Structure of Mappings in Several Complex Variables and Cauchy-Riemann Geometry

多复变量映射结构与柯西-黎曼几何

基本信息

批准号：
1600701
负责人：
Peter Ebenfelt
金额：
$ 19.45万
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：
2016
资助国家：
美国
起止时间：
2016-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600701&HistoricalAwards=false
关键词：
Structure Mappings Several Complex Variables

项目摘要

The study of invariant geometries is an important part of mathematics. Different types of invariant geometries arise in different areas of mathematics, e.g., in several complex variables (SCV), partial differential equations (PDE), and algebraic, complex, and differential geometry. In this research project, the principal investigator Peter Ebenfelt will investigate a particular geometry that arises in the study of SCV, and which has also deep connections with contemporary topics in mathematical physics such as quantum field theory and string theory, as well as applications in, e.g., systems engineering and control theory. Tools that are needed for this study come from a variety of different areas in mathematics, such as complex analysis, PDE, and differential geometry, and the techniques and tools developed in this study influence these areas as well. Ebenfelt expects that the project will also 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.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 Cauchy-Riemann (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. He will consider the context of 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 constitute the flat models in the theory of Levi nondegenerate hypersurfaces. The proposed work should also provide insight into how the local CR geometry of such hypersurfaces (in principle completely encoded in the CR curvature tensor) affects various properties, such as notions of nondegeneracy and rigidity, of their CR maps. Ebenfelt will also continue his study of CR maps between generic submanifolds of infinite type by investigating the prolongation of the system defining CR maps to a singular Pfaffian system on the jet bundle. He will, in particular, focus on understanding the recently discovered phenomenon that for infinite type hypersurfaces the biholomorphic, formal, and smooth CR equivalence classifications are different. He will also study a conjecture regarding finite jet determination of local automorphisms. These investigations are 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 also continue his work on normal forms for infinite type hypersurfaces in complex 2-space. Finally, Ebenfelt will study CR invariants on unit circle bundles over Kähler manifolds, specifically Cartan's umbilical tensor in the 3-dimensional case and those arising in the expansions of the Bergman and Szegö kernels. He intends to develop new methods for detecting umbilical points on general 3-dimensional CR manifolds, and resolve an open problem regarding existence of umbilical points on compact CR manifolds embedded in complex 2-space.

不变几何的研究是数学的重要组成部分，不同类型的不变几何出现在数学的不同领域，例如，在多个复变量（SCV）、偏微分方程（PDE）以及代数、复数和微分几何中。在这个研究项目中，首席研究员 Peter Ebenfelt 将研究 SCV 研究中出现的一种特殊几何形状，该几何形状与量子场论和弦理论等数学物理学的当代主题也有深刻的联系，例如以及系统工程和控制理论等领域的应用本研究所需的工具来自数学的各个不同领域，例如复分析、偏微分方程和微分几何，以及本研究中开发的技术和工具。 Ebenfelt 预计该项目还将为研究生和博士后提供有趣的研究主题。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。Peter 的这个数学研究项目的目标。埃本费尔特要学习Ebenfelt 将考虑复杂簇中的通用实数子流形（更一般地说，柯西-黎曼 (CR) 结构）的几何、解析和代数方面及其映射，以及有关 CR 映射的存在性、唯一性和规律性的问题。他将考虑将 Levi 非简并超曲面的 CR 映射为高维非简并超二次曲面的问题。这项研究将增强我们对 CR 的理解。超二次曲面的子流形结构，构成了 Levi 非简并超曲面理论中的平面模型。所提出的工作还应该深入了解此类超曲面的局部 CR 几何结构（原则上完全编码在 CR 曲率张量中）如何影响各种属性，例如他们的 CR 图的非简并性和刚性的概念 Ebenfelt 还将通过研究系统的延展来继续研究无限类型的通用子流形之间的 CR 图。他将特别关注最近发现的现象，即对于无限类型超曲面，双全纯、形式和平滑的 CR 等价分类是不同的。关于局部自同构的有限射流确定，这些研究有望为无限类型流形之间的 CR 映射的性质提供新的线索，并导致更好地理解在此背景下出现的普法夫系统。 Ebenfelt 还将继续研究复杂二维空间中无限型超曲面的范式。最后，Ebenfelt 将研究 Kähler 流形上单位圆丛的 CR 不变量，特别是 3 维情况下的嘉当脐张量以及展开中出现的张量。他打算开发用于检测通用 3 维 CR 流形上的脐点的新方法，并解决该问题。关于嵌入复数 2 空间中的紧凑 CR 流形上脐点的存在性的开放问题。