First Steps in Several Complex Variables: Reinhardt Domains (EMS Textbooks in Mathematics) by Marek Jarnicki and Peter Pflug (2008) Hardcover

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. Cover ......Page 1 Title ......Page 4 Preface......Page 6 Contents......Page 8 Introduction......Page 10 Summable families......Page 15 Domains of convergence of power series......Page 22 Maximal affine subspace of a convex set I......Page 29 Reinhardt domains......Page 38 Domains of convergence of Laurent series......Page 50 Holomorphic functions......Page 56 Balanced domains......Page 64 Extension of holomorphic functions......Page 67 Natural Fréchet spaces......Page 72 Domains of holomorphy......Page 81 Envelopes of holomorphy......Page 93 Holomorphic convexity......Page 98 Plurisubharmonic functions......Page 109 Pseudoconvexity......Page 126 Levi problem......Page 137 Hyperconvexity......Page 139 Smooth pseudoconvex domains......Page 151 Complete Kähler metrics......Page 155 Introduction......Page 169 Cartan theory......Page 186 Biholomorphisms of bounded complete Reinhardt domains in C^2......Page 189 Biholomorphisms of complete elementary Reinhardt domains in C^2......Page 205 Miscellanea......Page 216 General theory......Page 229 Elementary Reinhardt domains......Page 234 Maximal affine subspace of a convex set II......Page 239 H^infinity-domains of holomorphy......Page 245 A^k-domains of holomorphy ......Page 248 Lp_h-domains of holomorphy......Page 250 Introduction......Page 260 Holomorphically contractible families of functions......Page 262 Hahn function......Page 278 Examples I – elementary Reinhardt domains......Page 286 Holomorphically contractible families of pseudometrics......Page 302 Examples II – elementary Reinhardt domains......Page 319 Hyperbolic Reinhardt domains......Page 322 Carathéodory (resp. Kobayashi) complete Reinhardt domains......Page 326 The Bergman completeness of Reinhardt domains......Page 334 Bibliography......Page 342 Symbols......Page 354 List of symbols......Page 358 Subject index......Page 364 "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 readers may learn much about this area without encountering too many technical difficulties." "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."--Jacket