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

NSF提案DMS 0104049摘要 (Leon Simon \& Brian白色联合私家侦探Leon Simon计划研究与极小子流形和能量极小化映射的奇异集的结构有关的各种问题，包括他最近关于极小超曲面的奇异例子的构造的工作的扩展。 目前的目标是构造具有奇异集的极小子流形的例子，其中包括“间隙”和其他现象，他之前关于奇异集结构的一般工作尚未确定这种可能性。 他还建议完成工作的奇异集的模-2电流表明，在所有维度，top-dimensionalpart奇异集局部在于一个有限的工会连续微分子流形。 Simon还提出了几个与奇点渐近性有关的问题。 Brian白色计划研究系数在度量群中的极小化链的正则性和奇异结构如何依赖于群及其度量。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中的奇点。 他将继续他的调查2维极小曲面，特别是关于分支点和关于总boundarycurvature。理解奇点，以及奇点是如何形成的，是一个基本要素，在我们的整体理解许多物理和几何现象。 例如，在宇宙学中，时空奇点（如“黑洞”）扮演着重要的角色。 就像在研究拓扑学和几何学中自然出现的“经典”对象一样，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，每一种不同的背景下都有自己的有效技术的集合，并且在变分几何的背景下开发和应用这些技术是本研究建议的重点。具体地说，西蒙和白色建议继续他们的reffecting走向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energyminimizing地图的背景下。 这种技术很可能适用于研究其他具有几何和物理意义的物体-例如研究不混溶流体界面、皂膜和流体的平衡自由表面。

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.