Mathematical Sciences: Geometry of Hyperbolic 3-Manifolds

研究者将研究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3流形的几何无穷端点的几何性质。他计划使用的方法是基于双曲流形的褶皱表面技术。他打算发展一种褶皱表面的微积分，这将使他能够获得它们的几何形状和它们在3流形中收敛到无穷远的速度的估计。令人惊讶的是，尽管我们生活在一个三维空间，一个所谓的三维流形中，因此我们对这些几何物体有一种天生的直觉，但最终这并没有把我们带得像我们所期望的那样远，因为那些在高维流形中通过代数计算已经解决的问题，在三维情况下仍然令人困惑。其中最著名的是庞加莱在世纪之交提出的关于三维球体的著名猜想，正是在这个猜想中，最初的三维情况是唯一仍然开放的。研究者正在研究各种关于三维流形的问题，这些流形上有一些奇怪的距离概念，即所谓的双曲度量，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确的关联。

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.