A number of related problems arising in the mathematical theory of several complex variables will be addressed in this project. They are concerned primarily with the theory of analytic functions of several variables and the domains on which the functions are defined. The first question concerns curves in two complex dimensions on which restrictions of holomorphic functions can be expressed as a sum of two functions of a single variable. The requirement for this is that the two coordinates projections of the curve have multiplicity equal to one. Work will be done in determining what the form of the function can be if the multiplicities are known but are not equal. A second line of investigation will consider the higher dimensional form of a famous theorem of Carleman which relates the quadratic norm of the derivative of a holomorphic function with the integral of (the modulus of) the derivative along the boundary of a domain. This result is the general isoperimetric inequality which allows for multiple coverings of the domain. A first step in this direction will be to consider n-sheeted covers of a disk with boundary which has finite linear measure. An interesting by-product of such a result would be to prove that such covers have finite measure, a fact which is not yet established. A continuing investigation will also be carried out on the structure of polynomial hulls in two complex variables. The polynomial hull of any compact set is the largest set to which all polynomials can be extended without increase in uniform norm. Hulls are easy to describe in one complex dimension and almost impossible to visualize in several. If the compact set lies over a disk in the first variable, and the fibers are convex, then the hull can be described precisely. The current work will seek to analyze the hulls when the fibers are no longer connected, for example when then are the join of two disjoint Jordan curves. Some form of two-sheeted cover of the disk is the anticipated result.
