Research in Differential Geometry and Topology

微分几何与拓扑研究

• 批准号: 0104044
• 负责人: William Meeks
• 金额: $ 6万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104044&HistoricalAwards=false
• 关键词: Research; Differential; Geometry; Topology

Abstract for DMS - 0104044In this proposal the research will study the geometry, asymptotic behavior, conformal structure and topology of properly embedded minimal surfaces in R3 . One of the main goals of the proposal is to classify all properly embedded minimal surfaces of genus zero and to describe the asymptotic geometry of all finite genus examples. Related theoretical techniques concerning compactness, regularity and convergence of minimal surfaces of locally bounded genus will be investigated as well. One hoped for application of this research is to classify all smooth finite group actions on S3 . Finally the research proposes to prove that Bryant surfaces in hyperbolic three-space are unknotted.Classical minimal surface theory has its roots in 18th and 19th century mathematicsand gives one of the first important examples in what is called the calculus of variations, first described by Euler. Physically minimal surfaces can be modeled locally as soap films on wires or by surfaces of least-area relative to local boundaries. These surfaces play an important role as a tool in the study ofthree-dimensional topology and Riemannian geometry. The research in this proposal concerns global properties of these surfaces and possible applications to basic research in three-dimensional topology and geometry.

摘要DMS -0104044在这个建议中，研究将研究的几何形状，渐近行为，共形结构和拓扑结构的适当嵌入极小曲面在R3。 该建议的主要目标之一是分类所有适当嵌入极小曲面的亏格为零，并描述渐近几何的所有有限亏格的例子。同时也将研究局部有界亏格极小曲面的紧性、正则性和收敛性等相关理论技巧。 希望这项研究的应用是分类所有光滑有限群作用在S3。 经典的极小曲面理论起源于18、19世纪的世纪代数学，并给出了由Euler首先描述的变分法中最重要的例子之一。 物理上的最小曲面可以局部建模为导线上的肥皂膜或相对于局部边界的最小面积曲面。 这些曲面在三维拓扑学和黎曼几何的研究中起着重要的作用。 在这个建议中的研究涉及这些表面的整体性质和可能的应用，在三维拓扑和几何的基础研究。