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

摘要：无叶理是三维流形研究的基础和基础对象。它们在3流形拓扑上产生了深刻的结果，也与3流形的几何化猜想有关。一个重要的工具是与双曲叶（一般情况下）的叶相关联的通用圆。利用这个工具，PI最近证明了在r覆盖的叶状面情况下，流形要么是环面，要么存在一个伪ananosov流横向到叶状面。这种流动具有优异的动力学特性，这与流形的拓扑结构有很强的关系。本项目的目的之一是分析各种叶族的普适圆构造，并寻找横向伪阿诺索夫流。第二个目标是研究叶理和横向流动作为双重对象，并了解其联合动力结构的深刻后果。该项目旨在研究一般的伪阿诺索夫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3流形中叶状和横向流动的几何行为-一般情况。重点将是大规模的几何行为在通用覆盖。要分析的一个重要问题是，横向到叶理的伪阿诺索夫流是否是准等距流——也就是说，它们是否能很好地测量距离。PI（和Lee Mosher）在无reeless有限深度叶理的横向流动中证明了这一性质。本研究的目的之一是利用流动的拟等地线性质来推导有关叶理横向于流动的渐近几何行为的信息。这在一般有限深度叶理的情况下特别有希望。三维流形是一个局部具有三维空间的物体，就像三维欧几里得空间。三维流形的二维叶化是将流形分解为二维对象，很像一本书的页面。好的叶形存在于大量的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叶理与流形的几何和拓扑结构之间的关系。几何测量距离，大致拓扑测量流形中“孔”的结构。最好看一下歧管的通用盖：例如，将圆柱体展开成无限地毯即可获得圆柱体的通用盖。万能盖是无界的物体，但它们携带着大量关于流形的信息，包含在“折叠”中允许恢复流形。该项目的一个重要目标是分析通用覆盖层中的叶理。特别令人感兴趣的是“叶子”的大尺度特性。我们将集中在流形是双曲的情况下，这是一般情况。最后，叶理有一个双重对象，它是叶理叶片的横向流动。流动和叶理作用共同在流形中产生丰富的动力结构，并产生许多深刻的后果。在许多情况下，在所有可能的横向流动中存在“最紧”。最紧密的流动应提供有关叶理结构和与歧管连接的信息。最紧的流称为伪阿诺索夫流。

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.