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.

超人提案DMS-0406060 PI：John M. Lee，华盛顿大学切向Cauchy-Riemann复形的Neumann问题和CREmbeding问题摘要项目技术说明：本研究将研究CR流形中光滑有界区域上切向Cauchy-Riemann方程的自然Neumann边值问题的存在性和正则性定理。 这个问题的所有已知存在性结果只适用于具有非常特殊的定义函数的域，即只依赖于单个CR-全纯函数的真实的和虚部的域。 本研究的核心思想是利用这样一个定义函数提供了一个由紧CR-子流形构成的余维2叶理（靠近边界但远离特征点）的事实，通过对紧叶上的Kohn-Laplacian的已知估计，可以将Neumann问题转化为平面区域上的一个椭圆边界问题（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复形可解的特征域、CR流形之间映射的正则性以及CR向量丛的局部标架的存在性等问题。复流形几何（复数而不是真实的数可用作坐标的几何对象）最近开始在数学和物理学中发挥着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“量子弦”，它们在称为卡-丘流形的亚微观复杂流形中振动。 主要的分析工具研究复杂的流形是柯西-黎曼方程，系统的偏微分方程的特点，除其他外，这些功能，有复杂的衍生物。 当人们研究复杂流形中的曲面时（比如弦论中出现的“膜”），柯西-黎曼方程需要被一个更复杂的系统所取代，这个系统叫做“切向柯西-黎曼方程”，我们才刚刚开始理解。 这一建议将为研究切向Cauchy-Riemann方程的可解性所涉及的一些深层次的分析问题提供新的方法，这些问题对于理解复流形中曲面的几何和分析具有重要意义。