The Neumann Problem for the Tangential Cauchy-Riemann Complex and the CR Embedding Problem

切向柯西-黎曼复形的诺伊曼问题和 CR 嵌入问题

• 批准号: 0406060
• 负责人: John Lee
• 金额: $ 10.8万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0406060&HistoricalAwards=false
• 关键词: Neumann; Problem; Tangential; Cauchy; Riemann

SupermanSProposal DMS-0406060PI: John M. Lee, University of WashingtonThe Neumann Problem for the Tangential Cauchy-Riemann Complex and the CREmbedding ProblemABSTRACTTechnical description of the project:The proposed research will study existence and regularity theorems for thenatural Neumann boundary problem for the tangential Cauchy-Riemann equationson smoothly bounded domains in CR manifolds. All known existence resultsfor this problem work only on domains with very special defining functions,namely those that depend only on the real and imaginary parts of a singleCR-holomorphic function. The key idea of this research is to use the factthat such a defining function provides a codimension-2 foliation (near theboundary but away from characteristic points) by compact CR-submanifolds.By using known estimates for the Kohn Laplacian on the compact leaves, onecan reduce the Neumann problem to a (generally non-coercive) ellipticboundary problem in a plane domain. These results are expected to haveapplications to such problems as the local CR embedding problem, localdeformations of CR structures, characterizing domains on which thetangential Cauchy-Riemann complex is solvable, regularity of maps between CRmanifolds, and the existence of local frames for CR vector bundles.Non-technical description:The geometry of complex manifolds (geometric objects in which complexnumbers instead of real numbers can be used as coordinates) has recentlybegun to play a surprisingly important role in both mathematics and physics.For example, in string theory, physicists postulate that the fundamentalparticles of matter are actually "quantum strings" that vibrate insidesub-microscopic complex manifolds called Calabi-Yau manifolds. Theprincipal analytic tool for studying complex manifolds is the Cauchy-Riemannequations, a system of partial differential equations that characterizes,among other things, those functions that have complex derivatives. When onestudies surfaces within complex manifolds (such as the "branes" that arisein string theory), the Cauchy-Riemann equations need to be replaced by amuch more complicated system called the "tangential Cauchy-Riemannequations," which we are just beginning to understand. This proposal willdevelop new techniques for studying some deep analytic questions surroundingthe solvability of the tangential Cauchy-Riemann equations, which areexpected to be of fundamental importance in understanding the geometry andanalysis of surfaces in complex manifolds.

项目技术描述：拟研究CR流形光滑有界区域上切向Cauchy-Riemann方程的自然Neumann边界问题的存在性定理和正则性定理。所有已知的存在性结果只适用于具有非常特殊定义函数的定义域，即只依赖于单个lecr全纯函数的实部和虚部的定义域。本研究的关键思想是利用这样一个定义函数通过紧化cr -子流形提供一个协维-2叶化（靠近边界但远离特征点）的事实。通过使用紧叶上已知的Kohn Laplacian估计，可以将Neumann问题简化为平面域上的椭圆边界问题（通常是非强制的）。这些结果有望应用于诸如局部CR嵌入问题、CR结构的局部变形、切向Cauchy-Riemann复可解的表征域、CR流形之间映射的正则性以及CR向量束的局部框架的存在性等问题。非技术描述：复数流形（用复数代替实数作为坐标的几何对象）的几何学最近开始在数学和物理学中发挥令人惊讶的重要作用。例如，在弦理论中，物理学家假设物质的基本粒子实际上是在称为Calabi-Yau流形的亚微观复杂流形中振动的“量子弦”。研究复流形的主要分析工具是柯西-黎曼方程，它是一个偏微分方程系统，除其他外，它描述了那些具有复导数的函数。当一个人研究复杂流形中的表面（比如弦理论中出现的“膜”）时，柯西-黎曼方程需要被更复杂的系统所取代，称为“切向柯西-黎曼方程”，我们才刚刚开始理解。这一建议将为研究围绕切向柯西-黎曼方程的可解性的一些深度解析问题开发新的技术，这些问题预计将对理解复杂流形中曲面的几何和分析具有根本的重要性。