Deformations of hyperbolic 3-manifolds

提议：DMS-0072062摘要：Bromberg教授计划研究固定三维流形上的双曲度量空间。将调查两种类型的问题。第一个是关于双曲锥流形的研究。有限体积锥流形是作为尖点双曲三维流形和闭双曲三维流形之间的纽带而自然产生的。Bromberg的目标是将无限体积双曲锥流形的空间参数化为非奇异双曲结构。第二个问题涉及开三维流形上完备的无限体积度量的形变空间的拓扑。Anderson和Canary已经证明了固定同伦类型流形上的这种度量空间具有一个非常复杂的拓扑，类似于在复动力学中研究的Mandelbrot集。在与J.Holt的合作中，Bromberg已经证明了，即使变形空间被限制在McMullen的固定同胚型工作的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地出现在这两个主题中。Bromberg还计划研究具有给定离散完整表示的投影结构。本项目属于低维拓扑与几何领域，主要研究的对象是三维流形，三维流形是指比曲面多一维的空间(2-流形)。三维流形，例如我们生活的空间，是数学中最基本的对象之一，但关于它们，有许多简单的问题我们无法回答。度量是度量三维流形上距离的一种方法，而双曲度量是一类特殊的度量，使得在每个点、每个方向上的几何局部看起来是相同的。W.瑟斯顿的工作表明，“大多数”三维流形都带有双曲度量，Ahlfors，Bers和其他人的早期工作表明，具有无限体积的双曲度量的开放三维流形实际上会有许多双曲度量。有一个庞大的项目来理解所有这些指标，这个项目就是其中的一部分。

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.