Laminations, Foliations and Flows in 3-Manifolds

3 流形中的叠层、叶状结​​构和流动

• 批准号: 0305313
• 负责人: Sergio Fenley
• 金额: $ 28.97万
• 依托单位: Florida State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0305313&HistoricalAwards=false
• 关键词: Laminations; Foliations; Flows; Manifolds

Abstract."Laminations, foliations and flows in 3-manifolds" Essential laminations and Reebless foliations are a basic and fundamental tool in the study of 3-manifolds. Their use yields deep results concerning 3-manifold topologyand they are also related to the geometrization conjecturefor 3-manifolds. One goal is to analyse the existencequestion for laminations in 3-manifolds, particulary forhyperbolic 3-manifolds. The PI has recently shown thereare infinitely many hyperbolic 3-manifolds which do not admit essential laminations. A natural question is how common are essential laminations. The project will investigate Dehn surgery on torus bundles over the circleand general surface bundles and also special types oflaminations/foliations. The project will also considermore general structures, such as loosesse laminations andwhich properties they have. Another part of the project isto understand the geometric behavior of foliations andtransverse pseudo-Anosov flows in hyperbolic 3-manifolds - which is a very large class of manifolds. The focus will be on the large scale geometric behavior in the universal cover, which is fundamental for such manifolds. An important question is whether such flows are quasigeodesic - this is true in certain cases by previous work of the PI and Lee Mosher. One goal is to use the quasigeodesic property for the flows to derive information about the asymptotic behavior of the foliation transverse to the flow. This is speciallypromising in the case of general finite depth foliations.Another goal is to connect these properties with the universalcircle of the foliation - establishing a link between the global structure of the foliation and the global geometry of theuniversal cover. Another project is to study properties oflimit sets of leaves of foliations in hyperbolic manifolds. The project aims to better understand 3-manifolds.3-manifolds are widely used in the sciences: for exampleknots and their properties are very relevant to DNAresearch. 3-manifolds are also used to get mappings of thebrain and in other visualization techniques. Understanding3-manifolds can lead to progres in these other areas.The project focuses on laminations and foliations, whichare an essential and currently extremely dynamic area inlow dimensional topology. The proposed projects will have an effect on some main research directions in the area.One important goal is to attract students and postdoctoral researchers in this exciting area.

抽象的。“3-流形中的层合、叶理和流动”基本层合和Reebless叶理是研究3-流形的基本工具。它们的应用产生了关于3-流形拓扑的深刻结果，并且它们也与3-流形的几何化关系有关。目的之一是分析三维流形，特别是双曲三维流形中的层的存在性问题。PI最近证明了有无穷多个双曲三维流形不允许本质叠层。一个自然的问题是，基本的叠层有多普遍。该项目将研究Dehn手术对环面束在圆和一般表面束和特殊类型的flaminations/foliations。该项目还将测试更一般的结构，如松散的层压结构和它们的性能。该项目的另一部分是了解双曲三维流形中的叶理和横向伪Anosov流的几何行为-这是一个非常大的流形类别。重点将放在大规模的几何行为的普遍覆盖，这是基本的，这样的流形。一个重要的问题是这样的流动是否是准短程线的--PI和Lee Mosher以前的工作在某些情况下是正确的。一个目标是使用的准短程线的流动，以获得有关的渐近行为的叶理横向流动的信息。另一个目标是将这些性质与叶理的泛圆联系起来--在叶理的全局结构和泛覆盖的全局几何之间建立联系。另一个项目是研究双曲流形中叶理叶的极限集的性质。该项目旨在更好地理解3-流形。3-流形在科学中有着广泛的应用：例如，结及其性质与DNA研究非常相关。3-流形也被用来获得大脑的映射和其他可视化技术。了解三维流形可以导致在这些其他领域的进展。该项目侧重于层积和叶理，这是一个基本的和目前非常动态的领域在低维拓扑。这些项目将对该领域的一些主要研究方向产生影响，其中一个重要目标是吸引学生和博士后研究人员进入这一令人兴奋的领域。