Regularity and Singularity in Geometric Variational Problems and in Geometric Flow Problems

几何变分问题和几何流问题中的正则性和奇异性

• 批准号: 0104049
• 负责人: Leon Simon
• 金额: $ 31.27万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104049&HistoricalAwards=false
• 关键词: Regularity; Singularity; Geometric; Variational; Problems

Abstract for NSF proposal DMS 0104049 (Leon Simon \& Brian White joint P.I.'s)Leon Simon plans to pursue various questions related to the structureof the singular sets of minimal submanifolds and energy minimizingmaps, including the extension of his recent work on construction ofsingular examples of minimal hypersurfaces. A current aim is toconstruct examples of minimal submanifolds with singular sets whichinclude ``gaps'' and other phenomena, the possibility of which is leftopen by his previous general work on the structure of the singularset. He is also proposing to complete work on the singular set ofmod-2 currents to show that, in all dimensions, the top-dimensionalpart of the singular set locally lies in a finite union ofcontinuously differentiable submanifolds. Simon also proposes topursue several questions related to asymptotics on approach tosingularities. Brian White plans to study how regularity propertiesand singular structure for minimizing chains with coefficients in ametric group depend on the group and its metric. This includes studyof immiscible fluid interfaces as a special case. He also plans toinvestigate singularities in the mean-curvature flow. He willcontinue his investigations of 2-dimensional minimal surfaces, inparticular concerning branch points and concerning total boundarycurvature.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. Likewisein the study of the ``canonical'' objects which arise naturally intopology and geometry, singularities arise in a very natural andunavoidable manner, and the understanding of these singularities is anabsolutely fundamental problem. As with most non-linear phenomena,there is not a single general theory which applies in a wide range ofdifferent contexts. Rather, each different context has its owncollection of effective techniques, and it is the development andapplication of such techniques in the context of the geometriccalculus of variations which is the focus of the present researchproposal. Specifically, Simon and White propose to continue theirefforts toward a complete understanding of singularities, and how theyare formed, in the context of area minimizing submanifolds and energyminimizing maps. Such techniques are likely to be applicable to thestudy of other objects of geometric and physical significance---forexample to the study of immiscible fluid interfaces, soap-films, andthe equilibrium free surfaces of fluids.

NSF提案DMS 0104049 （Leon Simon &amp； Brian White联合P.I.）摘要s)Leon Simon计划研究与最小子流形和能量最小化映射的奇异集结构相关的各种问题，包括他最近关于最小超曲面奇异例子构造的工作的扩展。目前的目标是构造奇异集的最小子流形的例子，其中包括“间隙”和其他现象，这是他之前关于奇异集结构的一般工作留下的可能性。他还建议完成关于模2电流的奇异集的工作，以证明在所有维度中，奇异集的顶维部分局部存在于连续可微子流形的有限并中。Simon还提出了几个与逼近奇点的渐近性相关的问题。Brian White计划研究在度量群中带系数的最小化链的正则性和奇异结构是如何依赖于群及其度量的。这包括作为特殊情况的非混相流体界面的研究。他还计划研究平均曲率流中的奇点。他将继续研究二维极小曲面，特别是分支点和总边界曲率。对奇点的理解，以及奇点是如何形成的，是我们对许多物理和几何现象的整体理解的一个基本要素。例如，在宇宙学中，时空的奇点(如：“黑洞”)扮演着重要的角色。同样，在研究拓扑和几何中自然产生的“规范”对象时，奇点以一种非常自然和不可避免的方式出现，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广泛的不同背景。相反，每个不同的背景都有自己的有效技术集合，而这些技术在几何变分的背景下的发展和应用是本研究计划的重点。具体来说，西蒙和怀特建议继续努力，在面积最小化子流形和能量最小化图的背景下，全面理解奇点，以及它们是如何形成的。这些技术可能适用于研究其他具有几何和物理意义的对象，例如研究不混相流体界面、皂膜和流体的平衡自由表面。