Deformation Spaces of Hyperbolic 3-manifolds

双曲3流形的变形空间

• 批准号: 0406976
• 负责人: Kenneth Bromberg
• 金额: $ 8.67万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0406976&HistoricalAwards=false
• 关键词: Deformation; Spaces; Hyperbolic; manifolds

DMS-0204763Kenneth BrombergThe investigator plans to study spaces of hyperbolic metrics on a fixed3-manifold. Unlike on a closed manifold, on an open 3-manifold there willtypically be a large space of hyperbolic metrics. The investigatorsgoal is to understand this space. Some questions are simple to state:Is the deformation space the closure of its interior? A more detaileddescription of the space is the ending lamination conjecture,which gives invariants that classify every hyperbolic metric onthe manifold. This classification is not a parameterization; the map tothe space of invariants is not continuous. The investigator isinterested in understanding these discontinuities and describing thetopology of the deformation space of metrics. Hyperbolic cone-metricsare a key tool used throughout this work. The philosophy is that verycomplicated geometry in a smooth hyperbolic structure can be exchangedfor a less complicated, but singular, hyperbolic cone metric.Three manifolds are mathematical spaces that locally look like theuniverse we live in. Mathematicians, beginning with Poincare, have beeninterested in classification question about three manifolds. Hyperbolicgeometry is also a very old field dating back to the middle of theeighteenth century. In the last twenty-five years, largely due to the workThurston, it has been realized that there are deep and beautifulconnections between the two topics.

DMS-0204763Kenneth Bromberg 研究人员计划研究固定 3 流形上的双曲度量空间。与闭流形不同，在开三流形上通常会有很大的双曲度量空间。调查人员的目标是了解这个空间。有些问题很容易表述：变形空间是其内部的封闭吗？对空间的更详细描述是最终层压猜想，它给出了对流形上的每个双曲度量进行分类的不变量。这种分类不是参数化；而是参数化。不变量空间的映射不是连续的。研究人员有兴趣了解这些不连续性并描述度量变形空间的拓扑。双曲锥度量是整个工作中使用的关键工具。其理念是，平滑双曲结构中非常复杂的几何形状可以替换为不太复杂但奇异的双曲圆锥度量。三个流形是局部看起来像我们生活的宇宙的数学空间。从庞加莱开始，数学家一直对分类问题感兴趣大约三个流形。双曲几何也是一个非常古老的领域，其历史可以追溯到十八世纪中叶。在过去的二十五年里，很大程度上由于瑟斯顿的工作，人们已经意识到这两个主题之间存在着深刻而美丽的联系。