This book provides a comprehensive introduction to the field of several complex variables in the setting of a very special but basic class of domains, the so-called Reinhardt domains. In this way the reader may learn much about this area without encountering too many technical difficulties. Chapter 1 describes the fundamental notions and the phenomenon of simultaneous holomorphic extension. Chapter 2 presents a fairly complete discussion of biholomorphisms of bounded (complete) Reinhardt domains in the two dimensional case. The third chapter gives a classification of Reinhardt domains of existence for the most important classes of holomorphic functions. The last chapter deals with invariant functions and gives explicit calculations of many of them on certain Reinhardt domains. Numerous exercises are included to help the readers with their understanding of the material. Further results and open problems are added which may be useful as seminar topics. The primary aim of this book is to introduce students or non-experts to some of the main research areas in several complex variables. The book provides a friendly invitation to this field as the only prerequisite is a basic knowledge of analysis.
Author(s): Marek Jarnicki and Peter Pflug
Year: 2008
Language: English
Pages: 370
Front Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface ......Page 6
Contents ......Page 8
1.1 Introduction ......Page 10
1.2 Summable families ......Page 15
1.3 Domains of convergence of power series ......Page 22
1.4 Maximal affine subspace of a convex set I ......Page 29
1.5 Reinhardt domains ......Page 38
1.6 Domains of convergence of Laurent series ......Page 50
1.7 Holomorphic functions ......Page 56
1.8 Balanced domains ......Page 64
1.9 Extension of holomorphic functions ......Page 67
1.10 Natural Frechet spaces ......Page 72
1.11 Domains of holomorphy ......Page 81
1.12 Envelopes of holomorphy ......Page 93
1.13 Holomorphic convexity ......Page 98
1.14 Plurisubharmonic functions ......Page 109
1.15 Pseudoconvexity ......Page 126
1.16 Levi problem ......Page 137
1.17 Hyperconvexity ......Page 139
1.18* Smooth pseudoconvex domains ......Page 151
1.19* Complete Kahler metrics......Page 155
2.1 Introduction ......Page 169
2.2* Cartan theory ......Page 186
2.3 Biholomorphisms of bounded complete Reinhardt domains in CZ ......Page 189
2.4 Biholomorphisms of complete elementary Reinhardt domains in C2 ......Page 205
2.5* Miscellanea ......Page 216
3.1 General theory ......Page 229
3.2 Elementary Reinhardt domains ......Page 234
3.3 Maximal affine subspace of a convex set II ......Page 239
3.4 M'-domains of holomorphy ......Page 245
3.5 Ak-domains of holomorphy ......Page 248
3.6 Li -domains of holomorphy ......Page 250
4.1 Introduction ......Page 260
4.2 Holomorphically contractible families of functions ......Page 262
4.3* Hahn function ......Page 278
4.4 Examples I - elementary Reinhardt domains ......Page 286
4.5 Holomorphically contractible families of pseudometrics ......Page 302
4.6 Examples II - elementary Reinhardt domains ......Page 319
4.7 Hyperbolic Reinhardt domains ......Page 322
4.8 Carathdodory (respKobayashi) complete Reinhardt domains ......Page 326
4.9* The Bergman completeness of Reinhardt domains ......Page 334
Bibliography ......Page 342
Symbols ......Page 354
List of symbols ......Page 358
Subject index ......Page 364
Back Cover......Page 370
