Mathematical Sciences: Domains of Holomorphy and Their Automorphisms

数学科学：全纯域及其自同构

• 批准号: 9201019
• 负责人: Kang-Tae Kim
• 金额: $ 3.5万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-15 至 1995-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9201019&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Domains; Holomorphy; Their

This project concerns mathematical research on problems arising in the area of several complex variables. It involves the function-theoretic and geometric properties of bounded pseudoconvex domains in n-dimensional complex space and their automorphism groups. The first goal of the research is to identify domains with noncompact automorphism groups by the local geometry of their boundaries. Particular interest is placed on boundaries which are not entirely smooth. The second component of the work is the extension of studies on the asymptotic behavior of the intrinsic invariant metrics defined on the bounded pseudoconvex domains. The research relates the boundary behavior of the automorphisms and the compactness of the automorphism groups. Finally, work will be done identifying the bounded symmetric domains by their boundary geometry among domains with noncompact automorphism groups. The projects are connected by the theme that the size of the automorphism groups and the boundary geometry of bounded domains are tied together. The study of several complex variables arose at the beginning of the century initially to examine those properties of classical function theory which generalize to several variables. It turned out that most properties do not generalize and research turned instead toward some of the more intriguing geometric and analytic questions which arise only when the domains are of dimension two or greater. This project continues that tradition.

这个项目涉及对几个复变量领域中出现的问题的数学研究。它涉及n维复空间中有界伪凸域及其自同构群的泛函理论和几何性质。研究的第一个目标是通过其边界的局部几何来识别具有非紧自同构群的域。人们特别关注的是并不完全平滑的边界。工作的第二部分是对定义在有界伪凸域上的内在不变度量的渐近行为的研究的推广。研究了自同构群的紧性与自同构群的边界性质之间的关系。最后，我们将利用具有非紧自同构群的区域之间的边界几何来识别有界对称域。这些项目通过自同构群的大小和有界域的边界几何联系在一起的主题联系在一起。对多个复变量的研究兴起于本世纪初，最初是为了检验经典函数论的那些推广到多个变量的性质。事实证明，大多数性质并不是泛化的，相反，研究转向了一些更有趣的几何和分析问题，这些问题只有在区域是二维或更大维时才会出现。这个项目延续了这一传统。