Mathematical Sciences: Limit Sets of Foliations in Hyperbolic 3-Manifolds and Anosov Flows

数学科学：双曲 3 流形和阿诺索夫流中的极限叶集

• 批准号: 9201744
• 负责人: Sergio Fenley
• 金额: $ 4.36万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-01 至 1995-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9201744&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Limit; Sets; Foliations

A codimension-one foliation F in a hyperbolic 3-manifold lifts to a foliation F' in the universal cover, which is hyperbolic 3- space. The investigator will study the asymptotic behavior of leaves of F' and how fast they approach their limit sets. He will also analyze to what extent the topological structure of F determines the geometry of the manifold. Finally, a non-singular flow is Anosov if it induces hyperbolic behavior transversally to the flow direction, so that in dimension 3 it determines two transverse codimension-one foliations. The investigator plans to examine the structure of the induced foliations in the universal cover. This is strongly related to homotopic properties of closed orbits of the flow. He also plans to study metric properties of flow lines, for example, whether all flow lines can be quasigeodesic. A foliation of a manifold is a way of filling the manifold with lower dimensional pieces. In the case of a codimension-one foliation, these pieces are of dimension one less than that of the given manifold. Think of an onion or an artichoke. The topology of a manifold is strongly related to the kind of foliation which it will support, and in skillful hands this relation has been forged into a powerful tool for investigating the topology of manifolds. Foliations are also intimately related to the types of flows possible on a manifold. Particularly in low dimensions, where the standard algebraic techniques of topology work less well, using foliations to investigate manifolds and the possible flows on them is a very welcome addition to the arsenal.

双曲三维流形中的余维一叶理F 到泛覆盖中的叶理F'，它是双曲的3- 空间 研究者将研究以下的渐近行为： 他们的速度，他们的速度，他们的速度。 他将 并分析了F的拓扑结构在多大程度上 决定了流形的几何形状。 最后，一个非奇异的 流是Anosov，如果它引起横向于 流动方向，因此在维度3中，它决定了两个 横余维一叶理 研究者计划 研究宇宙中诱导叶理的结构 掩护 这与闭的同伦性质密切相关。 流的轨道。 他还计划研究 流水线，例如，是否所有流水线都可以 准短程线 流形的叶理是填充流形的一种方法 低维度的碎片。 在余维1的情况下， 叶理，这些碎片的尺寸小于 给定流形。 想想洋葱或朝鲜蓟。 拓扑 一个流形的形状与它所处的叶理类型密切相关， 将支持，并在熟练的手，这种关系已经锻造 成为研究流形拓扑的有力工具。 叶序也与水流类型密切相关 可能在一个流形上。 特别是在低维情况下， 拓扑学的标准代数技术工作得不太好， 研究流形和流形上可能的流动 是一个非常受欢迎的武器库。