Mathematical Sciences: Mapping Problems in Complex Analysis

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Several important mathematical topics will be addressed in this project. Its primary focus is the theory of several complex variables with emphasis on (a) mappings between domains in spaces of the same or different dimensions, (b) extension to the boundary of such maps and (c) the resulting regularity one can expect. The basis for this work can be traced to the classical Riemann Mapping Theorem in one complex variable which states that any two finite and simply connected regions can be transformed into one another by univalent analytic mappings. The same question in several variables has two formulations. This is due to the observation that even elementary domains such as spheres and polydiscs cannot be transformed analytically into one another. The work considers those domains which can be analytically mapped onto one another in a one-one fashion and those which effectively can be mapped onto one another in a (more or less) finite-to-one-manner. The latter functions are called proper mappings. The research is concerned with questions of the following type. If a proper map transforms one domain into another, what conditions on the boundary or the map ensure that the map can be extended to the boundary? If an extension exists, what is its degree of smoothness? Considerable progress has been made on these questions when the two domains have curved and very regular boundaries. The object now is to reduce the assumptions on the boundaries and apply new results of complex analysis, geometry and partial differential equations to develop the theory to its fullest extent. The main problem to be addressed in the near term is considerable in itself: one must show that a continuous map between certain manifolds is actually infinitely differentiable. Results of this type are already known, but not enough to suggest a path to the best possible conclusions.