Boundary geometry and asymptotics in several complex variables

项目编号：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。本文将研究两种不同的偏微分方程组在二维复欧几里得空间中利用超曲面的列维形式实现实超曲面变形的行为。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，但是这些特殊的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不共轭不变的低阶项的包含），这些特征会引入新的现象和困难。相关的稳态系统在泛函理论和工程中有已知的应用，对上述时间相关版本的研究可能会导致对分析延拓的新见解。第二个要研究的主题是Bergman核函数的边界行为（在对角线外）和Bergman代表坐标在有角的域上的边界行为，对严格伪凸域的一般相交的情况特别感兴趣（相交球作为关键模型问题的情况）。巴雷特教授将研究涉及多个参数的各种问题（这些参数被认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）。一个课题是研究用于在参数空间中使表面平坦化的偏微分方程组；有时这些方程会将表面推到一个平衡构型（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就会断裂（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（或没有平衡存在的文件）是重要的经典函数理论，也是工程学科被称为“h -∞控制理论”的中心主题。第二个要研究的主题是基于Stefan Bergman在复杂多参数空间中寻找区域的“理想形式”的方法。在单参数设置中，Bergman的方法告诉我们如何执行平滑出现在区域边界的外角的有用任务（这种平滑在经典空气动力学中是至关重要的）；拟议的研究将检查在多参数设置中拐角会发生什么。

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.