Real Hypersurfaces and Complex Analysis

1480 NOTICES OF THE AMS VOLUME 42, NUMBER 12 T he theory of functions (what we now call the theory of functions of a complex variable) was one of the great achievements of nineteenth century mathematics. Its beauty and range of applications were immense and immediate. The desire to generalize to higher dimensions must have been correspondingly irresistible. In this desire to generalize, there were two ways to proceed. One was to focus on functions of several complex variables as the generalization of functions of one complex variable. The other was to consider a function of one complex variable as a map of a domain in C to another domain in C and to study, as a generalization, maps of domains in Cn. Both approaches immediately led to surprises and both are still active and important. The study of real hypersurfaces arose within these generalizations. This article surveys some contemporary results about these hypersurfaces and also briefly places the subject in its historical context. We organize our survey by considering separately these two roads to generalization. We start with a hypersurface M2n−1 of R2n and consider it as a hypersurface of Cn, using an identification of R2n with Cn. We call M a real hypersurface of the complex space Cn to distinguish it from a complex hypersurface, that is, a complex n− 1 dimensional submanifold of Cn. This said, the dimensions in statements like M2n−1 ⊂ Cn