Mathematical Sciences: Geometry of Hyperbolic 3-Manifolds

数学科学：双曲 3 流形的几何

基本信息

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

项目摘要

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.

研究人员将攻击给定的三维流形上双曲线指标等轴测指标的问题，并研究此类指标的极限集拓扑。这两个问题涉及理解双曲线3个体的几何无限末端的几何形状。他计划使用的方法基于双曲线歧管中打褶表面的技术。他打算为打褶的表面开发一个微积分，这将使他能够获得其几何形状和速度在3个manifold中融合到无穷大的速度。令人惊讶的事实是，尽管我们生活在三维空间，这是一个所谓的3个manifold，因此对这种几何物体的自然直觉非常幸运，但最终，这并没有使我们无法预期的问题，因为这些问题是由代数量的较高歧管所解决的问题，这些问题仍在3- d-dimensional中仍然存在。其中最著名的是从本世纪之交的三维领域开始的众所周知的庞加雷（Poincare）猜想，那里的原始三维案例正是唯一仍开放的案例。研究人员正在提出有关三维流形的各种问题，其距离上有一些奇怪的距离，所谓的双曲线指标，但是这些问题一次又一次地证明与更熟悉的距离概念的歧管有明显的相关性。