Searching for Closed Minimal Surfaces on Compact Riemannian Manifolds

谢尔顿·张将继续他在最小表面上的工作。 这门学科的起源在于对肥皂膜的研究。 由于这些薄膜的表面张力， 最小的表面。也就是说，对于给定的边界， 所有具有此边界的曲面的面积。在数学 理论上，这个性质转化为一个曲率性质， 面现在，平均曲率为零的曲面被称为 作为最小的表面。张的研究关注的是 在给定曲面内存在极小曲面。这是一 非常自然和重要的延伸，因为这种最小的表面 提供了测地线概念的自然延伸，即， 最小长度路径，在表面上。 张已经在相当多的领域发展了非凡的专业知识， 几何测度论中难以触及的话题。许多最 最小曲面理论的令人兴奋的发展利用了 这个理论。他将以其他人最近的工作为基础， 积分电流和变量来解决这些问题。的 工作的主要重点将是研究的拓扑类型 这些最小曲面及其集合的大小 奇点可能是最难克服的障碍 将是找到适用于高余维的技术， 形势

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.