Mathematical Sciences: Geometry of Hyperbolic 3-Manifolds

研究人员将解决给定三维流形上双曲度量的等距分类问题，并研究这种度量的极限集的拓扑结构。这两个问题涉及到理解双曲三维流形的几何无穷端的几何。他计划使用的方法是基于双曲流形中的褶皱曲面技术。他打算开发一种关于褶皱曲面的微积分，这将使他能够获得关于它们的几何形状和它们在3-流形中收敛到无穷大的速度的估计。令人惊讶的是，尽管我们生活在一个三维空间，也就是所谓的三维流形，因此对这样的几何对象有一种自然的直觉，但最终这并没有带给我们我们可能预期的那样远的距离，因为已经通过代数计算解决的问题在三维情况下仍然令人困惑。其中最著名的是世纪之交关于三维球体的著名的庞加莱猜想，其中确切地说，原始的三维情况是唯一仍然开放的。研究人员正在探索关于具有稍微奇怪的距离概念的三维流形的各种问题，即所谓的双曲线度量，但这些问题一次又一次地被证明与具有更熟悉的距离概念的流形的情况有明显的相关性。

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.