id: 00046615
dt: b
an: 00046615
au: Dineen, Seán
ti: The Schwarz lemma.
so: Oxford Mathematical Monographs. Oxford: Clarendon Press. x, 248 p.
\sterling 25.00 (1989).
py: 1989
pu: Oxford: Clarendon Press
la: EN
cc:
ut: intrinsic metrics on complex manifolds of several and infinitely many
variables; infinite dimensional holomorphy; Schwarz-Pick lemma; Schwarz
lemma for subharmonic functions; Schwarz lemma for plurisubharmonic
functions; complex Green function; Poincaré distance; Infinitesimal
Finsler pseudometrics; Holomorphic curvature; Schwarz-Pick systems;
Carathéodory and Kobayashi pseudodistances; Hyperbolic manifolds;
pseudoconvex domains; Ahlfors-Schwarz lemma; generalized Hessian;
algebraic metric of Harris; holomorphic characterization of Banach
spaces containing $c\sb 0$; Irreducibility; cotype; Fixed point
theorems; Holomorphic retracts; analytic Radon-Nikodym property
ci: Zbl 0484.46044
li:
ab: 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 ${\bbfR}\sp n$ and
${\bbfC}\sp n$, 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 $JB\sp*$ 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
$c\sb 0$ (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.
rv: L.Nachbin