Mathematical Sciences: Holomorphic Mappings and Projections

在这个项目的工作结合几何和函数理论在几个复杂的变量来研究几何如何影响域上定义的全纯函数的性质。尽管最近在理解复杂几何不过度简并的领域（即所谓有限型领域）方面取得了进展，但一般的知识状态仍然有限。在定义域上定义的最易于管理的全纯函数中，那些平方可积的函数可能是最有用的。通过对核函数Bergman核的积分过程，可以将一个定义域上的任何平方可积函数转化为它最近的全纯邻居。每个域都有一个唯一的核，积分将每个函数投影到最佳全纯近似上。关于定义域上全纯函数的最重要的问题可能是确定投影是否保留定义域边界上可微的函数。它们的像在边界上也是可微的吗？即使是经验丰富的数学家也会认为这种事件的可能性是意外。事实并非如此。如果边界是光滑的，弯曲的形状（伪凸），那么投影总是可微的。我们将继续努力将目前的结果扩展到非伪凸的领域。要分析的比较容易处理的域是哈托格域。这些区域具有一定的圆形对称性。在哈托格域中发现了函数理论的几个反例。最后，这项工作将应用于分析几个复杂变量域之间的生物全纯映射的一般问题。众所周知，在许多情况下，这些地图对各自的边界有平滑的扩展。对于这些平滑扩展存在的域的完整描述可能是目前在几个复杂变量中研究的最重要的问题。

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.