convolution operator.

PSH Subharmonic and plurisubharmonic functions

373

PSH 18. Let ui,u2 E PSH(G). If ui = u2 almost everywhere in G (w.r.t. Lebesgue measure), then ul - u2.

PSH 19 (Hartogs Lemma). Let (u,,)x , C PSH(G) be locally uniformly bounded from above and suppose that lim sup , u, < M on G. Then for each s > 0 and for each compact set K C G there exists vo such that

sup u, <M+e, v> vo. K

A set P C C" is called pluripolar if for each a E P there exist a connected

neighborhood U. of a and a function ua E PSH(U0), ua 0 -oo, such that

Pf1UaCu;'(-oo). PSH 20. Any analytic set is pluripolar. Pluripolar sets have Lebesgue measure zero.

PSH 21 (Removable Singularities Theorem). Let P C G be a closed (in G) pluripo-

lar set and let u E PSH(G \ P) be locally bounded from above in G. Then the function u(z)u(z),

I

IimsupG\P3z..zu(z ),

zEG\P zEP

is psh on G. If G is connected, then G \ P is also connected. In particular, if P C G is a closed pluripolar set, then for any u E P S H (G)

lim sup u(z') = u(z),

G\P3z'-z

z E G.

PSH 22 (Schwarz Lemma). Let u be a log-sh function on E such that - the function u(A)/IAI is bounded near zero and - lim sup1AI

I _ u (A) < 1.

Then u(A) < 1). 1, A E E.

PSH 23 (Hadamard Three Circles Theorem). Let u be a log-sh function on the

annulus P := (A E C: r, < IAA < r2} with 0 < r, < r2 < +oo. Suppose that limsup,Al.yrf u(A) < Mf, j = 1, 2. Then

u (A) < MI

M2 r . A E P.

Appendix

374

PSH 24 (Oka Theorem). Let G C CI and let u E SH(G). Suppose that a: [0, 11 G is an IR-analytic curve. Then

limsup u(a(1)) = u(a(0)). r -o+

PSH 25. If U E PSH(C") is bounded from above, then u - const. If U E C2(G), then Cu: G x C" -+ C will denote the Levi form of u, that is, "

(Gu)(a; X) := E

a2U

aZ-aZk

(a)X'Xk

Observe that (,Cu) (a;

X) =

a2Ua.x (0),

axa

where ua. (A) := u(a +X X). In particular, in the case n = I we get (Cu) (a; X) _ a Au (a) I X I2, where 0 is the Laplace operator in R2.

PSH 26. Let u E C2(G, ]R). Then u E PSH(G) if (Lu)(a; X) > 0 for any a E G and X E C". PSH 27. Ifu E PSH(G), then Cu > 0 in the sense of distributions, i.e. fG u

(Z)(L )(z; X)d),(Z)

0,

rP E C(G, R+), X E C".

PSH 28. Let u : G -* [-oo, +oo] be a locally integrable function such that Cu > 0 in the sense of distributions. Then there exists a function u E PSH(G) such that u = u almost everywhere in G. (a) A function u E C2(G, IR) is said to be strictly plurisubharnwnic on G if

(Lu)(a;X)>0foranyaEGandXE(C").. (b) A function u E C(G, IR) is called strictly plurisubharmonic on G if for any relatively compact open subset Go of G there exists an E > 0 such that the function Go

z--+ u(z)-EIIZI12ispsh.

PSC Domains of holomorphy and pseudoconvex domains

375

PSH 29. A function It E C2(G, R) is strictly plurisubharmonic in the sense of (a) in the sense of W.

iff it is stricthv

PSH 30 (cf. IRic]). Let U E C(G. R) he a strictly

function and let

e: G - (0, +oe) be an arbitrary continuous function. Then there exists a strictly plurisubhannonic fuunction r E C'(G. I18) such that u < v < u + e. A simple proof of PSH 30 can be found in [For-Ste].

PSC Domains of holomorphy and pseudoconvex domains References: (Gun-Ros), IHor 21. (Kli 31, IKra fl, [Ran 21. A domain G in C" is called a domain of holomorphy if there exists a holomorphic function f on G such that for every pair (U,. U2) of non-empty open sets U. C C" with U, C U2 n G U2. U2 connected, the function f J L,, is never the restriction of an f E 0(U, ). Observe, that any domain in the complex plane is a domain of holomorphy.

If F = F(G) denotes a subfamily of 0(G). we say that G is an .F(G)-domain of holomorphy if the above definition holds with f E .F(G). For example. an H' (G)-domain of holonmrphy is a domain which admits a bounded holomorphic function which cannot be holomorphically extended through 8G. There is a long list of equivalent descriptions of domains of holomorphy. for example:

PSC 1 (Theorem of Thullen). A domain G C C" is a domain of holomorphy iff G is holomorphically convex, i.e. whenever K C G is compact. then its holomorphically

convex envelope K := Iz e G: If'(:)I

IIIIIK. f E 0(G)) is compact. too.

Observe that such a characterization fails to hold for H' -domains of holomorphy as the famous example of N. Sibony [Sib II has shown. The most important characterization of domains of holomorphy is based on a more geometric condition. A domain G C C" is said to be pseudoconvex if the 8G) is psh on G. function - log

PSC 2 (Solution of the Levi Problem). A domain G C C" is a domain of holomorphy iff G is pseucloconvet.

PSC 3. Let G be a domain in C". Then the following properties are equivalent: (i) G is pseudoconver, (ii) there exists a psh C" function it on G such that (z E G : u (;.) < k) CC G for every k E IL.

376

Appendix

(a E) CC G, then

(iii) whenever Va E C(E, G) fl O(E, G) with UaEI UaEi W- (t) cc G.

PSC 4. (a) If Gi C C'j is pseudoconvex, j = 1, 2, then G 1 x G2 is pseudoconvex.

(b) If G = U G,,, where G, C G,+1 is an increasing sequence of pseudoconvex I

domains, then v= G is pseudoconvex.

For certain classes of domains in C" the following characterizations of domains of holomorphy are known.

PSC 5. A Reinhardt domain G C C", 0 E G, is a domain of holomorphy iff G is

complete (i.e. if z E G, Ai E E for 1 < j < n, then (A,zi...... nZ,,) E G) and log G :_ {x E R": (ex' , ... , e^) E G) is convex in the usual sense. PSC 6. A balanced domain G in C" given by its Minkowski function h as G = G,, = { z E C": h (z) < 1) is a domain of holomorphy iff h is psh iff log h is psh. PSC 7. A Hartogs domain G = {z = (z', zn) E C"-' X C: Z' E G', IznI < e-'(Z') }, where G' is a domain of holomorphy in C"-' and V is an upper semicontinuous function on G', is a domain of holomorphy iff V is psh on G'. A bounded domain G C C" with smooth C2-boundary is called strongly pseudoconvex if n

ar

(Gr)(z;X)>0forallzEaGandXE(C").with Eaz (z)Xi=0, j=1

where r denotes an arbitrary function defining aG (i.e. r is a C2-function on

an open neighborhood U of aG satisfying U fl G = {z E U: r(z) < 0) and gradr(z) 36 0 for every z E aG). PSC 8. Let G be a strongly pseudoconvex domain in C". Then there exists a defining CZ function r on a neighborhood U of aG which is strictly psh on U.

PSC 9. Let l; be a boundary point of a strongly pseudoconvex domain G. Then there exists a biholomorphic mapping F : U -+ V, U a neighborhood of and V a neighborhood of 0, such that F(U fl G) is strictly convex. PSC 10. Any pseudoconvex domain G can be exhausted by an increasing sequence of strongly pseudoconvex domains G CC G with real analytic boundary.

PSC Domains of holomorphy and pseudoconvex domains

377

A bounded domain G C C" is called hyperconvex if there exists a continuous negative psh function u on G such that whenever e < 0, then (z E G: u(z) < e} is relatively compact in G; u is called an exhaustion function of G. PSC 11. Any domain of holomorphy G C C" is the union of an increasing sequence of hyperconvex subdomains.

Let G be a bounded domain in C". A boundary point i; E dG is a peak point I and

with respect to F C C(G) if there exists a function f E F with f

If(z)I
point with respect to 001). For domains in C2 even more is known. PSC 13 (cf. [Bed-For I J). If G is a bounded pseudoconvex domain in C2 with real analytic boundary, then any boundary point i; E dG is a peak point with respect to

A°(G) := C(G) n 0(G).

0 It seems to be still an open question whether PSC 13 remains true in higher dimensions. 0 Now, we repeat the main result of Hormander's 8-theory. Let G be an arbitrary

domain in C" and let tp E PSH(G). By LZ(f?, exp(-cp)) we denote the space of functions in S2 which are square integrable with respect to the measure a-°dk,

where dal denotes, as usual, Lebesgue measure. Let L(°,)(G,exp(-(p)) be the space of (0. q) forms a = >iJl=q ajdzi with coefficients ai E L2(G, exp(-(P)); means that summation is done only over strictly increasing multi-indices J = (ji. , jq) and dzj = dz1, A ... A d;,,,,. Then L(O.q)(G, exp(-(p)) is a Hilbert space with respect to the following scalar product (G.exp(-a))

f

a = >j-1 a1d7J E L(01)(G, exp(-(p)) the formula Ja =

da

- J A d;i r=1 J=1 a`J

d_

Appendix

378

defines a closed densely defined operator 8: L(0.1)(G, exp(-(p)) -> L(o.2)(G, exp(-(P))

PSC 14. Let G be a pseudoconvex domain and let 9 E PSH(G). Assume that a (0, 1) form a = :' ajdzj E L20 1) (G, exp(-(p)) satisfies (i)

aj E COO (G),

(ii)

1 < j < n,

501 = 0.

Then there exists a C°° function u E C°°(G) with du = r,"=1 It dz; = a and

f lu(z)IZ(1 + II
For the convenience of the reader we also quote the results from the theory of Stein manifolds we used in Chapter 1:I. Let M be a (connected) complex manifold. M is said to be a Stein manifold if

(i) O(M) separates the points of M, (ii) M is CA(M)-convex,

(iii) for any point p E M there exists a holomorphic coordinate system near p that is given by global holomorphic functions on M.

Observe that a domain G C C" is a Stein manifold iff G is a domain of holomorphy.

PSC 15. Any open Riemann surface is a Stein manifold.

The next fundamental theorem says that, in fact, the theory of Stein manifolds takes place in C". To be precise, we have PSC 16 (Embedding Theorem). Let M be an n-dimensional Stein manifold. Then

there exists a holomorphic embedding F: M -+ C'+', i.e. there is a closed submanifold M' C CZ"+i and a biholomorphism F: M -* M'. PSC 17. Let M be a closed submanifold of C". Then there exist a neighborhood V of M and a holomorphic retraction p : V -)- M.

For a proof see [Gun-Ros], Ch. VIII, C, Th. 8. PSC 18 (cf. [Siu]). Let M be a closed submanifold of V. Then any open neighborhood V of M contains a domain of holomorphy G with M C G, i.e. M admits a neighborhood basis of domains of holomorphy.

AUT Automorphisms

379

We mention that PSC 18 is a very special case of a general result in [Siu).

AUT Automorphisms References: [Nar 2). [Rud 21. Let G be a domain in C". a E G. Set

Aut(G) Aut,,,(G)

(F: G -- G: F is biholomorphic). IF E Aut(G): F(a) =:0).

The group Aut(G) is called the group ofautomorphisms of G. We say that the group Aut(G) acts transitively on G if for any Z'. Z" E G there exists an automorphism

F E Aut(G) such that F(:') = z". Automorphisms of the unit disc

Let E be the unit disc in C. For a E E define

AUT 1. (a) Aut(E) = (e'"h,,: 0 E R, a E E). (e'" id: 8 E R) is equal to the group of rotations. (b)

(c) The group Aut(E) acts transitively on E; (h")-) = h_,,, h"(a) = 0.

Automorphisms of the unit polydisc

AUT 2.

(a) Aut(E") = (E" a (Z).....:n) - (

..... H E R. at..... an E E. a is an a r b i t r a r y pennutation o f (I .... , n)). E" 3 (;.).....we) (b) Auto(ER) 01 ..... N E R. a is an arbitrary permutation of (1..... n)). H,

Appendix

380

(c) The group Aut(E") acts transitively on P.

Automorphisms of the unit Euclidean ball Let B" be the unit Euclidean ball in C". For a E B. define

ha (Z)

=

1

IIaIl2

1 - IIal12(llall2z - (z, a)z) - IIaII2a + (z, a)a 1 - (z, a)

ifa00,

ho(z) : = id, where 11 11 stands for the Euclidean norm and

for the standard complex scalar

product in C". We have 1 - (ha(z), ha(w)) -_

(1 - (a,a))(1 - (z, w))

(1 - (z,a))(l - (a,w))'

Z,wEB"(aEB").

AUT 3. (a) Aut(B") = (U o ha: U is a unitary operator on C" and a E B"). (b) Auto(B,,) is the group of unitary operators on C".

(c) The group Aut(B") acts transitively on B"; (ha)-I = h_a h,, (a) = 0,

h,(a)(X) =

1

IIaIl2

1 - IIaI12(IIaII2X - (X, a)a) + (X, a)a 1 - IIaI12

XEC"(a#0).

GR Green function and Dirichlet problem References: [Hay-Ken], [Helm], [Kli 3], [Lan], [Nar 3].

Let G be any bounded domain in C. Then the Dirichlet problem for G is the following question:

given a function f E C(aG, IR), find a function h E C(G,1R) with hIaG = I such that h is harmonic in G. Because of the maximum principle for harmonic functions, the solution of the Dirichlet problem, if it exists, is uniquely determined. A domain is called regular with respect to the Dirichlet problem, or simply, a Dirichlet domain, if for any continuous boundary function f there exists a solution of the Dirichlet problem. In the case G = E we have the following explicit solution.

GR Green function and Dirichlet problem

381

GR 1 (Poisson Integral Formula). For any f E C(aE, R) the function I

h(z)

2n

2,r Jo

/ id f (e )

1-r2

d9 ifz=rexpicpE E

11-2rcnr

ifz E aE

f(z)

solves the Dirichlet problem.

More generally, the following result holds.

GR 2. A bounded domain G C C is a Dirichlet domain i ff even' boundary point i; of G is a local peak point for continuous subharmonic functions, i.e. for any E aG there exist a neighborhood V of i; and a function u E C(G fl v) fl SH(G fl v) such that

u(z) = 0.

(i)

lim Gnv3z-.

(ii)

lim sup u(z) < 0, Gnvaz-t

u(z) < 0,

(iii)

t E aG fl V. z E G n V.

GR 3 (Theorem of Bouligand). Let G be bounded and let a E aG. Suppose that there exist V = V(a) and U E C(G fl v) fl SH(G fl v) with lim

GnV BZ-a

u(z) = 0.

U(Z) < 0, z E G f1 V.

Then a is a local peak point for continuous subharmonic functions. Even more is true. Namely, there exists a harmonic function h on the whole G with

Iim h(z) = 0, limsuph(z) < 0,

b E (aG) \ (al.

G3; -+h

in particular, we have the following sufficient criterion.

GR 4. Any bounded domain G in C such that no connected component of C \ G reduces to a point is a Dirichlet domain. For a domain G C C fix a E G. The (classical) Green function of G with pole at a is a function g(; (a, ) : G \ (a) -+ R that satisfies the following three properties: 1) gc; (a, .) is harmonic on G \ (a I ; 2) g(; (a. ) + log I -aI extends to a harmonic function on G;

Appendix

382

3) there exists a polar set F C aG, such that: if E (aG) \ F, then limG3z- gG(a, z) = 0, if E F or = 00 E aG, then 9G (a. ) is bounded near GR S. If G C C is a donwin whose boundary is not a polar subset of C, then for ever), a E G the Green fiinction 9G (a, ) exists and is unique. Moreover, by the maximum principle for subharmonic functions we obtain

GR 6. Let G C C be a domain and let a E G be such that the Green function gG (a, ) exists. Then for any subharmonic function u : G -+ [-oo, 0) satisfying u(z) < C + log Iz - aI near a we have the following inequality u(z) < -gG(a, z),

zEG. Therefore,

GR 7. Let G be an arbitrary domain in G, a E G. If aG is polar, then gG(a, ) = 0 on G. If aG is not polar, then gG(a, ) = exp(-gG(a, )) on G, where gG denotes the "complex Green function"; cf. Chapter IV. In particular, g' (a, ) is of class C°" in G.

GR 8. For G = B(0, R) the Green function is given by 1

gG(a,z)=log(RI

R2 - az

z-a I)' z0a.

GR 9. Let G C C be a domain such that 9G (a, ) exists for every a E G. Then the Green function is symmetric, i.e. 9G W, z") = 9G (Z", z') whenever z', z" E G,

z'0z" We conclude this part of the appendix with the Riesz representation theorem.

GR 10 (Riesz Theorem). Let U E SH(G), u $ -oo. Then there exists a unique Borel measure µ on G such that for any compact subset K C G with int(K) : m we have it (Z) =

flog Iz - Idi.() + h(z),

Thus many of the local properties of subharmonic functions can be deduced from those of logarithmic potentials.

MA Monge-Ampere operator

383

MA Monge-Ampere operator References: [Ceg], [Kli 31.

Let d = 8 + a denote the operator of exterior differentiation in C". Define d' := i(a - a). Let G be an open set in C". Then the Monge-Ampere operator is an operator acting on PSH(G) n L'°(G, loc), that assigns to each function u E PSH(G) n L°°(G, loc) a non-negative Borel measure (ddeu)" on G. In the case where u E PSH(G)nC2(G), the definition of (ddc'u)" is elementary, namely

(dd`u)" _ (det

8:iazk] j,k=i..... n [_3u

)

(Lebesgue measure).

In the general case, if u i .... , u" E P S H (G) n L'(G,1oc ), then (for k = 1,

... , n)

dd`ui A - A dd`uk is defined inductively as a positive (k, k)-current of order 0 by the formula

A dd`uk A X = f ukdd` u, A c

IG dd`ui A

.

A dd`'uk_I A dd`'X,

where X is an arbitrary test form in G of bidegree (n -k, n -k); see [Kli 3§ 3.4, A dd'*u. for details. Then we set (dd`'u)" := dd"U A n-times

MA 1 (Domination Principle, cf. [Bed-Tay]). Let G be a bounded open subset of C" and let u+, u_ E PSH(G) n L°C'(G) be such that (dd`u+)" (dd`u_)" in G, lim inf (u+(:) - u_ (;.)) > 0 for all 4 E aG. G3:-Z

Then u+>u_onG. A plurisubharmonic function u : G --* R is said to be maximal if for any relatively compact open subset Go of G and for every function v upper semicontinuous

on do and plurisubharmonic in Go, if v < u on 8Go. then v < u in Go.

MA 2. Let u E PSH(G) n L"°(G, loc). Then u is maximal ,j)'(dd(u)" = 0 in G.

Appendix

384

H Hardy spaces References: [Durl, [Garl. [Coil, [Rud 11.

Let 0 < p < +oo. We say that a function h E O(E) is of class Hr(E) if sup O
UO

Ih(re'")Ied8

I

< +oo.

H1 . HOO(E)CH1'(E)CHy(E)forany0
E 8E (w.r.t. Lebesgue measure on CE) the function h has non-tangential limit at , i.e. the limit lim h(A) =

lim

h(A)

exists and is independent of a > 1, where

(A E E: IA - I < all - IAI)[ Moreover, h' E Le(d E).

H 3 (Cauchy and Poisson Integral Formulas). If h E H I (E). then

h(A)

21ri

1'.

2n

Jn

A E E,

/ere + A

2;r

1

h(A) =

Add,

Re ( eie

) h"(e'")dO,

A E E.

In particular, - if I h' 1 < M almost everywhere on c7 E, then Ih 1 < M on E;

- if h' E R almost everywhere on a E, then h = const E R. H 4 (Blaschke Products). Let (a,, )x 1 C E be such that M

F, (l-IaI)<+00.

(*)

H Hardy spaces

385

Pitt I := { v : a, j4 01 and let k : = #(N \ 1). Then the product

it,. I - a,. PEi

I - [t,.

1a,.l

X E E.

(**)

is locally unijorntly convergent. B E H' (E). I B* I = I almost everywhere on i) E, and the zeros of B (counted with rnultiplicities) coincide with (a,,);',` i.

H 5. If h E H r (E ). It # 0. then the zeros of h (counted with nmltiplicities) satisfi (*) In particular, using (**) one can define the Blaschke product Bf, for It.

H 6 (Decomposition Theorem). Let h E H"(E). It 0 0. Then log lh*I E L1(i)E). 2r

log Ih(0)I

<2n-

Jo

(* * *)

log Ih"(e'")Id9.

It=cB,, where c E 3 E is a constant,

Sh()`) := exp

(- L

e1" - da(0)

(a is a non-negative Bore! measure. singular w.r.t. Lebesgue measure). and

(t

Qh(a) := eXp

r2 :r

2n (

+ e'" - h log

\

Ih-(e"')IdHI)

.

h E E.

Moreover,

- Sh E H"'(E); - IS,, I = I almost everywhere on a E w. r.t. Lehesgue measure. - SS = 0 almost everywhere on 8E t'.r.t. the measure a; - the function BhSh is constant iff tine have equality in condition (* * *).

H 7 (Identity Principle). Let h E H"(E). Suppose that h* = 0 on a set of positive Lebesgue measure. Then h - 0. H 8 (F.& M. Riesz Theorem). Let p be a complex Borel measure on (0.27r ). Then the follon-ing conditions are equivalent:

Appendix

386

(i) f(2" f (e`")dµ(9) = 0 for all f Eqt) fl O(E); (ii) there exists h E H' (E) such that dµ(9) = MTh*(e'd)d9. H 9 (Hardy-Littlewood Theorem). Let h E O(E) and let 0 < a < 1. Then h ct7endx U u elz to an a-Holder-continuous function on k iff there exists a constant .11 that

(l

-

MIXI)_a

I

A E E.

References

[Aba 21

M. Abate, Boundary behavior of invariant distances and complex geodesics, Atti Acc. Lincei Rend. Fis., Ser. VIII 80 (1986). 100-106. M. Abate. The complex geodesics of the classical domains, preprint (1986).

[Aba 3]

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Mediterranean Press, 1989. M. Abate, A characterization of hyperbolic manifolds. Proc. Amer. Math. (1993) (to appear). M. Abate & G. Patrizio. Uniqueness of complex geodesics and characterization of circular domains, manuscripta math. 74 (1992). 277-297. M. Abate & G. Patrizio. Holomorphic curvature of Finsler metrics, preprint (1992). J. Agler, Operator theory and the Carathe dory metric, Invent. Math. 101 (1990), 483-500.

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P. R. Ahem & R. Schneider, Isometrics of H'. Duke Math. J. 42 (1975). 321-326. L. V. Ahifors, An extension of Schwarz's Lemma, Trans. Amer. Math. Soc. 43 (1938), 359-364. L. V. Ahlfors & A. Beurling, Conformal invariants and function theoretic null sets, Acta Math. 83 (1950), 101-129. G. J. Aladro, Some consequences of the boundary behavior of the Caratheodory and Kobayashi metrics and applications to normal holomorphic functions, Ph. D. Dissertation, Pennsylvania State University, 1985. J. d'Angelo. A note on the Bergman kernel. Duke Math. J. 45 (1978), 259-

[Azu 11

265. K. Azukawa, Hyperboliciry of circular domains, T6hoku Math. J. 35 (1983).

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[Azu 21 [Azu 31

403-413. K. Azukawa, Bergman metric on a domain of Thullen type. Math. Rep. Toyama Univ. 7 (1984), 41-65. K. Azukawa, Two intrinsic pseudo-metrics with pseudoconrer indicatrices and starlike domains, J. Math. Soc. Japan 38 (1986), 627-647.

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[Sib 3]

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[Sta I1 ISta 21

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List of symbols

General symbols

N - the set of natural numbers, 0 gE N Z - the ring of integer numbers Q - the field of rational numbers R - the field of real numbers C - the field of complex numbers Re z, Im z - real and imaginary parts of z E C A" - the Cartesian product of n-copies of the set A, e.g. C"

A.:=A\{0},e.g. C., (C"). A+:= {aEA:a>0),e.g.Z+, R+

A_:={aEA:a<0} A>o:= (a E A: a > 0), e.g. R>o

A.o:=(aEA:a<0)

Rte := R U {fop}

A+B:=(a+b:aEA,bEB), where A,BCX,(X,+) is agroup where ACC,BCX,X is a C-vector space (z, w) = (Z. w)n := Ei=, zj wi, z = (Zt, ... , Zn), w = (w1, ... , wn) E C" (, - the Hermitian scalar product in C"

z=(ZI....,Zn),w=(w1....,wn)EC" llzll = Ilzlln := ((z, z))"2 = (Izi I2 + ... +

z = (Z,.... , z") E C"

11 - the Euclidean norm in C" diam A the diameter of the set A C C" with respect to the Euclidean distance 11

dist(A, B) := inf{Ila - bIl : a E A, b E B) dist(a, B) := disc({a}, B) cony A the convex hull of the set A Bd(a, r) {x E X : d(x, a) < r}, a E X, r > 0, where (X, d) is a pseudometric space

Bd(a, r)

{x E X : d(x, a) < r}, a E X, r > 0, where (X, d) is a pseudometric

space

Bq(a. r)

{x E X : q(x - a) < r), a E X, r > 0, where (X, q) is a normed

space

Bq(a, r) space

{x E X: q(x - a) < r}, a E X, r > 0, where (X, q) is a nonmed

List of symbols

401

B(a, r)

BII II(a, r), a E C". r > 0 B(a, r) := BII II (a, r), a E C", r > 0

3,,:=B(0.1)CC"

B. - the unit Euclidean ball in C" E := B, = (A E C: IAI < I} - the unit disc (Einheitskreis) top G - the Euclidean topology of G, G C C"

Illlln:=supflf(a)I:aEA}, f: A ->C Ck(SZ1, S22) - the space of all CR-mappings F: SZ, -+ 512; k E Z+ U {oo} U (w} Ck(Q) := C* (Q, C) L I (S2. loc) - the space of all locally integrable functions on S2 L22(S1) - the space of all square integrable functions on S2

L2(1. exp(-V)) - the space of all square integrable functions with the weight exp(-(p ) O(1j, S2,) - the space of all holomorphic mappings F: 1? -3. S2,

O(1) := O(1. C) L2(S1) - the space of all square integrable holomorphic functions on S1 H0O(S2) - the space of all bounded holomorphic functions on S2 Aut(G) - the group of all automorphisms of G C C"

Auta(G) := {h E Aut(G): h(a) = a} sh:=subharmonic

SH(G) - the set of all subharmonic functions on G psh:=plurisubharmonic

PSH(G) - the set of all plurisubharmonic functions on G gr, - the (classical) Green function of G - locally uniform convergence dr(z) - the real gradient of r at z

a - the a-operator

G - the Levi form Symbols in Chapter I the M6bius distance

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he.n ...............................3 .

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List of symbols

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Symbols in Chapter II

c - the Mobius pseudodistance for G

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CG - the Carathe odory pseudodistance for G . . . . . . . . yG - the Caratheodory-Reiffen pseudometric for G . . . . . Gh = {h < 1) - the balanced domain with Minkowski function h top cG the Carathe odory topology for G . . . . . . . . .

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L c - the cG length

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Ln: - the yG-length . . . . . . . . . . . . . - the inner Caratheodory pseudodistance for G CG f yG - the integrated form of yG . . . . . . . .

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S(G) - the set of all admissible exponents of G log A - the logarithmic image of A . . . . .

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Symbols in Chapter III

kG - the Lempert function for G . . . k* := tank kG . . . . . . . . . . . kG - the Kobayashi pseudodistance for G

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71 73

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k := tanhkG

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km) LAt, - the kG-length . . . . . . . . . . . xG - the Kobayashi-Royden pseudometric for G hG - the Hahn pseudometric . . . . . . .

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82 90 . 99 . 100 . 101 . 101

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f xG - the intergrated form of xG f xG - the intergrated form of xG

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101 101

Symbols in Chapter IV .

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6, 00, Oh, 0,, Ob d = (dG)GEmo ICG(a) GG(a)

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S = (SG )GEOo

115

403

List of symbols

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SG' gc - the complex Green function for G AG - the Azukawa pseudometric for G SG

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116 116

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116 120

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SG - the Sibony pseudometric for G . . . the k-th MSbius function for G yG't' - the k-th Reiffen pseudometric for G (dd`u)" - the Monge-Ampere operator . tic

Z(G)

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139 139 139 140 140 140

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141

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144

9X(G)

Ld - the d-length . . . . . . . d` - the inner pseudodistance for d L8 - the 8-length . . . . . . . J'S - the integrated form of 8 . . Dd - the derivative of d . . . . S - the Buseman pseudometric for S d'

..............................146 f8 ..............................146 5Dd...............................146 ..............................147 Symbols in Chapter VI )LI(G) - the L2(G)-scalar product II IIV(ci - the L2(G)-norm . . . . KG - the Bergman kernel for G . . PG - the Bergman projection . . .

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BG ..............................185 .

,0c - the Bergman pseudometric for G hG - the Bergman pseudodistance for G

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..............................188 MG ..............................190 6

exp(-(p)) - the space of all (0, q)-forms with coefficients in . . . . . . . . . . . . . . . . . . . L-(G, exp(-(p)) . . PG - the Skwarczyfiski pseudodistance for G . . . . . . . . . . Mc . . . . . . . . . . . . . . . . . . . . . . . . . . . L2

(,.,)(G,

194 195 197

List of symbols

404

Symbols in Chapter VIII

qG - the Minkowski function for G . . . q(, - the dual Minkowski subnorm for G . v(z) - the unit outer normal of G at z E aG r - the normalized Lebesgue measure on aE L" - the Lie ball in C" . . . . . . . . .

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e(p) - the complex ellipsoid

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246 . 247 . 248 . 250 . 259

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261

Symbols in Chapter X

gradr :-

the complex gradient of r

"

IIaII(; :_ F-;_l IIajIIG - the supremum norm of the form a

311 311

Index

A

Carath6odory ball

A-hyperbolicity

admissible exponent for G Ahlfors-Schwarz Lemma analytic capacity analytic ellipsoid Azukawa pseudometric

207

54

Carathe odory completeness Carathcodory pseudodistance

l1 221

Caratheodory-Reiffen pseudometric

3._2

Carathe odory topology

116.. LL8

213

b-finite compactness

213 1411

complex SG-geodesic

L 7-5

Bergman completeness Bergman kernel Bergman projection Bergman pseudodistance Bergman pseudometric Big Picard Theorem

2311

121 1Z1

L8ft

22

Cauchy inequalities

B b-completeness

balanced domain balanced pseudoconvex domain

11t

12

Cauchy Integral Formula Classical Schwarz Lemma Classical Schwarz-Pick Lemma complex dG-geodesic

-hyperbolicity

41

21lt

35

384

351 352 231

complex ellipsoid complex extreme point

232 264 252

complex geodesic

254

complex Green function l1 LL8 cone condition (general outer cone condition) 182 cone condition for n-circled domains

186 12

62

contractible family of functions

Ill

Blaschke product

384

Bohr theorem Borel-Caratheodory Lemma Bouligand theorem Brody hyperbolicity

358 56

contractible family of pseudodistances

381 202

contractible family of pseudometrics

Buseman pseudometric

145

contraction

C c-completeness c-finite compactness

c-hyperbolicity c-isometry c'-hyperbolicity CG

)-length

cG`)-rectifiability Cl-pseudodistance

213 213

28 22

40 35 35 L4fi

112 LL2

D a-closed (0, I)-form S-hyperbolicity S-length d-hyperbolicity d-length d-rectifiability dG-completeness dG-finite compactness

1

3!J 114 140 LL4

140 L40 213 213

406

Decomposition Theorem derivative

Dirichlet domain Dirichlet problem domain of holomorphy Domination Principle dual Minkowski subnorm

Index

385

Embedding Theorem Euclidean rectifiability exhaustion function extremal disc

323

3$4

Hahn pseudometric Hardy-Littlewood Theorem

99 386

375

Hardy space

384

383 242

Hartogs Lemma for plurisubharmonic functions 323 Hartogs Theorem 96 Hausdorff measure 82 holomorphic contractibility 12 holomorphic covering 368 holomorphic curvature holomorphic extension 362 homogeneous finite diagonal type

MOO

E elementary n-circled domain

Hadamard Three Circles Theorem

141

54

318 36 371 81, 22

extremal function for cG (z', z") L6

extremal function for yG (z: X)

365

Hopf lemma hyperbolic distance hyperconvex domain

L6

extremal function for

322 5

80.322

132

I

F Fatou theorem

362

finite type

364

Forelli theorem

347

G y-contraction y-hyperbolicity y-isomctry y-length ye-length Gelfand topology Gelfand transform

2 28

inner distance inner pseudodistance integrated form irrational type irreducible element of S(G) isometry

38 6 L4Q

101. 141 54 52

I

2 285 3 36 220 220

generalized Laplacian Generalized Poincare Theorem

356 25

generalized product property

309

General Schwarz-Pick Lemma Green function group of automorphisms

Al

L2

379

H H'`'-convexity H'1-domain of holomorphy H"`-extension

Identity Principle for HP 385 inner Caratheodory pseudodistance

K k-completeness k-finite compactness k-hyperbolicity k(; -length

k-th MSbius function k-th Reiffen pseudometric r--indicatrix x-isomctry x-hyperbolicity xG-length .,;.,C;-length

213 213 83 82 L16. 118 116. 118 92 285 207 Lot LOU

218 217

kernel function L69 Kobayashi-Buseman pseudometric

18

Loll

407

Index

Kobayashi completeness

22"3

Kobayashi pseudodistance Kobayashi-Royden pseudometric

352

non-tangential limit

23

0 44

374

Oka Theorem

L Landau theorem Lempert function Lempert's Theorem

243

p-rectifiability

Levi form

324

p-segment

P 14 21

-312 Levi polynomial 375 Levi Problem Lie ball 259 1159 Lie norm 22 Liouville domain Liouville Theorem 11 Little Picard Theorem au 311 local peak function 55 logarithmic image logarithmically plurisubharmonic func3751 tion Lu Qi-keng domain 1$4

p-isometry p-length

7 5

5

peak function

5 217

peak point

377

Picard theorem pluripolar set plurisubharmonic function

313

4 3211 5

Poincare distance

Poincare model Poincare Theorem

5 22

pointwise S-hyperbolicity

LL4

Poisson Integral Formula

3&L 384 42, 106. 296

product-formula product-property

pscudoconvex domain

42, 106. 29-6 375

M in-contractible family of functions

pseudometric

112

U2 in-contractible family of pseudodisLL2 tances m-contraction 2 2 m-isometry 3 m-segment maximal plurisubharmonic function 385 Maximum Principle for plurisubhar-

R rational type

54

monic functions

monic functions

Minkowski function Minkowski subnorm Mdbius distance MSbius function Mi bius pseudodistance

321

L&)46 241 3 115. 118 15

Mange-Ampere operator

393

Montel theorem

352

N

n-circled domain

53

regularity with respect to the Dirichlet problem

Reiffen pseudometric Reinhardt domain

380 116. LL& 54

removable singularities of holomorphic 3L2 functions removable singularities of plurisubharRiesz theorem Riesz, F.& M., Theorem Robinson's Fundamental Lemma

323 382

385 156

S

207 S-hypcrbolicity 14 Schottky theorem Schwarz Lemma for subharmonic 373 functions Schwarz-Pick Lemma 2

408

Index

Schwarz-Pick Lemma for the annulus 154 43

Sibony domain 116, 118 Sibony pseudometric 195 Skwarczyfiski pseudodistance 220 spectrum of H "*(G) square integrable holomorphic 170 function Stein manifold 378 strictly convex domain 257 strictly plurisubharmonic function 374 strongly convex domain 264 strongly linearly convex domain

subharmonic function

370

T taut domain theta function 'fhullen domain Thullen Theorem transitive action of a group U UFD-ring

77 162 173 375

379

Uniformization Theorem universal covering

177 368 368

W weak dG-completeness weak dG-finite compactness

213 213

360

strongly pseudoconvex domain 310, 376

ISSN 09:38-6572 ISBN :3-11-01:32.1-(i