Mathematical Sciences: Geometric Problems in Several ComplexVariables

在几个复杂变量领域的数学研究的两个一般方向是这个项目的特点。两者在表述上都可以看作是几何的，但在使用的方法上却可以看作是解析的。第一个问题是长期努力的结果，目的是理解几个复杂变量空间中球之间的适当映射的本质。一维圆盘间的全纯映射（连续到边界）只能通过有限Blaschke积来实现。在更高的维度中，唯一的（适当的）映射被证明是自同构。当人们发现，当范围是一个高维球时，从低维球到高维球有明显新的、有趣的固有映射，这个问题就重新开始了。他们可能不会平滑到边界，但如果他们是，那么人们可以说他们什么时候必须是理性的。当映射是多项式时，它就完全类似于有限Blaschke积。这项工作的重点将是对那些非多项式的有理固有映射进行分类。人们相信它们也可以被分解成简单的组成部分。第二部分研究了复杂n维空间中的实超曲面和柯西-黎曼流形的几何问题。这在描述上更具技术性。我们的目标是以一种内在的方式来表述真实超曲面的多重度和接触阶数的一些外在概念，并将其扩展到CR流形。关于真实超曲面几何的问题是理解Levi形式简并点附近曲面几何的更大努力的一部分。目前在几个复杂变量中进行的一些最有前途的研究集中在边界具有简并的域上，例如这里所研究的域。

Two general directions of mathematical research in the field of several complex varibles characterize this project. Both may be viewed as geometric in their statement, but analytic in the methods employed. The first question is the result of a long-term effort to understand the nature of proper mappings between balls in spaces of several complex variables. A holomorphic mapping beween discs in one dimension (continuous up to the boundary) can only be achieved by a finite Blaschke product. In higher dimension, the only (proper) maps were shown to be automorphisms. The question was reopened when it was discovered that when the range was a ball of higher dimension, there were distinctly new and interesting proper maps from the lower to the higher dimensional balls. They may not be smooth up to the boundary, but if they are then one can say when they must be rational. When the map is actually polynomial, then it factors completely analogously to the finite Blaschke product. The thrust of this work will be to take up the classification of those rational proper maps which are not polynomial. It is believed that they too admit a factorization into simple component parts. The second part of the work concerns the geometry of real hypersurfaces in complex n-dimensional space and Cauchy-Riemann manifolds. This is more technical in its description. The goal is to formulate some of the extrinsic notions of multiplicities and orders of contact for real hypersurfaces in an intrinsic manner that will extend to CR manifolds. Questions about the geometry of real hypersurfaces are part of a larger effort to understand surface geometry near points where the Levi form is degenerate. Some of the most promising research taking place in several complex variables at the present time centers on domains whose boundaries have degeneracies such as those studied here.