Lecture Notes on the Local Equivalence Problems For Real Submanifolds in Complex Spaces

§1. Global and Local Equivalence Problems There is a classical theorem in complex analysis, called the Riemann mapping theorem, which states that any simply connected domain in C is either holomorphically equivalent to C or to the unit disk. For more general domains in C, He-Schramm showed [HS] that if ∂D has countably many connected components, then D is holomorphically equivalent to a circle domain whose boundaries are either points or circles. These results give a nice picture on the holomorphic structures for domains in C. When one goes to higher dimensions, a natural question is then to investigate the complex structure for domains in C for n ≥ 2. More precisely, given two domains in C, one would like to know if there is a biholomorphic map between them. This the so-called global equivalence problem in several complex variables. Along these lines of investigations, substantial progress has been made in the past 30 years ([Fe], [CM], [BSW], etc.). However, we are still a certain big distance away from getting a relatively complete picture as in the one complex variable. An approach to the study of the equivalence problem is to attach holomorphic invariants to each given domain. Since domains in C are open complex manifolds, many (interior) invariants which are crucial for the study of compact complex manifolds are difficult even to define. As already observed by Poincaré about 100 years ago, the interior complex structure of a domain D in C for n > 1 is closely related to the partial complex structure in its boundary, which is the so-called CR structure. Hence, the classification of the complex structures for domains in C may be reduced to the equivalence problem for the boundary CR structures. Indeed, this idea has been proved to be fundamental through the work of Cartan, Tanaka, Chern-Moser, etc.. And it indeed led to the solutions to many questions.