Mathematical Sciences: Geometry of Hyperbolic 3-Manifolds

数学科学：双曲 3 流形的几何

• 批准号: 9201466
• 负责人: Francis Bonahon
• 金额: $ 9.21万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-01 至 1996-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9201466&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Hyperbolic; Manifolds

The investigator will attack the problem of the isometric classification of hyperbolic metrics on a given 3-dimensional manifold, as well as study the topology of limit sets of such metrics. These two problems involve understanding the geometry of geometrically infinite ends of hyperbolic 3-manifolds. The approach which he plans to use is based on the technique of pleated surfaces in hyperbolic manifolds. He intends to develop a calculus for pleated surfaces which will enable him to obtain estimates on their geometry and the speed at which they converge to infinity in the 3-manifold. It is a surprising fact that although we live in a three dimensional space, a so-called 3-manifold, and so are blessed with a natural intuition about such geometric objects, in the end this does not carry us as far as we might have expected,for questions which have been settled by algebraic calculations for higher dimensional manifolds still remain baffling in the 3-dimensional case. The most famous of these is the celebrated conjecture of Poincare from around the turn of the century concerning 3- dimensional spheres, where precisely the original 3-dimensional case is the only one still open. The investigator is pursuing a variety of questions about 3-dimensional manifolds with slightly strange notions of distance on them, so-called hyperbolic metrics, but time and time again these questions have been shown to have clear relevance to the case of manifolds with a more familiar notion of distance.

研究者将着手解决等距问题 三维双曲度量的分类 流形，以及研究拓扑的极限集，这样的 指标. 这两个问题涉及到理解 双曲三维流形的几何无限端点 的 他计划使用的方法是基于褶皱的技术， 双曲流形中的曲面 他打算发展一种微积分 这将使他能够获得的估计， 它们的几何形状以及它们在 3-流形。 这是一个令人惊讶的事实，虽然我们生活在一个三 三维空间，一个所谓的三维流形，所以是幸运的， 对这种几何物体的自然直觉，最终， 并没有像我们预期的那样， 这已经解决了代数计算为更高 三维流形在三维空间中仍然令人困惑 案子 其中最著名的是著名的猜想， 彭加勒从世纪之交开始，关于3- 三维球体，而正是原来的三维 这是唯一一个还没结的案子 调查人员正在追查一名 关于三维流形的各种问题， 奇怪的距离概念，所谓的双曲度量， 但这些问题一次又一次地被证明 明确相关的情况下，流形与更熟悉的 距离的概念。