Deformations of hyperbolic 3-manifolds

提案：DMS-0072062摘要：Bromberg教授计划研究固定3流形上的双曲度量空间。将调查两类问题。第一个是双曲锥流形的研究。有限体积锥流形是连接尖点双曲3-流形和闭双曲3-流形的一种自然现象。Bromberg的目标是参数化无限体积双曲锥流形的空间，并利用这种参数化来理解非奇异双曲结构。第二个问题涉及开3-流形上完全的无限体积度量的变形空间的拓扑。安德森和Canary已经表明，空间的这种度量的流形上的固定同伦有一个非常复杂的拓扑结构类似的曼德尔布罗特集研究的复杂动力学。在与J. Holt的联合工作中，Bromberg证明了即使变形空间被限制为固定同胚类型的3-流形上的度量，这种复杂的行为仍然存在。复杂的投射结构自然出现在这两个主题中。Bromberg还计划研究具有给定离散完整性表示的投影结构。本项目属于低维拓扑与几何领域，主要研究对象是三维流形，即比曲面（2-流形）多一维的空间。三维流形，例如我们生活的空间，是数学中最基本的对象，但有许多关于它们的简单问题我们无法回答。度量是在三维流形上测量距离的一种方法，而双曲度量是一类特殊的度量，使得几何在每个点和每个方向上局部看起来都是一样的。瑟斯顿表明，“大多数”3流形进行双曲度量和早期的工作Ahlfors，Bers和其他人已经表明，一个开放的3流形，有一个无限体积双曲度量实际上将有许多双曲度量。有一个庞大的计划来理解所有这些指标，这个项目是其中的一部分。

Proposal: DMS-0072062Abstract:Professor Bromberg plans to study spaces of hyperbolic metrics on a fixed3-manifold. Two types of questions will be investigated. The first ofthese is a study of hyperbolic cone-manifolds. Finite volumecone-manifolds arise naturally as a link between cusped hyperbolic3-manifolds and closed hyperbolic 3-manifolds. Bromberg's goal is toparameterize spaces of infinite volume hyperbolic cone-manifolds and usethis parameterization to understand non-singular hyperbolic structures.The second question involves the topology of deformation spaces ofcomplete, infinite volume metrics on open 3-manifolds. Anderson and Canaryhave shown that the space of such metrics on a manifold of fixed homotopytype has a very complicated topology akin to that of the Mandelbrot setstudied in complex dynamics. In joint work with J. Holt, Bromberg hasshown that this complicated behavior persists even if the deformationspace is restricted to metrics on a 3-manifold of fixed homeomorphism typegeneralizing work of McMullen. Complex projective structures naturallyappear in both of these topics. Bromberg also plans to study projectivestructures with a given discrete holonomy representation. This project lies in the field of low dimensional topology and geometry.The main objects studied are three-manifolds which are spaces with onemore dimension than a surface (a 2-manifold). Three-manifolds,such as the space that we live in, are some of the most basicobjects in mathematics yet there are many simple questions about them thatwe cannot answer. A metric is a way of measuring distance on a 3-manifoldand a hyperbolic metric is one of a special class of metrics such that atevery point and in every direction the geometry locally looks the same.The work of W. Thurston has shown that "most" 3-manifolds carry ahyperbolic metric and earlier work of Ahlfors, Bers and others has shownthat an open 3-manifold that has an infinite volume hyperbolic metric willin fact have many hyperbolic metrics. There is a vast program tounderstand all such metrics of which this project is a piece.