Geometry of Real Submanifolds in Complex Space and CR Structures

复空间中实子流形的几何与CR结构

• 批准号: 0401215
• 负责人: Peter Ebenfelt
• 金额: $ 13万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401215&HistoricalAwards=false
• 关键词: Geometry; Real; Submanifolds; Complex; Space

DMS 0401215PI: Peter EbenfeltUC San DiegoGeometry of real submanifolds in complex space . . .Abstract:The principal investigator will study geometric, analytic, and algebraic aspects of generic real submanifolds in complex manifolds (or, more generally, of manifolds with a CR structure) and their mappings. He will investigate the existence, uniqueness, and regularity of CR mappings between given CR manifolds, as well as geometric questions that arise in connection with this study.More specifically, the proposer will focus on these problems in the context of studying CR embeddings of a strictly pseudoconvex hypersurface into another of higher dimension. In contrast to the case in which the manifolds are of the same dimension, little is known here and there are a number of unresolved but very basic questions. The PI will also study geometric properties ofCR mappings between generic submanifolds of higher codimension, as well as continuing his study of CR mappings between real hypersurfaces of infinite type (in the sense of Kohn and Bloom--Graham) by investigating closer the prolongation of the system defining CR mappings to a singular Pfaffian system on the jet bundle. Another problem that the PI plans to study is that of normal forms for real hypersurfaces in complex manifolds.The study of real submanifolds in complex manifolds is central to the theory of several complex variables, and has close connections to other areas of mathematics, such as partial differential equations, differential geometry, and algebraic geometry, as well as to contemporary topics in mathematical physics. A real submanifold of a complex manifold inherits, from its ambient manifold, a partial complex structure, called a CR (for Cauchy-Riemann) structure, which in general is much more rigid than the complex structure of the ambient manifold. Much of the research in this project is motivated by the desire to classify such CR structures up to equivalences which leave the ambient manifold invariant. This is one of the most fundamental questions in this field.

DMS 0401215PI：Peter Ebenfelt 加州大学圣地亚哥分校复杂空间中真实子流形的几何。 。摘要：主要研究者将研究复流形（或更一般地，具有 CR 结构的流形）中的通用实子流形的几何、解析和代数方面及其映射。他将研究给定 CR 流形之间 CR 映射的存在性、唯一性和规律性，以及与本研究相关的几何问题。更具体地说，提议者将在研究 CR 嵌入的背景下重点关注这些问题。严格的伪凸超曲面到另一个更高维度。与流形具有相同维度的情况相反，这里知之甚少，并且存在许多未解决但非常基本的问题。 PI 还将研究更高余维的通用子流形之间 CR 映射的几何特性，并通过更仔细地研究系统定义 CR 映射到射流束上的奇异普法夫系统。 PI 计划研究的另一个问题是复流形中实超曲面的范式问题。复流形中实子流形的研究是复数变量理论的核心，并且与其他数学领域有密切联系，例如偏微分方程、微分几何和代数几何，以及数学物理学的当代主题。复流形的实子流形从其环境流形继承了部分复形结构，称为 CR（柯西-黎曼）结构，它通常比环境流形的复形结构刚性得多。该项目中的大部分研究都是出于将此类 CR 结构分类为等价性的愿望，从而使环境流形保持不变。这是该领域最基本的问题之一。