Local Geometric Properties of Real Submanifolds in Complex Space.

摘要

A fundamental theorem in the theory of one complex variable is the Riemann mapping theorem which asserts that any proper simply connected domain omega in the complex plane is biholomorphically equivalent to the unit disk, i.e. there exists an invertible holomorphic mapping sending omega onto the unit disk. Moreover, it is a classical result that if omega has a smooth C(sup infinity) boundary, then the mapping extends smoothly to the boundary aloha omega. Similarly, if aloha omega is real-analytic, then the mapping extends holomorphically to an open neighborhood of straight line over omega, the closure of omega. The classification (up to biholomorphic equivalence) of domains in C(sup N), N > or = 2, is much more complicated. Already early in the 20th century, Poincare (Po 1907) showed that the bidisk is not biholomorphically equivalent to the unit ball in C(sup 2), even though both are bounded domains which are diffeomorphic to each other (and even real analytically equivalent). It was also known to him that two domains, chosen at random in a class of bounded diffeomorphic domains, are very unlikely to be biholomorphically equivalent.