Deformation Spaces of Hyperbolic 3-manifolds

研究者计划研究固定3流形上的双曲度量空间。与封闭流形不同，开放的3-流形通常有很大的双曲度量空间。研究人员的目标是了解这个空间。有些问题很简单：变形空间是其内部的封闭吗？空间的一个更详细的描述是结束分层猜想，它给出了对流形上的每个双曲度量进行分类的不变量。这种分类不是参数化；不变量空间的映射是不连续的。研究者感兴趣的是理解这些不连续和描述度量的变形空间的拓扑结构。双曲锥度规是整个工作中使用的关键工具。其原理是，一个光滑的双曲结构中非常复杂的几何结构可以用一个不那么复杂但奇异的双曲锥度规来交换。三流形是数学空间，局部看起来像我们生活的宇宙。从庞加莱开始，数学家们一直对三流形的分类问题感兴趣。双曲几何也是一个非常古老的领域，可以追溯到18世纪中期。在过去的25年里，主要由于瑟斯顿的工作，人们已经意识到这两个主题之间存在着深刻而美丽的联系。

DMS-0204763Kenneth BrombergThe investigator plans to study spaces of hyperbolic metrics on a fixed3-manifold. Unlike on a closed manifold, on an open 3-manifold there willtypically be a large space of hyperbolic metrics. The investigatorsgoal is to understand this space. Some questions are simple to state:Is the deformation space the closure of its interior? A more detaileddescription of the space is the ending lamination conjecture,which gives invariants that classify every hyperbolic metric onthe manifold. This classification is not a parameterization; the map tothe space of invariants is not continuous. The investigator isinterested in understanding these discontinuities and describing thetopology of the deformation space of metrics. Hyperbolic cone-metricsare a key tool used throughout this work. The philosophy is that verycomplicated geometry in a smooth hyperbolic structure can be exchangedfor a less complicated, but singular, hyperbolic cone metric.Three manifolds are mathematical spaces that locally look like theuniverse we live in. Mathematicians, beginning with Poincare, have beeninterested in classification question about three manifolds. Hyperbolicgeometry is also a very old field dating back to the middle of theeighteenth century. In the last twenty-five years, largely due to the workThurston, it has been realized that there are deep and beautifulconnections between the two topics.