Mathematical Sciences: Partial Differential Equations

数学科学：偏微分方程

• 批准号: 8908167
• 负责人: C. Robin Graham
• 金额: $ 3.96万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-15 至 1992-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8908167&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Partial; Differential; Equations

Work will continue on research projects involving the interplay of mathematical analysis, geometry and algebra. Questions arising in conformal and biholomorphic geometry and Fuchsian partial differential equations will be the focus of the research. The first project is a continuing collaboration aimed at the construction of all scalar invariants of a conformal or Cauchy- Riemann manifold. A complete solution for odd dimensions in the conformal case appears to be accessible at this time. The problem has been reduced to a purely algebraic problem in invariant theory. A partial solution in even dimensions has also been found. The ultimate goal of this work is to give an explicit invariant description of the asymptotic expansion of the Bergman kernel of a smooth bounded strictly pseudoconvex domain in complex n-dimensional space. A related project involves the application of representation-theoretic methods to parabolic invariant theory. The objective is the same as above except that an alternate approach is to be analyzed using Verma modules which have the same structure as the space of jets of a conformal structure. A formulation of the problem in terms of Verma modules brings more structure and tools to bear as well as providing a large class of test problems on which to develop methods. The third project concerns the study of existence, uniqueness and regularity of Fuchsian differential equations described by Einstein metrics and harmonic maps on conformally compact manifolds. These equations are given by nonlinear systems of elliptic type which degenerate at the manifold boundary. As a first goal, work will be done in finding conformally compact metrics having constant negative Ricci curvature, with the conformal class specified in advance. This would extend the model case of the hyperbolic metric of constant negative curvature defined on a ball. Here, the conformal class is the standard metric of the sphere.//

将继续开展涉及数学分析、几何和代数相互作用的研究项目。共形几何、双全纯几何和富氏偏微分方程中的问题将是研究的重点。第一个项目是一个持续的合作，目的是构造共形流形或柯西-黎曼流形的所有标量不变量。此时，保形情况下奇数维的完整解似乎是可行的。这个问题已经归结为不变量理论中的一个纯代数问题。在偶数维上也找到了部分解。本文的最终目的是给出n维复空间中光滑有界严格伪凸域的Bergman核的渐近展开式的一个显式不变刻画。一个相关的项目涉及表示论方法在抛物不变量理论中的应用。目标与上面相同，除了使用与共形结构的喷流空间具有相同结构的Verma模块进行分析之外。按照Verma模块对问题进行表述会带来更多的结构和工具，并提供一大类测试问题来开发方法。第三个项目是研究共形紧流形上由爱因斯坦度量和调和映射描述的富克斯微分方程解的存在唯一性和正则性。这些方程是由在流形边界退化的椭圆型非线性系统给出的。作为第一个目标，我们将致力于寻找具有常负Ricci曲率的共形紧度量，并预先指定共形类。这将推广定义在球上的常负曲率的双曲度规的模型情况。这里，保形类是球面的标准度量。//