Mathematical Sciences: CR Manifolds and Holomorphic Embeddings

数学科学：CR 流形和全纯嵌入

• 批准号: 8913354
• 负责人: Sidney Webster
• 金额: $ 9.01万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1992-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8913354&HistoricalAwards=false
• 关键词: Mathematical; Sciences; CR; Manifolds; Holomorphic

The principal investigator will analyze the problem of existence and regularity of holomorphic embeddings of certain CR manifolds of real hypersurface type. These are manifolds having the same internal structure as is induced on the boundary of a smooth, strongly pseudoconvex domain in complex Euclidean space. Key elements of the method to be applied are Henkin's solution operators for the tangential Cauchy-Riemann equations. For the local problem he will improve previous estimates and reduce, as much as possible, the considerable derivative loss in his existing arguments. The problem of global embeddings will be further analyzed. The study will lead to a deeper understanding of the concept of boundary of a complex manifold and of properties of certain over-determined systems of linear, first order, partial differential equations. An endless curve in the plane which wraps up on itself is an example of an immersion of the real line which is not an embedding. An embedding always manages to steer clear of itself. Holomorphic embeddings require that the function, which injects the curve or surface into the target manifold, be smooth in the complex analytic sense. Theorems from the study of CR or complex analytic manifolds have always seemed miraculous by the standards of traditional real analysis. The results to be completed by this project will be no different.

主要研究实超曲面型CR流形的全纯嵌入的存在性和正则性问题。这些流形的内部结构与复欧氏空间中光滑、强伪凸域边界上的流形相同。所用方法的关键元素是切向Cauchy-Riemann方程的Henkin解算子。对于本地问题，他将改进先前的估计，并尽可能减少他现有论点中相当大的派生损失。我们将进一步分析全局嵌入的问题。这一研究将使我们更深入地理解复流形的边界的概念，以及某些一阶超定线性偏微分方程组的性质。平面上的一条自身缠绕的无穷无尽的曲线是一种对实线的浸入，而不是嵌入。嵌入总是设法避开它自己。全纯嵌入要求将曲线或曲面注入目标流形的函数在复解析意义下是光滑的。从传统的实分析的标准来看，从CR或复解析流形的研究中得到的定理似乎总是神奇的。这个项目完成的结果也不会有什么不同。