Mathematical Sciences: Geometry and Topology of Foliations and Flows in 3-Manifolds

点击翻译按钮获取中文摘要

9612317 Fenley Foliations are a fundamental tool in the study of the topology of 3-manifolds. A Reebless two dimensional foliation F in a closed 3-manifold M lifts to a foliation (F1) by topological planes in the universal cover. This project proposes to study the topological structure of F1, the geometric properties of F1, and how these relate to the topology and geometry of M. Of special interest is the study of the large scale geometry and asymptotic behavior (when M is hyperbolic) of leaves of F1. A second topic to be considered consists of flows in 3-manifolds and the large scale geometry of flow lines when lifted to the universal cover. The principal investigator has constructed many examples of metrically efficient (called quasigeodesic) pseudo-Anosov flows in hyperbolic 3-manifolds and now proposes to study the consequences of the quasigeodesic property for the geometry of the stable and unstable foliations associated to these flows and the finite depth foliations to which these quasigeodesic flows are almost transverse. A foliation is a decomposition of an object of dimension n (called a manifold) into a disjoint union of objects of dimension less than n (called the leaves of the foliation), much the same as a book (dimension 3) is the union of its pages (of dimension 2). In mathematics these objects are quite common; for instance, a flow without fixed orbits is a foliation of dimension 1 (the leaves are the flow lines and they have dimension 1). A general principle of manifold theory is the following: the smaller the dimension, the easier to understand. Hence foliations are very useful; understand the leaves and how they are put together to form the manifold and one partially understands the manifold. This project attempts to understand how the geometry of the leaves interacts with the geometry of the manifold; in the case of a book, the leaves sit flat in the manifold, but in general their geometric behavior is much more complex. The principal investigator hopes to describe this geometric behavior for a special class of manifolds called hyperbolic manifolds.