Searching for Closed Minimal Surfaces on Compact Riemannian Manifolds

搜索紧致黎曼流形上的闭合极小曲面

• 批准号: 8800526
• 负责人: Sheldon Chang
• 金额: $ 3.72万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-07-01 至 1990-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8800526&HistoricalAwards=false
• 关键词: Searching; Closed; Minimal; Surfaces; Compact

Sheldon Chang will continue his work on minimal surfaces. The origins of this subject lie in the study of soap films. Because of the surface tension in these films they naturally form minimal surfaces. That is, for a given boundary they have minimum area of all surfaces with this boundary. In the mathematical theory this property translates into a curvature property of the surface. Nowadays surfaces with zero mean curvature are referred to as minimal surfaces. Chang's research is concerned with the existence of minimal surfaces within a given surface. This is a very natural and important extension since such minimal surfaces provide a natural extension of the notion of geodesic, that is, minimum length path, in the surface. Chang has developed remarkable expertise in the rather inaccessible topic of geometric measure theory. Many of the most exciting developments in minimal surface theory have made use of this theory. He will build on recent work of others concerning integral currents and varifolds to attack these problems. The major focus of the work will be to study the topological type of these minimal surfaces and the size of their set of singularities. Probably the most difficult hurdle to overcome will be to find techniques which apply to the high codimensional situation.

谢尔顿·张将继续他在极小曲面上的工作。这门学科的起源在于肥皂片的研究。由于这些薄膜中的表面张力，它们自然形成最小的表面。也就是说，对于给定的边界，它们具有具有该边界的所有曲面的最小面积。在数学理论中，这一特性转化为曲面的曲率特性。如今，平均曲率为零的曲面被称为极小曲面。张的研究关注的是给定曲面内极小曲面的存在。这是一个非常自然和重要的扩展，因为这样的最小曲面提供了测地线概念在曲面中的自然扩展，即最小长度路径。张在几何测度论这个相当难理解的话题上发展了非凡的专业知识。极小曲面理论中许多最激动人心的发展都利用了这一理论。他将在其他人最近关于积分电流和各种电流的工作的基础上来解决这些问题。这项工作的主要焦点将是研究这些极小曲面的拓扑类型及其奇点集的大小。可能最难克服的障碍将是找到适用于高协维度情况的技术。