Geometry of Real Submanifolds in Complex Space and CR Structures

DMS 0401215PI: Peter EbenfeltUC San diego在复空间的实子流形几何…摘要：首席研究员将研究复杂流形（或者更一般地说，具有CR结构的流形）中的一般实子流形及其映射的几何、解析和代数方面。他将研究给定CR流形之间CR映射的存在性、唯一性和规律性，以及与本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嵌入的背景下关注这些问题。与流形具有相同维度的情况相反，这里所知甚少，并且有许多未解决但非常基本的问题。PI还将研究高余维的一般子流形之间的CR映射的几何性质，并通过更密切地研究将CR映射定义为射流束上的奇异Pfaffian系统的系统的延伸，继续研究无限型实超曲面之间的CR映射（在Kohn和Bloom- Graham的意义上）。PI计划研究的另一个问题是复杂流形中实超曲面的正规形式。复流形中的实子流形的研究是几个复变量理论的核心，并且与其他数学领域密切相关，如偏微分方程、微分几何和代数几何，以及数学物理中的当代主题。复杂流形的真正子流形从其环境流形继承了部分复杂结构，称为CR （Cauchy-Riemann）结构，该结构通常比环境流形的复杂结构刚性得多。这个项目中的大部分研究都是出于将CR结构分类到使环境流形不变的等价的愿望。这是这个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.