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 提案DMS-9803493主要研究者：Leon Simon和Brian白色 莱昂·西蒙提出追求与极小子流形和能量最小化映射的奇异集的结构有关的各种问题，包括将他最近关于面积最小化子流形的奇异集的光滑性的工作扩展到更高的维度和其他类别的子流形。 具体来说，他建议考虑的问题是否顶维部分的奇异集局部在于有限工会顺利子流形，他最近建立了3和4维模2最小化子流形的余维2。 西蒙还建议继续他的努力，以发展方法产生的例子，奇异集，并追求几个问题有关的渐近方法的奇异性。 此外，他建议研究的问题有关的结构的分支集的能源最小化地图。 布赖恩白色计划研究如何正则性和奇异结构最小化链的系数在度量组取决于组和它的度量。 这包括作为特殊情况的不混溶流体界面的研究。 他还计划研究平均曲率流和相关双曲线流中的奇点，这应该更准确地模拟真实的肥皂膜的动力学。 他还将继续他的调查分支点在2维极小曲面。 对奇点的理解，以及奇点是如何形成的，是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，时空的奇点（例如“黑洞”）扮演着重要的角色。 同样，在研究拓扑学和几何学中自然出现的“规范”对象时，对奇点的理解是绝对基本的。 与大多数非线性现象一样，没有一个单一的有序理论可以在广泛的不同背景下应用。 相反，每个不同的背景下有自己的有效的技术，它是发展和应用这些技术的背景下的几何变分法，这是本研究建议的重点。 具体来说，西蒙和白色建议继续努力，以一个完整的理解奇点，以及它们是如何形成的，在面积最小化子流形和能量最小化地图的背景下。 这样的技术可能适用于其他对象的几何和物理意义的研究-例如，不混溶流体界面，肥皂膜和流体的平衡自由表面的研究。