This book presents many of the main developments of the past two decades in the study of real submanifolds in complex space, providing crucial background material for researchers and advanced graduate students. The techniques in this area borrow from real and complex analysis and partial differential equations, as well as from differential, algebraic, and analytical geometry. In turn, these latter areas have been enriched over the years by the study of problems in several complex variables addressed here. The authors, M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, include extensive preliminary material to make the book accessible to nonspecialists.
One of the most important topics that the authors address here is the holomorphic extension of functions and mappings that satisfy the tangential Cauchy-Riemann equations on real submanifolds. They present the main results in this area with a novel and self-contained approach. The book also devotes considerable attention to the study of holomorphic mappings between real submanifolds, and proves finite determination of such mappings by their jets under some optimal assumptions. The authors also give a thorough comparison of the various nondegeneracy conditions for manifolds and mappings and present new geometric interpretations of these conditions. Throughout the book, Cauchy-Riemann vector fields and their orbits play a central role and are presented in a setting that is both general and elementary.
Table of Contents:
Ch. I Hypersurfaces and Generic Submanifolds in C[superscript N] 3
Ch. II Abstract and Embedded CR Structures 35
Ch. III Vector Fields: Commutators, Orbits, and Homogeneity 62
Ch. IV Coordinates for Generic Submanifolds 94
Ch. V Rings of Power Series and Polynomial Equations 119
Ch. VI Geometry of Analytic Discs 156
Ch. VII Boundary Values of Holomorphic Functions in Wedges 184
Ch. VIII Holomorphic Extension of CR Functions 205
Ch. IX Holomorphic Extension of Mappings of Hypersurfaces 241
Ch. X Segre Sets 281
Ch. XI Nondegeneracy Conditions for Manifolds 315
Ch. XII Holomorphic Mappings of Submanifolds 349
Ch. XIII Mappings of Real-algebraic Subvarieties 379