Mathematical Sciences: The Topology and Geometry of Hyperbolic 3-Manifolds

数学科学：双曲 3-流形的拓扑和几何

• 批准号: 9626578
• 负责人: Richard Canary
• 金额: $ 6.3万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626578&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topology; Geometry; Hyperbolic

9626578 Canary Professor Canary will explore a variety of conjectures concerning hyperbolic 3-manifolds and the deformation theory of Kleinian groups. Several of these conjectures are motivated by Marden's tameness conjecture, which predicts that every hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, i.e., homeomorphic to the interior of a compact 3-manifold. Professor Canary has previously established that topological tameness has strong consequences for the geometry of a hyperbolic 3-manifold. He will also study the relationship between the algebraic limit and the geometric limit of a sequence of Kleinian groups. This study is inspired by Thurston's ending lamination conjecture, which provides a conjectural classification of all hyperbolic 3-manifolds. A 3-manifold is a mathematical space such that about any point there is a neighborhood which can be identified with a ball in 3-dimensional space. One can imagine building 3-dimensional manifolds by gluing together 3-dimensional blocks. Of course, this gluing would have to be an abstract gluing in general, not one which could be done in 3-dimensional space. For example, consider the 3-manifold obtained by taking the unit cube and gluing the top to the bottom, the front to the back, and the left side to the right side. A Riemannian metric is a way of measuring distances and angles in a 3-manifold. For instance, the world we live in is a 3-manifold with a Riemannian metric. In the 1970's, Wiliam Thurston conjectured that every 3-manifold can be cut up, in a canonical way, so that each piece has a Riemannian metric of one of 8 geometric types. He proved his conjecture for large classes of three-manifolds. The 3-manifolds which admit seven of the eight geometric types of Riemannian metrics are completely classified and well-understood. 3-manifolds that admit the eighth type of metric are called hyperbolic and are currently a subject of intense interest and act ivity in the fields of geometry and topology. Professor Canary is investigating the relationship between the topology and the geometry of hyperbolic 3-manifolds. ***

小行星9626578 金丝雀教授将探讨各种有关双曲3-流形和变形理论的克莱因集团的apturtures。 其中几个猜想的动机是马尔登的驯服猜想，该猜想预测每个具有有限生成基本群的双曲3-流形在拓扑上是驯服的，即， 与紧致三维流形的内部同胚。 Canary教授先前已经建立了拓扑驯服对双曲三维流形的几何有很强的影响。 他还将研究之间的关系的代数极限和几何极限的一系列克莱因集团。 本研究的灵感来自于Thurston的终结层猜想，它提供了所有双曲3-流形的一个几何分类。 三维流形（英语：3-manifold）是一个数学空间，使得任何点都有一个邻域，可以用三维空间中的球来识别。 人们可以想象通过将三维块粘合在一起来构建三维流形。 当然，这种粘合通常是抽象的粘合，而不是在三维空间中可以完成的粘合。 例如，考虑通过将单位立方体和顶部粘合到底部，前部粘合到后部，左侧粘合到右侧而获得的3-流形。 黎曼度量是一种测量三维流形中距离和角度的方法。 例如，我们生活的世界是一个具有黎曼度量的三维流形。 在1970年代，威廉·瑟斯顿（英语：William Thurston）提出，每一个三维流形都可以用一种规范的方式被分割，使得每一片都有一个黎曼度量（英语：Riemannian metric），是8种几何类型中的一种。 他证明了他的猜想大类三流形。 3-流形的八个几何类型的黎曼度量承认七个完全分类和理解。 3-允许第八类度量的流形被称为双曲流形，并且目前是几何和拓扑学领域中的一个强烈兴趣和活跃的主题。 Canary教授正在研究双曲三维流形的拓扑和几何之间的关系。 ***