The Topology and Geometry of Hyperbolic 3-Manifolds

摘要：Canary教授提出研究双曲型3流形的变形空间。在他的第一个项目中，他将研究双曲3-流形同伦等价于固定紧3-流形M的空间AH(M)的整体拓扑，其内部的分量由拓扑数据枚举，每个分量由解析数据参数化。在M具有不可压缩边界的情况下，Anderson， Canary和McCullough给出了当内部的两个分量具有相交闭包时的完整表征。Canary建议进一步研究交点轨迹的拓扑结构，从而更好地理解空间。通过沙利文字典，对AH(M)的研究类似于对复杂动力学中的曼德尔布罗特集的研究。Canary还提出研究几何有限双曲3流形的拟共形变形空间的边界，以及曲面群的任意几何极限的结构。Canary研究了3流形的拓扑结构和几何结构之间的关系。三维流形是一个数学空间，使得在任何一点上都有一个区域看起来像三维空间中的一个区域。黎曼度规是一种在三维流形中测量距离和角度的方法。一旦有了流形上的黎曼度规就可以讨论它的几何了。例如，我们生活的世界是一个带有黎曼度规的3流形。双曲度规是一种特别好的黎曼度规，自从它在19世纪首次被发现以来，就被拓扑学家和几何学家广泛研究。最近，物理学家一直在讨论具有双曲度量的宇宙几何模型。宇宙模型的度量选择与我们的宇宙是膨胀还是收缩的问题有关。Canary的大部分研究都涉及对同一拓扑流形上不同双曲度量空间的研究。

Proposal: DMS-9971554Principal Investigator: Richard D. Canary Abstract: Professor Canary proposes to study deformation spaces of hyperbolic 3-manifolds. In his first project, he will will study the global topology of the space AH(M) of hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M. The components of its interior are enumerated by topological data and each component is parameterized by analytic data. In the case that M has incompressible boundary, Anderson, Canary and McCullough have given a complete characterization of when two components of the interior have intersecting closures. Canary proposes to further investigate the topology of the intersection locus and thus develop a better understanding of the space. This study of AH(M) is analogous, via the Sullivan dictionary, to the study of the Mandelbrot set in complex dynamics. Canary also proposes to study boundaries of quasiconformal deformation spaces of geometrically finite hyperbolic 3-manifolds, as well as the structure of arbitrary geometric limits of surface groups. Canary studies the relationship between the topology and the geometry of a 3-manifold. A 3-manifold is a mathematical space such that about any point there is a region which looks like a region in 3-dimensional space. A Riemannian metric is a way of measuring distances and angles in a 3-manifold. Once one has a Riemannian metric on a manifold one can talk about its geometry. For example, the world we live in is a 3-manifold with a Riemannian metric. A hyperbolic metric is an especially nice type of Riemannian metric, which has been extensively studied by topologists and geometers since it was first discovered in the nineteenth century. Recently, physicists have been discussing models for the geometry of our universe which have hyperbolic metrics. The choice of metric for one's model of the universe is related to the issue of whether our universe is expanding or contracting. Much of Canary's research involves the study of spaces of different hyperbolic metrics on the same topological manifold.