Mathematical Sciences: Applications of Minimal Surfaces to Curvature and Topology of Riemannian Manifolds and Submanifolds

数学科学：最小曲面在黎曼流形和子流形曲率和拓扑中的应用

• 批准号: 9002203
• 负责人: John Moore
• 金额: --
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-07-01 至 1993-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9002203&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Applications; Minimal; Surfaces

The principal investigator will develop a Morse theory of minimal two-spheres analogous to that of closed geodesics. A goal of the project is to construct large numbers of minimal two- spheres of low area using a result of the investigator concerning Morse theory at low energies. The investigator will also exploit the relationship between minimal two-spheres in Riemannian manifolds and holomorphic vector bundles over the Riemann sphere. And an extension of Hilbert's theorem to n-dimensional submanifolds of constant negative curvature in odd-dimensional Euclidean space will also be investigated. This goal of this project is to understand minimal surfaces with low area. Soap bubbles are examples of minimal surfaces which minimize energy. The principal investigator will use Morse theory to study these surfaces and their generalizations to higher dimensions. This theory is an extension or our concept of elevation, where basins and mountain passes undergo separate mathematical interpretations.

首席研究员将开发一个莫尔斯理论的最小两个领域类似的封闭测地线。该项目的一个目标是利用低能量下关于莫尔斯理论的研究结果来构造大量的低面积的最小二球。研究者还将利用黎曼流形中的极小两球面与黎曼球面上的全纯向量丛之间的关系。并且将Hilbert定理推广到奇维欧氏空间中的n维常负曲率子流形。这个项目的目标是了解低面积的最小表面。肥皂泡是最小化能量的最小表面的例子。首席研究员将使用莫尔斯理论来研究这些表面和他们的推广到更高的维度。这个理论是我们的海拔概念的延伸，盆地和山口经历了单独的数学解释。