Degenerate Microlocal Methods in Geometric Analysis

NSF奖摘要DMS 0805529 (Rafe Mazzeo): PI的研究重点是涉及分层空间上退化椭圆或抛物方程的几何分析中的一些问题。一个主题是寻找紧迭代锥边空间类上的正则度量；特殊的低维情况包括对具有二次奇异点的黎曼曲面的新的分析，更重要的是，对空间形式的三维多面体的一些鲁棒变形结果。与此密切相关的是一个以一般秩非紧对称空间为模型的非紧爱因斯坦空间的变形理论的研究项目；这是庞加莱-爱因斯坦度规理论的推广，它在最近的许多共形几何论文中发挥了重要作用，也在弦理论中，它作为AdS/CFT对应的一部分出现。另一个项目涉及这些庞加莱-爱因斯坦空间的最小子流形，特别是三维凸紧双曲流形。本文的目的是定义和研究具有嵌入渐近边界曲线的适当嵌入最小曲面空间上的重归一化面积泛函的变分性质；这似乎是研究具有边界曲面的微分几何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演化方程。在几何简单性方面的第一种情况是曲率流从封闭嵌入曲线的设置推广到曲线网络的设置；目的是建立一个良好的流的存在性理论，检验非唯一性问题，并证明Steiner网络的长期存在性。为了把这项工作放在更广泛的背景下，几何分析中的大多数工作都是在光滑的几何物体的背景下进行的，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而频繁地出现，因此将几何分析的方法和结果扩展到这类更广泛的对象中是很自然的。然而，分析和偏微分方程的适当工具并不存在于这种一般性中，因此需要的主要工作是将这些技术扩展到具有奇点的空间。这是PI在其整个职业生涯中所做的主要努力。他现在正在研究的具体问题涉及到一些精细的问题，比如这些奇异空间上的最优度量或形状的存在和本质，以及不断将这样一个空间变形为这些最优形状之一的进化方程的研究。在光滑空间上，这些问题一直是几何分析研究的一些主要方向，但在这种奇异环境下，它们的类似问题还没有得到系统的研究。另一个问题涉及研究具有恒定负曲率的空间内一类最优（最小）曲面的表面积的重整化版本。这是一个作用泛函，在弦理论中得到了深入的研究。这位PI工作的最后一部分是教育性质的：他是一个住宿暑期项目（SUMaC）的创始人和主任，每年在他的机构举行，该项目针对高度积极和有才华的高中生，鼓励他们继续学习数学。这个项目现在已经运行了14年。

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.