Mappings of real submanifolds in complex space, CR geometry, and analytic PDE

复空间中真实子流形的映射、CR 几何和解析 PDE

• 批准号: 1001322
• 负责人: Peter Ebenfelt
• 金额: $ 16.44万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001322&HistoricalAwards=false
• 关键词: Mappings; real; submanifolds; complex; space

The goal of this project is to study geometric and analytic aspects of generic real submanifolds in complex manifolds (or, more generally, of manifolds with a CR-structure). Particular attention will be paid to the structure of mappings between such manifolds. Examples of generic manifolds include (smooth) boundaries of open subsets of complex Euclidian space. Important information about proper mappings between open sets can be gleaned from their restrictions to the boundary. Basic questions that will be investigated include the existence, uniqueness, and regularity of CR-mappings between given CR-manifolds, as well as geometric questions that arise in connection with this study. The objective is to gain a deeper understanding of mappings in CR-geometry, and their role in complex analysis and PDE. One part of the project will focus on these problems in the context of nontrivial CR-mappings of a Levi-nondegenerate hypersurface into another of higher dimension. The principal investigator expects that this research will lead to a better grasp of how the local CR-geometry of such hypersurfaces (which, in principle, is completely encoded in the Chern-Moser CR-curvature tensor) affects geometric properties of CR-mappings (e.g., various notions of nondegeneracy and rigidity). The principal investigator will also study geometric and analytic properties of CR-mappings between more general CR-manifolds. A particular analytic property of interest is that of finite jet determination. The equidimensional case is by now fairly well understood. The principal investigator intends to study the situation where the target manifold has a higher dimension than that of the source. This situation appears to be drastically different from the equidimensional one. The principal investigator anticipates that this study will involve the development of substantially new methods, which in turn will enhance our understanding of mappings into higher dimensional spaces.The study of real submanifolds in complex manifolds is central to complex analysis and to other areas of mathematics and physics. In this research, tools from a wide range of areas such as real and complex analysis, partial differential equations, and algebraic geometry are used and further developed. The principal investigator is hopeful that the investigations carried out in this project will enhance our understanding of the geometry of real submanifolds and partial differential equations in complex space, which will benefit research in adjacent areas of mathematics as well as in areas of theoretical physics. He expects the project to provide interesting research topics for graduate students. The seminar activity that results from the project should prove stimulating for both students and other researchers.

这个项目的目标是研究复流形中的一般真实的子流形的几何和分析方面（或者更一般地说，具有CR结构的流形）。特别注意将支付给这种流形之间的映射的结构。一般流形的例子包括复欧几里得空间的开子集的（光滑）边界。关于开集之间真映射的重要信息可以从它们对边界的限制中得到。将调查的基本问题包括存在性，唯一性和规则性的CR-映射之间的CR-流形，以及几何问题，出现在与本研究。目标是获得更深入的了解CR-几何中的映射，以及它们在复分析和偏微分方程中的作用。该项目的一部分将集中在这些问题的背景下的非平凡CR映射的列维非退化超曲面到另一个更高的维度。首席研究员预计，这项研究将导致更好地掌握这种超曲面的局部CR几何（原则上，它完全编码在Chern-Moser CR曲率张量中）如何影响CR映射的几何性质（例如，非简并性和刚性的各种概念）。主要研究者还将研究更一般的CR-流形之间的CR-映射的几何和分析性质。感兴趣的一个特殊的分析性质是有限喷流的确定。等维情形现在已经相当好地理解了。主要研究者打算研究目标流形的维数高于源流形的情况。这种情形似乎与等维情形截然不同。主要研究者预计，这项研究将涉及到大量的新方法的发展，这反过来将提高我们的理解映射到高维spaces.The研究的真实的子流形在复杂的流形是中央复杂的分析和其他领域的数学和物理。在这项研究中，从广泛的领域，如真实的和复杂的分析，偏微分方程，代数几何工具的使用和进一步发展。主要研究者希望，在这个项目中进行的调查将提高我们的理解几何的真实的子流形和偏微分方程在复杂的空间，这将有利于研究在邻近地区的数学以及理论物理领域。他希望该项目能为研究生提供有趣的研究课题。该项目所产生的研讨会活动应该对学生和其他研究人员都有激励作用。