ABSTRACT Proposal: DMS-9622594 PI: Pinchuk The proposed research will be carried out in three directions. The first is the study of the conjecture that in complex n-space any proper holomorphic mapping between bounded domains with real analytic boundaries extends to a proper holomorphic mapping between neighborhoods of their closures. In dimension 1 this is a fundamental result known as the Schwarz Reflection Principle. For n>1 the conjecture has been proved only in special cases. Pinchuk will use a geometric approach, which is based on the technique of Segre varieties, and is a powerful new tool to study various problems in domains with real analytic boundaries. The goal of the first part of the proposal is to develop the geometric reflection principle for higher dimensions and prove the conjecture above in general. The second direction is to apply the technique of Segre varieties together with other geometric methods to study rigidity phenomena of holomorphic mappings. Namely, Pinchuk plans to describe domains with noncompact automorphism groups. He will also attack the related problem that in rather general situations proper holomorphic self-mappings are biholomorphic. The third direction is the investigation of the so called Abhyankar Equations (AEs) with respect to the Jacobian Conjecture (JC). The JC (in the complex case, which is the principal one) claims that any locally inevitable (biholomorphic) polynomial mapping is globally inevitable. The JC is one of the most intriguing problems in mathematics. It has been intensively studied by many mathematicians (including very famous ones) since 1939. A number of partial results as well as faulty proofs has been published. Most of the experts believed and still believe that the JC is true. Therefore a recent counterexample to the real version of the JC, which was constructed by Pinchuk (using the Aes) came as a surprise. The JC has equivalent formulations in different areas of mathematics. Among others, its resolution ( positive or negative) will have implications for the finiteness of inversion formulas, for the existence of global solutions of certain differential equations, and for singularities of algebraic sets. The other problems of this proposal are natural, important, and known as hard problems in several complex variables. Their solution will became the essential part of the theory of holomorphic mappings.
