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

超命题DMS-0406060PI：John M.Lee，华盛顿大学切向Cauchy-Riemann复形的Neumann问题和CREmbedding问题项目的技术描述：拟议的研究将研究CR流形上切向有界区域上切向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区域，即那些仅依赖于单个CR-全纯函数的实部和虚部的区域。这项研究的核心思想是利用这样的定义函数通过紧致CR子流形提供余维2层(靠近边界但远离特征点)的事实，通过利用紧叶上Kohn Laplace算子的已知估计，可以将Neumann问题归结为平面区域上的(一般非强制的)椭圆边界问题。这些结果有望应用于局部CR嵌入问题，CR结构的局部变形，刻画切向Cauchy-Riemann复形可解的区域，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。非技术描述：复流形(其中复数而不是实数可以用作坐标的几何对象)的几何最近开始在数学和物理中发挥着令人惊讶的重要作用。例如，在弦论中，物理学家假设物质的基本粒子实际上是在被称为Calabi-Yau流形的亚复微观流形中振动的“量子弦”。研究复流形的主要分析工具是柯西-黎曼方程，这是一个偏微分方程组，其中刻画的是那些具有复导数的函数。当一个人研究复杂流形中的曲面时(如弦理论中出现的“膜”)，柯西-黎曼方程需要被称为“切向柯西-黎曼方程”的更复杂的系统所取代，这是我们刚刚开始理解的。这一建议将为研究切向Cauchy-Riemann方程的可解性方面的一些深层分析问题提供新的方法，这些问题有望对理解复杂流形中曲面的几何和分析具有重要意义。

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.