Mathematical Sciences: Geometric Function Theory in Several Complex Varaiables

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This project continues mathematical work on problems in the study of functions and domains in the space of several complex variables. The first involves a combination of ordinary differential equations and several complex variables where one is interested in holomorphic vector fields, realized as systems of differential equations. The solution vectors provide examples of one parameter groups of automorphisms of the full n-dimensional space. When the vector field is complete, the automorphisms can be continued indefinitely. One goal of this project is to determine the extent to which every holomorphic vector field can be approximated by a complete one. A second line of investigation focuses on the image of two-dimensional space under a holomorphic embedding. More precisely, work will be done in determining the type of algebraic variety which can be omitted by such an embedding. It is known that one complex line may be omitted and three complex lines cannot. Additional work will be done in an effort to understand how the example of Duval allows for a Lagrangian disk to be embedded in two-dimensional space which is not polynomially convex. Yet, in the appropriate sense, allpolynomial hulls can always be explained in terms of holomorphic disks. The difference must lie somewhere between the idea of a holomorphic disk and the older idea of an H-infinity disk. Questions of this type have been studied for decades, so progress is expected to be slow. Several complex variables arose at the beginning of the century as a natural outgrowth of studies of functions of one complex variable. It became clear early on that the theory differed widely from its predecessor. The underlying geometry was far more difficult to grasp and the function theory had far more affinity with partial differential operators of first order. It thus grew as a hybrid subject combining deep characteristics of differential geometry and differential equations. Many of the fundamental structures were defined in the last three decades. Current studies still concentrate on understanding these basic mathematical forms.