Mathematical Sciences: Geometric Problems in Several ComplexVariables

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Two general directions of mathematical research in the field of several complex varibles characterize this project. Both may be viewed as geometric in their statement, but analytic in the methods employed. The first question is the result of a long-term effort to understand the nature of proper mappings between balls in spaces of several complex variables. A holomorphic mapping beween discs in one dimension (continuous up to the boundary) can only be achieved by a finite Blaschke product. In higher dimension, the only (proper) maps were shown to be automorphisms. The question was reopened when it was discovered that when the range was a ball of higher dimension, there were distinctly new and interesting proper maps from the lower to the higher dimensional balls. They may not be smooth up to the boundary, but if they are then one can say when they must be rational. When the map is actually polynomial, then it factors completely analogously to the finite Blaschke product. The thrust of this work will be to take up the classification of those rational proper maps which are not polynomial. It is believed that they too admit a factorization into simple component parts. The second part of the work concerns the geometry of real hypersurfaces in complex n-dimensional space and Cauchy-Riemann manifolds. This is more technical in its description. The goal is to formulate some of the extrinsic notions of multiplicities and orders of contact for real hypersurfaces in an intrinsic manner that will extend to CR manifolds. Questions about the geometry of real hypersurfaces are part of a larger effort to understand surface geometry near points where the Levi form is degenerate. Some of the most promising research taking place in several complex variables at the present time centers on domains whose boundaries have degeneracies such as those studied here.