Mathematical Sciences: Holomorphic Mappings and Projections

点击翻译按钮获取中文摘要

Work on this project combines geometry and function theory in several complex variables to study how the geometry of a domain influences the properties of holomorhpic functions defined on the domain. Although progress has been made recently in understanding domains whose complex geometry is not excessively degenerate, so-called domains of finite type, the general state of knowledge is still limited. Among the most manageable classes of holomorphic functions defined on a domain, those which are square integrable are probably the most useful. It is possible to convert any square integrable function on a domain into its nearest holomorphic neighbor by a process of integration against a kernel function - the Bergman kernel. There is a unique such kernel for each domain and integration projects each function onto the best holomorphic approximant. Perhaps the most important question related to holomorphic functions on domains is that of determining whether or not the projection preserves functions which are differentiable on the boundary of the domain. Are their images also differentiable on the boundary? Even experienced mathematicians would consider the likelihood of such an event an accident. The facts are otherwise. If the boundary has a smooth, curved shape (pseudoconvex) then the projections are always differentiable. Work will continue in an effort to extend present results to domains which are not pseudoconvex. Among the more tractable domains to be analyzed are the Hartogs domains. These are domains possessing some circular symmetry. Several counterexamples in function theory have been found in Hartogs domains. Ultimately, this work will be applied to the general problem of analyzing biholomorphic mappings between domains in several complex variables. These maps are known to have smooth extensions to the respective boundaries in many instances. A complete description of the domains for which these smooth extensions exist is probably the most important question under investigation in several complex variables at this time.