Mathematical Sciences: Holomorphic Mappings and Projections

这个项目的工作结合了几个复变量中的几何和函数理论，以研究区域的几何如何影响定义在区域上的全纯函数的性质。虽然最近在理解其复杂几何不是过度退化的域(即所谓的有限型域)方面取得了进展，但一般的知识状态仍然是有限的。在定义在区域上的最易管理的全纯函数类中，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，可以将区域上的任何平方可积函数转换为其最近的全纯邻域。对于每个域，都有一个唯一的这样的核，并且积分将每个函数投影到最佳全纯逼近上。也许与区域上的全纯函数相关的最重要的问题是确定投影是否保持在区域边界上可微的函数。他们的图像在边界上也是可区分的吗？即使是经验丰富的数学家也会认为发生这种事件的可能性是一种意外。事实并非如此。如果边界具有光滑、弯曲的形状(伪凸)，则投影总是可微的。将继续努力将目前的结果推广到非伪凸的区域。在要分析的更容易处理的域中，有Hartogs域。这些区域具有一定的圆对称性。在Hartogs域中已经找到了函数论中的几个反例。最终，这项工作将被应用于分析多个复变元域之间的双全纯映射的一般问题。众所周知，在许多情况下，这些地图具有对各自边界的平滑延伸。这些光滑延拓存在的区域的完整描述可能是目前正在研究的几个复变量中最重要的问题。

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.