Mathematical Sciences: Domains of Holomorphy and Their Automorphisms

该项目涉及在几个复杂变量领域中出现的问题的数学研究。研究了n维复空间中有界伪凸区域及其自同构群的泛函性质和几何性质。研究的第一个目标是通过边界的局部几何来识别具有非紧自同构群的区域。特别关注的是不完全平滑的边界。第二部分是对有界伪凸域上固有不变度量的渐近行为研究的扩展。研究了自同构群的边界行为和紧性。最后，利用非紧自同构群域间的边界几何来确定有界对称域。这些项目通过自同构群的大小和有界域的边界几何形状捆绑在一起的主题联系在一起。复数变量的研究兴起于本世纪初，最初是为了检验经典函数理论的那些可推广到复数变量的性质。事实证明，大多数性质并没有推广，研究转向了一些更有趣的几何和分析问题，这些问题只有在二维或更大的域才会出现。这个项目延续了这一传统。

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.