Foliations, flows, and 3-manifolds: Topology and geometry

提案：DMS-0071683项目名称：“叶状体、流动和三维流形：拓扑和几何“PI：Sergio R.摘要：无肋叶理是三维流形研究中的一个基本对象。它们在3-流形拓扑学上产生了深刻的结果，并与3-流形的几何化猜想有关。 一个重要的工具是与具有双曲叶的叶理相关联的通用圆（一般情况）。 使用这个工具，PI最近证明，在R-覆盖叶理的情况下，然后要么流形是环形的，或者有一个伪Anosov流横向叶理。这样的流具有优良的动力学性质，这与流形的拓扑结构有很强的关系。这个项目的一个目标是分析各种叶理族的通用圆构造，并寻找横向伪Anosov流。第二个目标是研究叶理和横向流动的双重对象，并了解其联合动力学结构的深刻后果。该项目旨在研究一般的伪Anosov流和横向叶理所施加的附加结构。该项目的第三个目标是了解双曲三维流形中的叶理和横向流动的几何行为-一般情况。 重点将放在大规模的几何行为的通用覆盖。要分析的一个重要问题是，横截叶理的伪Anosov流是否是准测地线，也就是说，它们是否能很好地测量距离。PI（和Lee Mosher）在横向于Reebless有限深度叶理的流动情况下证明了这一性质。本研究的一个目的是利用准短程线性质的流动，以获得有关的渐近几何行为的叶理横向流动的信息。这在一般有限深度叶理的情况下特别有希望。3-流形是局部具有3维的对象，就像3维欧氏空间一样。一个三维流形的二维叶理是流形分解成二维物体，就像一本书的书页。好的叶理存在于大类的3-流形中，它们提供了关于流形的非常有用的信息。该项目的目标是了解与流形的几何和拓扑结构的叶理的关系。几何学测量距离，拓扑学粗略地测量流形中“洞”的结构。最好看看流形的万有覆盖：例如，圆柱的万有覆盖是通过将其展开成无限地毯而得到的。 泛覆盖是无界对象，但它们携带了关于流形的大量信息，包含在允许恢复流形的“折叠”中。该项目的一个重要目标是分析普遍覆盖层中的叶理。特别令人感兴趣的是叶理的“叶子”的大尺度特性。 我们将集中讨论流形是双曲流形的情况--这是一般情况。最后，对于叶理有一个双重的对象，它是叶理的叶子的横向流动。 流动和叶理共同产生了丰富的动力学结构的流形与许多深刻的后果。在许多情况下，在所有可能的横向流动中存在“最紧”的流动。 最紧密的流动应该提供有关叶理结构和与流形连接的信息。最紧的流称为伪Anosov流。

Proposal: DMS-0071683Project title: "Foliations, flows and 3-manifolds: Topology and geometry"PI: Sergio R. FenleyABSTRACT: Reebless foliations are a basic and fundamental object in the study of 3-manifolds. They yield deep results on 3-manifold topology and are also related to the geometrization conjecture for 3-manifolds. An important tool is the universal circle associated to a foliation with hyperbolic leaves (the generic case). Using this tool, the PI has recently proved that in the case of R-covered foliations, then either the manifold is toroidal or there is a pseudo-Anosov flow transverse to the foliation. Such a flow has excellent dynamical properties and this gives a strong relationship with the topology of the manifold. One objective of this project is to analyze the universal circle construction for various families of foliations and to search for transverse pseudo-Anosov flows. A second goal is to study foliation and transverse flows as dual objects and to understand the deep consequences of their joint dynamical structure. The project aims to study general pseudo-Anosov flows and the additional structure imposed by a transverse foliation. A third goal of the project is to understand geometric behavior of foliations and transverse flows in hyperbolic 3-manifolds - the generic case. The focus will be on the large scale geometric behavior in the universal cover. An important question to be analyzed is whether the pseudo-Anosov flows transverse to foliations are quasigeodesic - that is, whether they measure distances well. This property has been proved by the PI (and Lee Mosher) in the case of a flow transverse to a Reebless finite depth foliation. One objective of this study is to use the quasigeodesic property for flows to derive information about the asymptotic geometric behavior of the foliation transverse to the flow. This is specially promising in the case of general finite depth foliations.A 3-manifold is an object that locally has 3-dimensions, like 3-dimensional Euclidean space. A 2-dimensional foliation of a3-manifold is a decomposition of the manifold into 2-dimensional objects, much like the pages of a book. Good foliations exist in large classes of 3-manifolds and they yield very useful information about the manifold. The goal of the project is to understand the relationship of the foliation with geometric and topological structures of the manifold. Geometry measures distance and roughly topology measures the structures of "holes" in the manifold. It is best to look at the universal cover of the manifold: for example the universal cover of a cylinder is obtained by unrolling it into an infinite carpet. Universal covers are unbounded objects but they carry a lot of information about the manifold, contained in the "folding" allowed to recover the manifold. One important goal of the project is to analyse the foliation in the universal cover. Of particular interest is the large scale properties of the "leaves" of the foliation. We will concentrate in the case where the manifold is hyperbolic -this is the generic situation. Finally there is a dual object to a foliation which is a transverse flow to the leaves of the foliation. The flow and foliation jointly produce a rich dynamical structure in the manifold with many deep consequences. There are many cases when there is a "tightest" amongst all possible transverse flows. The tightest flow should give information about the structure of the foliation and connections with the manifold. The tightest flow is called a pseudo-Anosov flow.