Mappings in Several Complex Variables and CR geometry

多个复杂变量和 CR 几何中的映射

• 批准号: 1301282
• 负责人: Peter Ebenfelt
• 金额: $ 22.4万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301282&HistoricalAwards=false
• 关键词: Mappings; Several; Complex; Variables; CR

The goal of this mathematics research project by Peter Ebenfelt is to study geometric, analytic, and algebraic aspects of generic real submanifolds in complex varieties (more generally, of CR structures) and their mappings. Ebenfelt will consider questions regarding existence, uniqueness, and regularity of CR maps, as well as related questions that arise in connection with this study. Ebenfelt will consider CR maps of a Levi nondegenerate hypersurface into a nondegenerate hyperquadric of higher dimension. This study will enhance our understanding of the CR submanifold structure of the hyperquadrics, which themselves constitute the flat models in the theory of Levi nondegenerate hypersurfaces. The work should also provide insight into how the local CR geometry of such hypersurfaces (in principle completely encoded in the CR curvature tensor) affects properties, such as e.g. various notions of nondegeneracy and rigidity, of their CR maps. There are highly interesting and nontrivial problems even in the special case where both the source and target manifolds are hyperquadrics (e.g. the Gap Conjecture). Ebenfelt will continue his study of CR maps between generic submanifolds of infinite (commutator) type by investigating the prolongation of the system defining CR maps to a singular Pfaffian system on the jet bundle. The PI will, in particular, focus on a conjecture in this context regarding finite jet determination of local automorphisms. The study is expected to shed new light on the nature of CR maps between infinite type manifolds, and lead to a better understanding of the Pfaffian systems arising in this context. Ebenfelt will continue his work on normal forms for infinite type hypersurfaces in complex 2-space.; results in this direction will yield results on the finite jet determination problem and, possibly, on the problem of convergence of formal mappings in this context. Ebenfelt will also study CR maps between more general CR manifolds: one topic of interest is that of transversality, and he will try to improve recent transversality results (joint with with Duong and Baouendi--Rothschild) for CR maps into a higher dimensional space. Current conditions for transversality in this context involve eigenvalues of the Levi form. Ebenfelt believes that "deeper" invariants of the CR structures are actually involved, as in the equidimensional case. This study will likely require development of substantially new methods, which in turn will benefit the theory of CR maps 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 mathematics research project, 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 investigations carried out in this project will benefit research in adjacent areas of mathematics as well as in areas of theoretical physics. The methods and techniques developed will be useful in other areas of mathematics, and likely also in physics (e.g., string theory) and engineering (e.g., control theory; systems engineering). Ebenfelt expects that the project will provide interesting research topics for graduate students and postdocs. The seminar activity that results from the project should be stimulating for both students and other researchers.

这个数学研究项目的目标是彼得Ebenfelt是研究几何，分析和代数方面的一般真实的子流形在复杂的品种（更一般地说，CR结构）和他们的映射。 Ebenfelt将考虑关于CR映射的存在性、唯一性和规律性的问题，以及与本研究相关的问题。Ebenfelt将考虑从Levi非退化超曲面到高维非退化超二次曲面的CR映射。本文的研究将加深我们对超二次曲面的CR子流形结构的理解，超二次曲面本身构成Levi非退化超曲面理论中的平坦模型。这项工作还应该提供洞察如何局部CR几何这种超曲面（原则上完全编码在CR曲率张量）影响的属性，例如各种概念的非退化和刚性，他们的CR映射。即使在源流形和目标流形都是超二次曲面的特殊情况下，也有非常有趣和重要的问题（例如差距猜想）。 Ebenfelt将继续他的研究CR映射之间的通用子流形的无限（换向器）类型的调查延长系统定义CR映射到一个奇异的Pfavian系统的喷丛。PI将，特别是，集中在一个猜想，在这方面的有限喷射确定的局部自同构。该研究有望揭示无限型流形之间CR映射的性质，并导致更好地理解在此背景下产生的Pfidian系统。Ebenfelt将继续他关于复2-空间中无限型超曲面的规范形式的工作。在这个方向上的结果将产生关于有限喷流确定问题的结果，并且可能产生关于在这个上下文中形式映射的收敛问题的结果。 Ebenfelt还将研究CR映射之间的更一般的CR流形：一个感兴趣的主题是，横截性，他将试图改善最近的横截性结果（联合与Duong和Baouendi-罗斯柴尔德）的CR映射到一个更高的维度空间。在这种情况下，横截性的当前条件涉及Levi形式的特征值。 Ebenfelt认为，CR结构的“更深”不变量实际上是相关的，就像在等维情况下一样。这项研究可能需要开发新的方法，这反过来将有利于CR映射理论到高维空间。复流形中的真实的子流形的研究是复分析和其他数学与物理领域的核心。在这个数学研究项目中，工具从广泛的领域，如真实的和复杂的分析，偏微分方程，代数几何使用和进一步发展。该项目中进行的调查将有利于邻近数学领域以及理论物理领域的研究。开发的方法和技术将在数学的其他领域有用，也可能在物理学（例如，弦理论）和工程（例如，控制理论;系统工程）。 Ebenfelt预计，该项目将为研究生和博士后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。