Mathematical Sciences: Geometry and Analysis of 3-Dimensional CR-Structures

数学科学：3 维 CR 结构的几何和分析

• 批准号: 9623040
• 负责人: Charles Epstein
• 金额: $ 10.91万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623040&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Analysis; Dimensional

ABSTRACT Proposal: DMS-962304 PI: Epstein CR-geometry is the natural odd dimensional analogue of complex geometry. From work of Kohn, Boutet de Monvel, and Harvey and Lawson it is known that any compact strictly pseudoconvex CR-manifold of dimension 5 or greater can be realized as the boundary of a compact normal Stein space. A CR-manifold with such a realization is called embeddable. It has been known since the 1960s that the situation in 3-dimensions is quite different: the generic perturbation of an embeddable CR-structure is not embeddable. Our goal is to describe the set of embeddable CR-structures on a compact three manifold as a subset of the set of all CR-structures. For the case of deformations of the unit 3-sphere the small embeddable perturbations are essentially an infinite dimensional and infinite codimensional analytic submanifold of the set of all CR-structures. The research outlined in this proposal is directed towards extending such results to more general classes of 3-manifolds. In earlier work the investigator introduced a stratification of the set of embeddable CR-structures. Locally the stratification is defined by formally analytic relations. A major thrust of the proposed work is to analyze these equations using methods arising from the Nash-Moser implicit function theorem. It is hoped that it can be shown that the strata have a transverse analytic structure. General position arguments and a generalization of a result of Kiremidjian being sought by the investigator and G. Henkin would then show that the stratification has only finitely many distinct strata. This would imply that the set of embeddable structures is a closed subset in a reasonable topology. Using extremal determinants analogous to those used by Osgood, Phillips and Sarnak it is hoped to define a natural exhaustion function for the space of embeddable structures. The importance of mathematics in physics, economics, chemistry, engineering, etc. stems from the fact that EQUATIONS describe the rela tionships among the variables that arise in these fields. Making predictions is thereby reduced to solving the equations. One therefore needs criteria to determine whether the equations have solutions, and algorithms for solving them. In practice this can only be done approximately so it is important to have an estimate for the size of he errors one is making. In this proposal we consider a FAMILY of related equations that arise naturally in analytic geometry. These equations often do not have solutions. Even though the equations in the family are closely related, the property of solvability can change very wildly as one moves through the family. Our principal aim is to understand this instability and give criteria that describe the well behaved members of the family. There are similarities between the issues that arise in this analysis and problems encountered in image reconstruction techniques used, e.g., in CAT scans. It is hoped that a better understanding of the case at hand will provide insight into other cases where similar instabilities arise.

摘要提案：DMS-962304 PI：爱泼斯坦 CR-几何是复几何的自然奇维模拟。 从Kohn、Boutet de Monvel、Harvey和Lawson的工作中可以知道，任何5维或更大维的紧致严格伪凸CR-流形都可以实现为紧致正规Stein空间的边界。 具有这样一个实现的CR-流形称为可嵌入的。 自20世纪60年代以来，人们已经知道，在3维的情况是完全不同的：一般扰动的嵌入CR-结构是不可嵌入的。我们的目标是把紧致三流形上的可嵌入CR-结构集描述为所有CR-结构集的子集。对于单位3-球面的形变情形，小的可嵌入扰动本质上是所有CR-结构的集合的无穷维和无穷余维解析子流形。 在这项建议中概述的研究是针对扩展这些结果更一般的类3-流形。在早期的工作中，研究人员介绍了一个分层的一组嵌入式CR结构。局部的分层定义的形式分析关系。所提出的工作的一个主要推力是分析这些方程使用的方法所产生的纳什-莫泽隐函数定理。希望能说明地层具有横向解析结构。一般立场的论点和Kiremidjian的一个结果的推广正在寻求的研究者和G。然后，亨金将表明，分层只有100多个不同的层。这意味着可嵌入结构的集合是合理拓扑中的闭子集。使用类似于Osgood，菲利普斯和Sarnak所使用的极值行列式，希望定义一个自然的耗尽功能的空间嵌入结构。 数学在物理学、经济学、化学、工程学等方面的重要性，源于方程描述了这些领域中出现的变量之间的关系。因此，做预测就简化为解方程。因此，我们需要判断方程是否有解的标准，以及求解方程的算法。在实践中，这只能近似地完成，所以重要的是要有一个估计的大小，他的错误之一。在这个建议中，我们考虑在解析几何中自然出现的一族相关方程。这些方程往往没有解。尽管方程族中的方程是密切相关的，但可解性的性质可以随着方程族的移动而发生非常大的变化。我们的主要目的是理解这种不稳定性，并给出描述家庭中表现良好的成员的标准。在该分析中出现的问题与所使用的图像重建技术中遇到的问题之间存在相似之处，例如，在CAT扫描中。我们希望，更好地了解手头的情况将提供洞察力的其他情况下，类似的不稳定性出现。