Geometry of Real Submanifolds in Complex Space and CR Structures

DMS 0401215 PI：Peter EbenfeltUC San Diego复空间中真实的子流形的几何。.摘要：主要研究者将研究几何，分析和代数方面的一般真实的子流形在复杂的流形（或更一般地说，与CR结构的流形）和他们的映射。他将研究给定CR流形之间的CR映射的存在性、唯一性和正则性，以及与本研究相关的几何问题。更具体地说，提出者将在研究严格伪凸超曲面到另一个高维超曲面的CR嵌入的背景下关注这些问题。与流形具有相同维数的情况相反，这里所知甚少，有许多未解决但非常基本的问题。PI还将研究几何性质ofCR映射之间的通用子流形的更高的余维，以及继续他的研究CR映射之间的真实的超曲面的无限型（在意义上的科恩和布卢姆-格雷厄姆）通过调查更密切的延长系统定义CR映射到一个奇异Pfiran系统的射流束。PI计划研究的另一个问题是复流形中真实的超曲面的规范形。复流形中真实的子流形的研究是多复变理论的核心，与数学的其他领域，如偏微分方程，微分几何和代数几何以及当代数学物理的主题有着密切的联系。复流形的一个真实的子流形从它的周围流形继承了一个部分复结构，称为CR（柯西-黎曼）结构，它通常比周围流形的复结构刚性得多。在这个项目中的大部分研究的动机是希望这样的CR结构分类的等价离开周围的流形不变。这是这一领域最基本的问题之一。

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.