Boundary geometry and asymptotics in several complex variables

多个复变量中的边界几何和渐近

• 批准号: 0072237
• 负责人: David Barrett
• 金额: $ 7.5万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-01 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072237&HistoricalAwards=false
• 关键词: Boundary; geometry; asymptotics; several; complex

AbstractAward: DMS-0072237Principal Investigator: David E. BarrettProfessor Barrett will investigate various topics in complexanalysis. One topic to be studied is the behavior of twodifferent systems of partial differential equations implementingdeformation of a real hypersurface in two-dimensional complexeuclidean space by the Levi-form of the hypersurface. Thesystems are analogous, respectively, to harmonic-mapping heatflow and to the Ricci-flow on the space of conformal metrics, butthese particular systems have special features (the role ofLorentzian geometry in the target space and the inclusion oflower-order terms which are not conjugation invariant in thesource space) that introduce new phenomena and difficulties. Theassociated steady-state system has known applications to functiontheory and engineering, and the study of the time-dependentversions given above may lead to new insights into analyticcontinuation. A second topic to be studied is the boundarybehavior of the Bergman kernel function (off the diagonal) andBergman representative coordinates on domains with corners, withparticular interest in the case of generic intersections ofstrictly pseudoconvex domains (the case of intersecting ballsserving as a key model problem).Professor Barrett will investigate various problems involvingmultiple parameters (the parameters are understood to lie in theso-called complex number system, a widely-used extension of thestandard number system). One topic involves the study of systemsof partial differential equations which serve to flatten asurface in the parameter space; sometimes the equations push thesurface to an equilibrium configuration (the situation issomewhat analogous to that of a soap film attached to a fixedwire boundary), but sometimes the surface breaks before reachingequilibrium (for example, this will happen if there is noavailable equilibrium configuration). The computation ofequilibrium configurations for these problems (or thedocumentation that no equilibrium exists) is important inclassical function theory, and is also a central topic in theengineering discipline known as "H-infinity control theory." Asecond topic to be studied is based on Stefan Bergman's method offinding a sort of "ideal form" for a region in complexmultiparameter space. In the one-parameter setting Bergman'smethod tells us how to perform the useful task of smoothing outcorners appearing in the boundary of the region (such smoothingis of fundamental importance for example in classicalaerodynamics); the proposed research will examine what happens tocorners in the multi-parameter setting.

摘要奖：DMS-0072237首席研究员：David E.Barrett教授将研究络合分析中的各种主题。要研究的一个主题是两个不同的偏微分方程组的行为，它们实现了二维复核空间中的实超曲面通过超曲面的Levi形式的变形。这些系统分别类似于调和映射热流和共形度量空间上的Ricci流，但这些特殊系统具有特殊的特征(洛伦兹几何在目标空间中的作用，以及在源空间中包含非共轭不变的低阶项)，这带来了新的现象和困难。相关的稳态系统在泛函理论和工程上都有已知的应用，而对上述含时版本的研究可能导致对分析继续的新的见解。第二个要研究的主题是Bergman核函数(在对角线外)和Bergman代表坐标在有角域上的边界行为，特别是在严格伪凸域的一般相交的情况下(相交球作为关键模型问题的情况)。Barrett教授将研究涉及多个参数的各种问题(参数被理解为位于所谓的复数系统中，它是标准数系统的广泛使用的扩展)。一个主题涉及研究参数空间中用来展平表面的偏微分方程组；有时方程将表面推向平衡构型(这种情况有点类似于固定金属丝边界上的肥皂膜)，但有时表面在达到平衡之前就会破裂(例如，如果没有可用的平衡构型，就会发生这种情况)。这些问题的平衡构型的计算(或关于不存在平衡的文献)是经典函数论中的重要内容，也是被称为“H-无穷控制理论”的工程学科的中心话题。第二个要研究的课题是基于Stefan Bergman的方法，为复杂的多参数空间中的一个区域寻找一种“理想形式”。在单参数设置中，Bergman的方法告诉我们如何执行对区域边界中出现的外角进行平滑的有用任务(这种平滑是非常重要的，例如在经典空气动力学中)；拟议的研究将检查在多参数设置中角点发生了什么。