Mathematical Sciences: Complex Geometry and Analysis

9408994伯恩斯该奖项继续支持与区域和边界相关的几何问题的数学研究，这些问题位于多个复变量空间中。将进行两个主要领域的研究。它们都涉及重整化特征类的解释以及它们与带边界的复流形上的分析问题的关系。第一个问题涉及复双曲流形的均匀化和结构。一致化问题是确定一个紧致的、连通的、球面的CR-流形的一个分支何时可以实现为以一组不连续的CR-自同构群为模的n-复变量球的区域。第二条研究线考虑某些线性偏微分方程组，目的是找到无限体积的黎曼流形或Kahler流形的有限特征数的解析解释。还将努力证明与最近关于格劳特管的结果相反的工作。也就是说，证明了这些流形的所有双全纯都是由等距诱导的。本世纪初出现了几个复变量，这是一个复变量函数研究的自然结果。很早就很明显，这一理论与其前身有很大不同。基本的几何学要难得多，而函数论与一阶偏微算符的亲和力要强得多。因此，它成长为一门混合学科，结合了微分几何和微分方程式的深刻特征。许多基本结构都是在过去30年里定义的。目前的研究仍然集中在理解这些基本的数学形式上。

9408994 Burns This award continues support for mathematical research on geometric problems associated with domains and there boundaries situated in spaces of several complex variables. Two major areas of study will be undertaken. Both are concerned with interpretation of renormalized characteristic classes and their relation to analytic problems on complex manifolds with boundary. The first involves uniformization and structure of complex hyperbolic manifolds. The uniformization question is that of determining when a component of a complex manifold which is a compact, connected, spherical CR-manifold can be realized as a domain of a ball in n-complex variables modulo a properly discontinuous group of CR-automorphisms. The second line of investigation considers certain linear partial differential equations with the goal of finding analytic interpretations of characteristic numbers which are finite for Riemannian or Kahler manifolds of infinite volume. Work will also be done in an effort to prove a converse to a recent result on Grauert tubes. Namely, to show that all biholomorphisms of these manifolds are induced by isometries. Several complex variables arose at the beginning of the century as a natural outgrowth of studies of functions of one complex variable. It became clear early on that the theory differed widely from it predecessor. The underlying geometry was far more difficult to grasp and the function theory had far more affinity with partial differential operators of first order. It thus grew as a hybrid subject combining deep characteristics of differential geometry and differential equations. Many of the fundamental structures were defined in the last three decades. Current studies still concentrate on understanding these basic mathematical forms.