Degenerate Microlocal Methods in Geometric Analysis

美国国家科学基金会奖DMS 0805529摘要(Rafe Mazzeo)：PI的研究集中在几何分析中的一些问题，涉及分层空间上退化的椭圆或抛物型方程。一个主题是在紧迭代锥边空间类上寻找正则度量；特殊的低维情形包括对具有圆锥奇异性的Riemann曲面的新分析，更重要的是，得到了空间形式中三维多面体的一些鲁棒变形结果。一个密切相关的项目是确定非紧爱因斯坦空间的形变理论，该理论以一般秩非紧对称空间为模型；这是Poincare-Einstein度规理论的推广，该理论在最近的许多论文中发挥了重要作用，在共形几何和弦理论中，它作为ADS/CFT对应的一部分出现。另一个项目涉及这些Poincare-Einstein空间的极小子流形，特别是三维凸余紧双曲流形。这里的目的是定义和研究具有嵌入渐近边界曲线的适当嵌入的极小曲面空间上的重整化面积泛函的变分性质，这似乎是从微分几何的角度研究具有边界的曲面的Willmore泛函的自然背景。最后，PI也一直在研究奇异空间上的几何发展方程。在几何简单性方面，第一种情况是将封闭嵌入曲线的曲率流推广到曲线网络的曲率流；在更广泛的背景下，几何分析中的大多数工作都是在光滑的几何对象的背景下进行的，即没有角、边或其他奇点。然而，具有奇点的空间在几何、物理和其他应用中自然地频繁地出现，因此将几何分析的方法和结果推广到这类更广泛的对象是很自然的。然而，分析和偏微分方程中的适当工具不存在于这种一般情况下，因此需要的主要工作是将这些技术扩展到具有奇性的空间。这是私家侦探在整个职业生涯中的一项主要努力。他目前正在研究的具体问题涉及精细化的问题，如这些奇异空间上最优度量或形状的存在和性质，以及将这样的空间不断变形为这些最优形状之一的演化方程的研究。在光滑空间上，这些问题一直是几何分析的主要研究方向，但它们在这种奇异背景下的类似问题还没有得到任何系统的研究。另一个问题涉及研究常负曲率空间中一类最优(极小)曲面的表面积的重整化形式。这是一个在弦理论中得到了深入研究的行动泛函。这位PI的工作的最后一部分是教育性质的：他是一个寄宿暑期项目(SUMAC)的创始人和主任，该项目每年在他的机构举行，该项目针对的是积极性高、有天赋的高中生，以鼓励他们继续学习数学。这一方案现已进入实施的第14个年头。

Abstract of NSF Award DMS 0805529 (Rafe Mazzeo):The PI's research focuses on a number of problems in geometric analysis involving degenerate elliptic or parabolic equations on stratified spaces. One theme is the search for canonical metrics on the class of compact iterated cone-edge spaces; special low-dimensional cases include a new analysis of Riemann surfaces with conic singularities, and more significantly, some robust deformation results for three dimensional polyhedra in space forms. Closely related is a project to determine the deformation theory of noncompact Einstein spaces modeled on noncompact symmetric spaces of general rank; this is a generalization of the theory of Poincare-Einstein metrics which has played a prominent role in many recent papers in conformal geometry, and also in string theory, where it appears as part of the AdS/CFT correspondence. Another project concerns minimal submanifolds of these Poincare-Einstein spaces, and in particular, in three-dimensional convex cocompact hyperbolic manifolds. The goal here is to define and study the variational properties of the renormalized area functional on the space of properly embedded minimal surfaces with embedded asymptotic boundary curves; this seems to be a natural context for studying the Willmore functional from differential geometry for surfaces with boundary. Finally, the PI has also been studying geometric evolution equations on singular spaces. The first case in terms of geometric simplicity is the generalization of the curvature flow from the setting of closed embedded curves to that of networks of curves; the goals are to establish a good existence theory for the flow, to examine questions of nonuniqueness and to prove long-time existence to a Steiner network.To put this work into a broader context, most work in geometric analysis is in the setting of geometric objects which are smooth, i.e. do not have corners, edges or other singularities.However, spaces with singularities appear naturally and very frequently in geometry, physics and other applications, and it is natural to try to extend the methods and results of geometric analysis to this broader class of objects. However, the appropriate tools from analysis and partial differential equations do not exist in this generality, so a major part of the work needed is to extend these techniques to spaces with singularities. This has been a major endeavour of the PI throughout his career. The specific problems on which he is now working involve refined questions such as the existence and nature of optimal metrics, or shapes, on these singular spaces, and the study of evolution equations which continuously deform such a space into one of these optimal shapes. On smooth spaces, these questions have been some of the principal directions of research in geometric analysis, but their analogues in this singular setting have not been studied in any systematic way. Another problem involves the study of a renormalized version of the surface area of a class of optimal (minimal) surfaces inside a space with constant negative curvature. This is an action functional which has been studied intensively in string theory.One final part of this PI's work is of an educational nature: he is the founder and director of a residential summer program (SUMaC), held at his institution each year, which is directed toward highly motivated and talented high school students to encourage their continued study of mathematics. This program is now in its fourteenth year of operation.