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.

该项目的目标是研究复流形（或更一般地，具有 CR 结构的流形）中通用实子流形的几何和分析方面。将特别关注这些流形之间的映射结构。通用流形的示例包括复杂欧几里得空间的开子集的（平滑）边界。关于开集之间正确映射的重要信息可以从它们对边界的限制中收集到。将研究的基本问题包括给定 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。目标是更深入地了解 CR 几何中的映射及其在复杂分析和偏微分方程中的作用。该项目的一部分将在列维非简并超曲面到另一个更高维度的重要 CR 映射的背景下关注这些问题。首席研究员预计这项研究将有助于更好地理解此类超曲面的局部 CR 几何（原则上完全编码在 Chern-Moser CR 曲率张量中）如何影响 CR 映射的几何特性（例如，各种非简并性和刚性的概念）。首席研究员还将研究更一般的 CR 流形之间 CR 映射的几何和分析特性。一个特别令人感兴趣的分析特性是有限射流测定。等维情况现在已经很好理解了。主要研究者打算研究目标流形的维数高于源流形的情况。这种情况似乎与等维情况截然不同。首席研究员预计这项研究将涉及实质上新方法的开发，这反过来将增强我们对映射到更高维空间的理解。对复流形中的实子流形的研究是复分析以及数学和其他领域的核心。物理。在这项研究中，使用并进一步开发了实分析和复分析、偏微分方程和代数几何等广泛领域的工具。首席研究员希望该项目中进行的研究能够增强我们对复空间中实子流形和偏微分方程几何的理解，这将有利于数学邻近领域以及理论物理领域的研究。他希望该项目能为研究生提供有趣的研究课题。该项目产生的研讨会活动应该会刺激学生和其他研究人员。