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

本项目的目的是研究复杂流形（或更一般地说，具有cr结构的流形）及其映射中的一般实子流形的几何和解析方面。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。目标是获得对cr几何中的映射的更深的理解，以及它们在复杂分析和PDE中的作用。项目的一部分将集中在这些问题的背景下，从一个列维非简并超曲面到另一个高维超曲面的非平凡cr映射。首席研究员期望这项研究将导致更好地掌握这些超曲面的局部cr -几何（原则上，它完全编码在chen - moser cr -曲率张量中）如何影响cr -映射的几何性质（例如，各种非简并和刚性的概念）。首席研究员还将研究更一般cr -流形之间cr -映射的几何性质。我们感兴趣的一个特殊几何性质是横向性。首席研究员将尝试将他最近的一些关于相同维数cr流形之间映射的横向性结果（与Rothschild联合工作）扩展到目标流形比源流形具有更高维数的情况。这种情况与等维情况明显不同。首席研究员预计，这项研究将涉及大量新方法的发展，这反过来将增强我们对高维空间映射的理解。复流形中的实子流形的研究是复分析和其他数学和物理领域的核心。在这项研究中，来自广泛领域的工具，如实分析和复分析，偏微分方程和代数几何被使用和进一步发展。首席研究员希望在这个项目中进行的研究将增强我们对复空间中实子流形和偏微分方程的几何的理解，这将有利于相邻数学领域以及理论物理领域的研究。他希望这个项目能为研究生提供有趣的研究课题。从这个项目中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。

The aim of this project is to study geometric and analytic aspects of generic real submanifolds in complex manifolds (or, more generally, in manifolds with a CR-structure) and their mappings. 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 obtain 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 properties of CR-mappings between more general CR-manifolds. A particular geometric property of interest is transversality. The principal investigator will try to extend some of his recent transversality results (joint work with Rothschild) for mappings between CR-manifolds of the same dimension to the situation where the target manifold has higher dimension than that of the source manifold. This case appears significantly different from the equidimensional case. 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.