Geometry of Real Submanifolds in Complex Space and CR Structures

在这个项目中，首席研究员（PI）研究了复杂流形中的实子流形的几何、解析和代数方面，更一般地说，研究了CR结构流形的几何、解析和代数方面。更具体地说，他专注于与局部分类问题相关的问题，该问题要求在一个区分点附近的流形上对CR结构进行局部描述直至等价。例如，对于复空间中的实子流形，我们想知道哪些实子流形通过局部生物全纯变换等价于它。PI研究这个问题的外在方法是寻找实子流形的正规形式，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转换组，或者更一般地说，两个给定结构之间的映射集。本课题的PI研究复空间中实子流形的局部稳定群，即保留实子流形及其上的给定区别点的局部生物全纯群。他研究了在什么条件下这个群体可以作为预定顺序的射流群体的子群体嵌入，并试图更详细地描述以这种方式出现的子群体。他寻找暗示矫顽性结果的条件，例如实解析子流形之间的所有形式映射的收敛性，或所有光滑CR映射的实解析性。他还进一步研究了将CR映射定义为Pfaffian系统的系统的扩展，并探讨了其应用。复变理论是一门迅速发展的数学学科，在当代数学物理（如量子场论和弦理论）和工程（如控制理论）中都有应用。复杂空间中实子流形的几何研究，如光滑域边界，是该理论的核心，也与偏微分方程和微分几何等其他数学领域有关。在这个项目中，我们研究了复杂空间中实子流形的几何问题及其映射，这些问题出现在保留环境空间复杂结构的等价分类问题中。

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.