Degenerate Microlocal Methods in Geometric Analysis

几何分析中的简并微局部方法

• 批准号: 0805529
• 负责人: Rafe Mazzeo
• 金额: $ 39.38万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805529&HistoricalAwards=false
• 关键词: Degenerate; Microlocal; Methods; Geometric; Analysis

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.

NSF奖DMS 0805529（Rafe Mazzeo）摘要：PI的研究重点是几何分析中的一些问题，涉及分层空间上的退化椭圆或抛物方程。一个主题是寻找规范度量类的紧凑迭代锥边空间;特殊的低维情况下，包括一个新的分析黎曼曲面与圆锥奇点，更重要的是，一些强大的变形结果三维多面体的空间形式。密切相关的是一个项目，以确定一般秩的非紧对称空间为模型的非紧爱因斯坦空间的变形理论;这是庞加莱-爱因斯坦度量理论的推广，在最近的许多共形几何论文中发挥了突出的作用，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-爱因斯坦空间的极小子流形，特别是在三维凸余紧双曲流形。这里的目标是定义和研究的变分性质的重整化面积功能的空间适当嵌入极小曲面嵌入渐近边界曲线，这似乎是一个自然的背景下研究Willmore功能从微分几何的表面边界。最后，PI还研究了奇异空间上的几何演化方程。就几何简单性而言，第一种情况是将曲率流从封闭嵌入曲线的设置推广到曲线网络的设置;目标是建立一个很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。为了将这项工作放到更广泛的背景下，几何分析中的大多数工作都是在设置光滑的几何对象，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且非常频繁地出现，并且尝试将几何分析的方法和结果扩展到这类更广泛的对象是自然的。然而，适当的工具，从分析和偏微分方程并不存在，在这种普遍性，所以一个主要部分的工作是需要将这些技术扩展到空间的奇异性。这是PI在其职业生涯中的一项重大努力。具体的问题，他现在的工作涉及精致的问题，如存在和性质的最佳指标，或形状，对这些奇异的空间，并研究发展方程不断变形这样的空间到这些最佳形状之一。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的类似问题还没有被系统地研究过。另一个问题涉及研究一类最优（最小）曲面在具有常负曲率的空间内的表面积的重整化版本。这是一个在弦理论中被深入研究的作用泛函。这位PI的工作的最后一部分是教育性质的：他是每年在他的机构举行的住宅暑期项目（SUMaC）的创始人和主任，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。该方案现已进入第十四个年头。