Geometry of Real Submanifolds in Complex Space and CR Structures

AbstractEberfelt在这个项目中，主要研究者（PI）研究几何，分析和代数方面的真实的子流形在复杂的流形，更普遍的是，流形与CR结构。更具体地说，他专注于与局部分类问题有关的问题，该问题要求在流形上接近一个显著点的CR结构的局部描述，直到等价。例如，对于复空间中的一个真实的子流形，人们想知道哪些其他的真实的子流形通过局部双全纯变换与它等价。PI研究这个问题extraditionally试图找到规范形式类的真实的子流形，并在本质上作为一个等价问题的微分方程组。 分类问题的一个重要部分是理解保持结构的变换群，或者更一般地说，两个给定结构之间的映射集。该项目的PI研究复空间中真实的子流形的局部稳定群，即保持真实的子流形和其上给定的特殊点的局部双全纯群。他研究在什么条件下这个群可以嵌入预定阶的喷流群的子群，并试图更详细地描述以这种方式出现的子群。他期待的条件，意味着advervity结果，如收敛的所有正式映射之间的真正解析子流形，或真正的解析所有顺利CR映射。他还研究了更密切的延长系统定义CR映射到Pfiran系统，并探讨其应用。理论的几个复杂的变量是一个迅速发展的主题，在数学中的应用在当代数学物理（如量子场论和弦理论）以及在工程（如控制理论）。复杂空间中真实的子流形的几何研究，例如域的光滑边界，是这个理论的核心，也与其他数学领域有关，如偏微分方程和微分几何。在这个项目中，我们调查的问题，关于几何的真实的子流形在复杂的空间和他们的映射中出现的分类问题，这样的等价物，保持复杂的结构的周围空间。

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.