Regularity questions in the geometric calculus of variations and in geometric flow problems

几何变分法和几何流问题中的正则性问题

• 批准号: 0406209
• 负责人: Leon Simon
• 金额: --
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0406209&HistoricalAwards=false
• 关键词: Regularity; questions; geometric; calculus; variations

Proposal DMS-0406209 Abstract ``Regularity questions in the geometric calculus of variationsand in geometric flow problems'' (Leon Simon and Brian White) Leon Simon plans to pursue various questions related to the structureof the singular sets of minimal submanifolds and energy minimizingmaps. More specifically Simon aims to establish first orderregularity (i.e. that the singular set lies in a locally finite unionof continuously differentiable submanifolds) for various multiplicity1 classes near points where there is a tangent cylinder with across-section which admits a calibration. Such a result would inparticular apply to mod-2 minimizers near ``top-dimensional'' singularpoints. Brian White plans to study regularity properties of meancurvature flow, including non-uniqueness properties and ``fattening.''In addition he will continue his work on the singular structure ofminimizing cones with coefficients in a metric group, and his work on2 dimensional minimal surfaces with particular emphasis on the studyof branch points. An understanding of singularities, and how singularities are formed,is a fundamental element in our overall understanding of many physicaland geometric phenomena. For example, in cosmology singularities ofspace-time (e.g. ``black holes'') play a fundamental role, and theunderstanding of singularity formation in geometric flow problems is akey ingredient in the approach of Hamilton, Perelman and others toThurston's Geometrization Conjecture. Likewise in the study of the``canonical'' objects which arise naturally in topology and geometry,singularities arise in a very natural and unavoidable manner, and theunderstanding of these singularities is an absolutely fundamentalproblem. As with most non-linear phenomena, there is not a singlegeneral theory which applies in a wide range of different contexts.Rather, each different context has its own collection of effectivetechniques, and it is the development and application of suchtechniques in the context of the geometric calculus of variationswhich is the focus of the present research proposal. Specifically,Simon and White propose to continue their efforts toward a morecomplete understanding of singularities, and how they are formed, inthe context of area minimizing submanifolds and energy minimizingmaps, and in the context of various geometric flow problems.

提议DMS-0406209《几何变分中的正则性问题和几何流问题中的正则性问题》(Leon Simon和Brian White)Leon Simon计划探讨与极小子流形和能量最小化映射的奇集的结构有关的各种问题。更具体地说，Simon的目标是建立各种重数类的一阶正则性(即奇异集位于连续可微子流形的局部有限单位中)，这些点附近有一个允许校准的切线柱面。这样的结果将特别适用于“顶维”奇点附近的mod-2极小值。Brian White计划研究平均曲率流的正则性，包括非唯一性和“肥化”性质。此外，他还将继续研究系数在度量组中的锥体最小化的奇异结构，以及他在二维极小曲面上的工作，重点是分支点的研究。对奇点的理解，以及奇点是如何形成的，是我们全面理解许多物理和几何现象的基本要素。例如，在宇宙学中，时空奇点(如“黑洞”)起着基础性的作用，而对几何流动问题中奇点形成的理解是哈密尔顿、佩雷尔曼等人解决瑟斯顿几何化猜想的一个关键因素。同样，在研究拓扑学和几何学中自然产生的“正则”对象时，奇点以一种非常自然和不可避免的方式出现，而对这些奇点的理解是一个绝对基本的问题。与大多数非线性现象一样，没有一个单一的通用理论适用于各种不同的背景，相反，每个不同的背景都有自己的有效技术集合，而这些技术的发展和应用是在几何变分的背景下进行的，这是本研究方案的重点。具体地说，Simon和White建议继续努力，在面积最小子流形和能量最小化映射的背景下，以及在各种几何流动问题的背景下，更全面地理解奇点及其形成方式。