Mathematical Sciences: Function Theory in Several Complex Variables

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DMS-9500881 Huang Huang will study the mapping problems for real hypersurfaces in complex spaces and try to prove various regularity results for Cauchy-Riemann mappings. He will investigate the smooth and analytic structure of a real submanifold in the complex space and the structure of its local hull of holomorphy. Also he will explore the Kobayashi extremal mappings of bounded domains in complex n-space and their applications to the study of holomorphic self-mappings. 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 it 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.