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

摘要：DMS-962304 PI: Epstein CR-geometry是复杂几何的自然奇维模拟。根据Kohn， Boutet de Monvel， Harvey和Lawson的工作，我们知道任何5维或更大的紧致严格伪凸cr流形都可以被实现为紧致正规Stein空间的边界。具有这种实现的cr流形称为可嵌入的。自20世纪60年代以来，人们已经知道三维中的情况是完全不同的：可嵌入cr结构的一般摄动是不可嵌入的。我们的目标是将紧三流形上的可嵌入cr -结构集描述为所有cr -结构集的子集。对于单位3球的变形，小的可嵌入微扰本质上是所有cr结构集合的无限维和无限协维解析子流形。本提案中概述的研究旨在将这些结果扩展到更一般的3-流形类。在早期的工作中，研究者介绍了一组可嵌入cr结构的分层。局部分层由形式解析关系定义。提出的工作的一个主要推力是分析这些方程使用的方法产生的纳什-莫泽隐函数定理。希望能证明该地层具有横向解析结构。研究者和G. Henkin所寻求的一般立场论证和对Kiremidjian结果的概括将表明，该层析只有有限多个不同的层。这意味着可嵌入结构集是合理拓扑中的封闭子集。使用类似于Osgood， Phillips和Sarnak使用的极值行列式，希望为可嵌入结构空间定义一个自然耗尽函数。数学在物理、经济、化学、工程等领域的重要性源于这样一个事实，即方程描述了这些领域中出现的变量之间的关系。因此，作出预测就简化为解方程。因此，我们需要确定方程是否有解的准则，以及求解它们的算法。在实践中，这只能近似地完成，因此对所犯错误的大小进行估计是很重要的。在这个建议中，我们考虑在解析几何中自然出现的一系列相关方程。这些方程通常没有解。即使家族中的方程是密切相关的，但随着一个人在家族中移动，可解性的性质也会发生很大的变化。我们的主要目的是了解这种不稳定性，并给出描述表现良好的家庭成员的标准。在这个分析中出现的问题和在使用的图像重建技术中遇到的问题之间有相似之处，例如在CAT扫描中。希望更好地了解目前的情况将有助于深入了解出现类似不稳定的其他情况。

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.