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: 9803493
• 负责人: Leon Simon
• 金额: $ 22.27万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803493&HistoricalAwards=false
• 关键词: Regularity; Singularity; Geometric; Variational; Problems

Abstract Proposal DMS-9803493 Principal Investigators: Leon Simon and Brian White Leon Simon proposes to pursue various questions related to the structure of the singular sets of minimal submanifolds and energy minimizing maps, including the extension of his recent work on smoothness of the singular set of area minimizing submanifolds to higher dimensions and to other classes of submanifolds. Specifically, he is proposing to consider the question of whether the top-dimensional part of the singular set locally lies in a finite union of smooth submanifolds, which he has recently established for 3 and 4 dimensional mod-2 minimizing submanifolds in codimension 2. Simon also proposes to continue his efforts to develop methods for generating examples of singular sets, and to pursue several questions related to asymptotics on approach to singularities. In addition he proposes to study questions related to the structure of the branching set of energy minimizing maps. Brian White plans to study how regularity properties and singular structure for minimizing chains with coefficients in a metric group depend on the group and its metric. This includes study of immiscible fluid interfaces as a special case. He also plans to investigate singularities in the mean-curvature flow and in a related hyperbolic flow that should more accurately model the dynamics of real soap films. He will also continue his investigations of branch points in 2 dimensional minimal surfaces. An understanding of singularities, and how singularities are formed, is a fundamental element in the overall understanding of many physical and geometric phenomena. For example, in cosmology singularities of space-time (e.g. "black holes") play a fundamental role. Likewise in the study of the "canonical" objects which arise naturally in topology and geometry, the understanding of singularities is absolutely fundamental. As with most non-linear phenomena, there is no single well ordered theory which a pplies in a wide range of different contexts. Rather, each different context has its own collection of effective techniques, and it is the development and application of such techniques in the context of the geometric calculus of variations which is the focus of the present research proposal. Specifically, Simon and White propose to continue their efforts toward a complete understanding of singularities, and how they are formed, in the context of area minimizing submanifolds and energy minimizing maps. Such techniques are likely to be applicable to the study of other objects of geometric and physical significance---for example to the study of immiscible fluid interfaces, soap-films, and the equilibrium free surfaces of fluids.

摘要建议DMS-9803493主要研究人员：莱昂·西蒙（Leon Simon）和布莱恩·怀特·莱昂·西蒙（Brian White Leon Simon）提议提出与最小的亚曼菲尔德（Minimal Submanifolds）的结构相关的各种问题，以及最小化图的结构，包括他最近将Submanifolds奇异平稳的工作范围扩展到更高的suipmanions and Submanifs submanif submanif of Submanif的奇异平稳性。 具体而言，他提议考虑一个问题，即单数集的顶级部分是否在当地是否在于有限的平滑亚策略结合，他最近为3和4个维度-2建立了这一问题，这是Condimension 2中的3和4维MOD-2。 此外，他建议研究与能量最小化图的分支集合相关的问题。 布莱恩·怀特（Brian White）计划研究如何在公制组中使用系数最小化链的规律性和奇异结构取决于该组及其度量。 这包括将不混溶的液体界面作为特殊情况的研究。 他还计划研究均值曲面流和相关双曲线流中的奇异性，该流动流应该更准确地对真肥皂膜的动力学进行建模。 他还将继续对2维最小表面的分支点进行调查。 对奇异性的理解以及奇异性的形成方式，是对许多物理和几何现象的总体理解的基本要素。 例如，在宇宙学中，时空（例如“黑洞”）起着基本作用。 同样，在对拓扑和几何形状自然出现的“规范”对象的研究中，对奇点的理解绝对是基本的。 与大多数非线性现象一样，没有单一的有序理论，可以在各种不同的环境中进行。 相反，每个不同的环境都有自己的有效技术收集，并且是在变化的几何计算的背景下的开发和应用，这是本研究建议的重点。 具体而言，西蒙（Simon）和怀特（White）提议在最小化亚货号和能量最小化地图的情况下，继续对奇点的完全理解以及如何形成。 这样的技术可能适用于对几何和物理意义的其他物体的研究 - 例如，研究不混溶的流体界面，肥皂膜和流体的平衡不足表面。