Mathematical Sciences: Complex Geometry and Analysis

9408994 Burns该奖项继续支持与域和位于多个复杂变量空间中的边界相关的几何问题的数学研究。将进行两个主要领域的研究。两者都是关于重归一化特征类的解释及其与复流形上有边界的解析问题的关系。第一部分涉及到复杂双曲流形的均匀化和结构。均匀化问题是确定复流形的一个紧的、连通的球形cr -流形的一个分量何时可以在n个复变量上模一个适当不连续的cr -自同构群中的一个球的定域。第二项研究考虑某些线性偏微分方程，目的是寻找无限体积黎曼流形或卡勒流形有限特征数的解析解释。研究人员还将努力证明与格劳尔特管最近的结果相反的结果。也就是说，证明这些流形的所有生物全纯态都是由等距引起的。本世纪初出现了几个复变量，作为一个复变量函数研究的自然产物。很明显，这个理论与它的前身有很大的不同。底层的几何结构更难掌握，而函数理论与一阶偏微分算子的关系要密切得多。因此，它成长为一门混合学科，结合了微分几何和微分方程的深层特征。许多基本结构是在过去三十年中确定的。目前的研究仍然集中在理解这些基本的数学形式。

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.