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.

摘要奖项：DMS-0072237 首席研究员：David E. Barrett Barrett 教授将研究复杂分析中的各种主题。 要研究的一个主题是两个不同的偏微分方程组的行为，通过超曲面的列维形式实现二维复欧几里得空间中实超曲面的变形。 这些系统分别类似于共形度量空间上的调和映射热流和里奇流，但这些特定系统具有特殊功能（洛伦兹几何在目标空间中的作用以及包含非共轭的低阶项）源空间中的不变性），引入了新的现象和困难。 相关的稳态系统在函数理论和工程中具有已知的应用，并且对上面给出的时间相关版本的研究可能会导致对解析连续性的新见解。 要研究的第二个主题是伯格曼核函数（偏离对角线）的边界行为和带角域上的伯格曼代表坐标，特别关注严格伪凸域的一般相交情况（相交球的情况作为关键模型问题） Barrett教授将研究涉及多个参数的各种问题（这些参数被理解为存在于所谓的复数系统中，这是标准数字系统的广泛使用的扩展）。 其中一个主题涉及偏微分方程组的研究，该方程组用于使参数空间中的表面平坦化；有时方程将表面推向平衡构型（这种情况有点类似于附着在固定线边界上的肥皂膜），但有时表面在达到平衡之前破裂（例如，如果没有可用的平衡构型，就会发生这种情况）。 这些问题的平衡配置的计算（或不存在平衡的证明）在经典函数理论中很重要，也是被称为“H-无穷控制理论”的工程学科的中心主题。 要研究的第二个主题是基于 Stefan Bergman 的方法，该方法为复杂多参数空间中的区域寻找一种“理想形式”。 在单参数设置中，伯格曼方法告诉我们如何执行平滑区域边界中出现的外角的有用任务（例如，这种平滑在经典空气动力学中至关重要）；拟议的研究将研究多参数设置下拐角会发生什么。