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The Schwarz lemma. (English) Zbl 0708.46046
Oxford Mathematical Monographs. Oxford: Clarendon Press. x, 248 p. £25.00 (1989).
The study and applications of intrinsic metrics on complex manifolds of several and infinitely many variables has proceeded in a variety of directions over the last decade. This very nice book is a basic selfcontained introduction to the theory of such metrics and, at the same time, reports on some new developments. It is of interest to graduate students and research workers in complex analysis, differential geometry, fixed point theory, functional analysis, and potential theory. The interdisciplinary nature of the subject is one of its most striking features. This book will provide an introduction to these new areas for specialists from the underlying areas. The research interests of the author lies in functional analysis and infinite dimensional holomorphy. This is reflected in the contents and style of the book. He does not discuss the Bergman metric as it relies on Lebesgue measure not available in infinite dimensions. He avoids in places the use of differential geometry language to make the material accessible to analysists. His choice of applications is mainly infinite dimensional. He assumes no knowledge of intrinsic metric.
The book is organized as follows. Part I (Chapters 1-9) deals with the basic theory. Part 2 (Chapters 10-12) deals with applications.
Chapter 1, The classical Schwarz lemma (The Schwarz lemma and the Schwarz-Pick lemma, A Schwarz lemma for subharmonic functions) discusses the Schwarz lemma for holomorphic and subharmonic functions. Chapter 2, A Schwarz lemma for plurisubharmonic functions (Potential theory on \({\mathbb{R}}^ n\) and \({\mathbb{C}}^ n\), A Schwarz lemma for plurisubharmonic functions) discusses complex potential theory, introduces the complex Green function and proves a Schwarz lemma for plurisubharmonic functions. Chapter 3, The Poincaré distance on the unit disc (Infinitesimal Finsler pseudometrics, Holomorphic curvature (1), The Poincaré distance and the Green function on D) discusses the basic properties of the Poincaré (or hyperbolic) distance. Chapter 4, Schwarz-Pick systems of pseudodistances (The infinitesimal Carathéodory and Kobayashi pseudometrics) defines Schwarz-Pick systems following Harris and obtains the fundamental properties of the Carathéodory and Kobayashi pseudodistances. Chapters 5, Hyperbolic manifolds (Extension theorems, Equicontinuous families of holomorphic mappings) and 6, Special domains (Balances pseudoconvex domains, Convex domains, A characterization of polydisc) are devoted to hyperbolic manifolds and special (say convex, pseudoconvex) domains respectively. Chapter 7, Pseudometrics defined using the (complex) Green functions (Inequalities satisfied by the (complex) Green functions, The infinitesimal pseudometrics of Sibony and Azukawa, Infinitesimal metrics on the annulus) discusses the infinitesimal pseudometric of Azukawa and Sibony. Chapter 8, Holomorphic curvature (Curvature in differential geometry, Holomorphic curvature (2), The Ahlfors-Schwarz lemma, Convex domains in the complex plane) introduces the generalized Hessian and uses it to study holomorphic curvature. Chapter 9, The algebraic metric of Harris (Bounded symmetric domains and \(JB^*\) triple systems, The algebraic inner product, The infinitesimal algebraic metric) contains a study of the algebraic metric of Harris on finite rank bounded symmetric domains. Chapter 8 contains a differential-geometric Schwarz lemma, Chapter 9 contains an algebraic- geometric Schwarz lemma, and in Chapter 11 it is presented a fixed point free Schwarz lemma.
After the theory developed in Part I, applications of it come in Part II with Chapter 10, A holomorphic characterization of Banach spaces containing \(c_ 0\) (Irreducibility and cotype), Chapter 11, Fixed point theorems (The Earle-Hamilton fixed point theorem, A fixed point free Schwarz lemma, Holomorphic retracts) and Chapter 12, The analytic Radon- Nikodym property (Closed bounded submanifolds, Geometric properties of Banach spaces, Carathéodory complete Banach manifolds).
Each chapter ends with notes and remarks. The book ends with a generous list of pertinent references. Among them the book “Complex analysis in locally convex spaces”, North-Holland Math. Stud. 57 (1981; Zbl 0484.46044) by the author is now the classical and best reference textbook on infinite dimensional holomorphy. There is a fairly detailed subject index at the end. This book is based on seminars at University College Dublin, Ireland and a course given at Federal University of Rio de Janeiro, Brazil during the 1987 summer. Although the author states that he tried to stay close to the format of that course rather than attempting to write a comprehensive research monograph, this volume of Oxford Mathematical Monographs is a valuable addition to the literature of the subject matter, its ramifications and applications, from the viewpoints of both research and teaching.
Reviewer: L.Nachbin

46G20 Infinite-dimensional holomorphy
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis