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: 0100110
• 负责人: Peter Ebenfelt
• 金额: $ 10.6万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100110&HistoricalAwards=false
• 关键词: Geometry; Real; Submanifolds; Complex; Space

AbstractEberfeltIn this project, the principal investigator (PI) studies geometric, analytic, and algebraic aspects of real submanifolds in complex manifolds and, more generally, of manifolds with a CR structure. More specifically, he focuses on questions that are related to the local classification problem, which asks for a local description of the CR structure on a manifold near a distinguished point up to equivalence. For instance, for a real submanifold in complex space, one would like to know which other real submanifolds are equivalent to it by a local biholomorphic transformation. The PI studies this problem extrinsically by trying to find normal forms in classes of real submanifolds, and intrinsically as an equivalence problem for systems of differential equations. An important part of the classification problem is to understand the group of transformations preserving the structure or, more generally, the set of mappings between two given structures. The PI of this project studies the local stability group of a real submanifold in complex space, i.e. the group of local biholomorphisms preserving the real submanifold and a given distinguished point on it. He investigates under what conditions this group can be embedded as a subgroup of the jet group of a predetermined order, and seeks to describe the subgroups that arise in this way in more detail. He looks for conditions that imply coercivity results such as e.g. convergence of all formal mappings between real-analytic submanifolds, or real-analyticity of all smooth CR mappings. He also investigates closer the prolongation of the system defining CR mappings to a Pfaffian system, and explores its applications.The theory of several complex variables is a rapidly developing subject in mathematics which has applications in contemporary mathematical physics (e.g quantum field theory and string theory) as well as in engineering (e.g. control theory). The study of the geometry of real submanifolds in complex spaces, such as e.g. smooth boundaries of domains, is central to this theory, and is also related to other areas of mathematics such as partial differential equations and differential geometry. In this project, we investigate questions regarding the geometry of real submanifolds in complex space and their mappings that arise in the classification problem of such up to equivalences that preserve the complex structure of the ambient space.

摘要 Eberfelt 在这个项目中，首席研究员 (PI) 研究复流形中实子流形的几何、解析和代数方面，更一般地说，研究具有 CR 结构的流形。更具体地说，他关注与局部分类问题相关的问题，该问题要求在靠近显着点的流形上对 CR 结构进行局部描述，直到达到等价。例如，对于复空间中的实子流形，人们想知道哪些其他实子流形通过局部双全纯变换与其等效。 PI 通过尝试在实数子流形类中找到范式来研究这个问题，本质上是微分方程组的等价问题。 分类问题的一个重要部分是理解保留结构的变换组，或者更一般地说，理解两个给定结构之间的映射集。该项目的PI研究复空间中实子流形的局部稳定性群，即保留实子流形及其上给定特征点的局部双全纯群。他研究了在什么条件下该群可以嵌入为预定顺序的射流群的子群，并试图更详细地描述以这种方式出现的子群。他寻找暗示矫顽力结果的条件，例如 实解析子流形之间所有形式映射的收敛性，或所有平滑 CR 映射的实解析性。他还更仔细地研究了定义 CR 映射的系统向普法夫系统的延拓，并探索了其应用。多复变量理论是数学中快速发展的学科，在当代数学物理（例如量子场论和弦理论）中具有应用。 ）以及工程（例如控制理论）。研究复杂空间中真实子流形的几何形状，例如域的平滑边界是该理论的核心，并且还与偏微分方程和微分几何等其他数学领域相关。在这个项目中，我们研究了有关复杂空间中真实子流形的几何形状及其映射的问题，这些问题在分类问题中出现，直到保留环境空间的复杂结构的等价性。