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

摘要“几何变分和几何流动问题中的正则性问题”（Leon Simon和Brian White） 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有关的各种问题。更具体地说，Simon的目标是建立一阶正则性（即奇异集位于连续可微子流形的局部有限联合），用于各种多重性1类的点附近，其中有一个切线圆柱体的横截面允许校准。这样的结果特别适用于“顶维”奇点附近的mod-2最小值。Brian White计划研究平均曲率流的规律性，包括非唯一性和“增肥性”。此外，他将继续研究度量群中带系数的最小化锥的奇异结构，以及他在2维最小曲面上的工作，特别强调分支点的研究。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”)扮演了一个基本的角色，而对几何流动问题中奇点形成的理解是汉密尔顿、佩雷尔曼等人实现瑟斯顿几何化猜想的关键因素。同样，在研究拓扑和几何中自然产生的“规范”对象时，奇点也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演算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，以及在各种几何流问题的背景下，更全面地理解奇点，以及奇点是如何形成的。

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.