Geometry of Real Submanifolds in Complex Space and CR Structures

DMS 0401215PI：Peter EbenFeltUC San Diego复空间中实子流形的几何。。摘要：主要研究人员将研究复流形(或更一般地，具有CR结构的流形)中一般实子流形的几何、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、唯一性和正则性，以及与此研究相关的几何问题。更具体地说，作者将在研究一个严格伪凸超曲面到另一个高维超曲面的CR嵌入的背景下集中讨论这些问题。与流形具有相同维度的情况相比，这里知之甚少，有许多尚未解决但非常基本的问题。PI还将研究高余维类属子流形之间CR映射的几何性质，并通过更密切地研究定义CR映射的系统到射丛上的奇异Pfaffian系统的延拓来继续他对无穷型实超曲面(在Kohn和Bloom-Graham意义下)之间的CR映射的研究。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几个复变量理论的核心，并与其他数学领域，如偏微分方程组、微分几何和代数几何，以及数学物理的当代主题有着密切的联系。复流形的实子流形从其环境流形继承了一个部分复结构，称为CR(柯西-黎曼结构)，它通常比环境流形的复结构要坚硬得多。这个项目中的许多研究都是出于这样一种愿望，即将这种CR结构分类到使环境流形保持不变的等价。这是该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.