Complex Analysis and CR Geometry

Cauchy-Riemann (CR) geometry is the study of manifolds equipped with a system of CR-type equations. Compared to the early days when the purpose of CR geometry was to supply tools for the analysis of the existence and regularity of solutions to the $\bar\partial$-Neumann problem, it has rapidly acquired a life of its own and has became an important topic in differential geometry and the study of non-linear partial differential equations. A full understanding of modern CR geometry requires knowledge of various topics such as real/complex differential and symplectic geometry, foliation theory, the geometric theory of PDE's, and microlocal analysis. Nowadays, the subject of CR geometry is very rich in results, and the amount of material required to reach competence is daunting to graduate students who wish to learn it. However, the present book does not aim at introducing all the topics of current interest in CR geometry. Instead, an attempt is made to be friendly to the novice by moving, in a fairly relaxed way, from the elements of the theory of holomorphic functions in several complex variables to advanced topics such as extendability of CR functions, analytic discs, their infinitesimal deformations, and their lifts to the cotangent space. The choice of topics provides a good balance between a first exposure to CR geometry and subjects representing current research. Even a seasoned mathematician who wants to contribute to the subject of CR analysis and geometry will find the choice of topics attractive.