Moduli space of bounded complete Reinhardt domains and complex plateau problem

One of the most fundamental problems in complex geometry is to determine when two bounded domains in Cn are biholomorphically equivalent. Even for complete Reinhardt domains, this fundamental problem remained unsolved for many years. Using the Bergmann function theory, we construct an infinite family of numerical invariants from the Bergman functions for complete Reinhardt domains in Cn . These infinite family of numerical invariants are actually a complete set of invariants if the domains are pseudoconvex with C 1 boundaries. For bounded complete Reinhardt domains with real analytic boundaries, the complete set of numerical invariants can be reduced dramatically although the set is still infinite. We shall also discuss the role of the Hilbert 14th problem in the construction of numerical biholomorphic invariants of complete Reinhardt domains in Cn .;Moreover, for n = 2, we can construct an infinite family of numerical invariants from the Bergman functions for such domains in An-variety {(x, y, z) ∈ C3 : xy = zn +1}. These infinite family of numerical invariants are actually a complete set of invariants for either the set of all bounded strictly pseudoconvex complete Reinhardt domain in An variety or the set of all bounded pseudoconvex complete Reinhardt domains with real analytic boundaries in An variety. In particular the moduli space of these domains in An variety is constructed explicitly as the image of this complete family of numerical invariants. It is well known that An variety is the quotient of cyclic group of order n + 1 on C2 . We prove that the moduli space of bounded complete Reinhardt domains in An variety coincides with the moduli space of the corresponding bounded complete Reinhardt domains in C2 .;Another natural fundamental questions of complex geometry is to study the boundaries of complex varieties. For example, the famous classical complex Plateau problem asks which odd dimensional real sub-manifolds of CN are boundaries of complex sub-manifolds in CN . Let X be a compact connected strictly pseudoconvex CR manifold of real dimension 2n - 1 in Cn+1 . For n ≥ 3, Yau used Kohn-Rossi cohomology groups to solve the classical complex Plateau problem in 1981. For n = 2, the problem has remained unsolved for over a quarter of a century. In this paper, we introduce a new CR invariant of X to solve this problem completely.