3 Dimensional Geometry, Heegaard Splittings and Rank of the Fundamental Group

3 维几何、Heegaard 分裂和基本群的秩

• 批准号: 0939587
• 负责人: Juan Souto
• 金额: $ 8.56万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-10-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0939587&HistoricalAwards=false
• 关键词: Dimensional; Geometry; Heegaard; Splittings; Rank

After the work of Perelman, most 3-manifolds are known to admit a hyperbolic metric, i.e. a metric of constant negative curvature. Unfortunately, with few exceptions, no further properties of this metric are known. In this project the P.I. will study how the topology of the 3-manifold and the properties of the hyperbolic geometry are related to each other. A first goal is to obtain explicite estimates of geometric data in terms of topological and combinatorial information such as the rank of the fundamental group of the Heegaard genus of the manifold. The second goal is to use these geometric information to reconstruct, under suitable assumptions, the hyperbolic metric itself.A 3-manifold is a mathematical object of fundamental interest. For example, the space we live in is a 3-manifold. A recent trend in the study of 3-manifolds is to encode as much of their fine, infinitely complicated, geometric structure in finite combinatorial models. A certain amount of information is lost in the process. The goal of the project is to quantify how much information actually gets lost. Obtaining concrete a priori estimates is then crucial. For example, they open the door to predictions in terms of finite models of phenomena occurig in 3-manifolds. Surprisingly, it seems very likely that in many situations sufficiently precise a priori estimates will allow to recover all the geometric information. Concrete estimates will make possible to have accurate computer simulations.

在佩雷尔曼（Perelman）的工作之后，已知大多数3个manifolds都接受双曲线度量，即恒定负曲率的度量。 不幸的是，除少数例外，该指标的进一步属性。在这个项目中将研究3个manifold的拓扑以及双曲几何形状的特性如何相互关联。第一个目标是从拓扑和组合信息（例如歧管的Heegaard属的基本组等级）中获得几何数据的显式估计。第二个目标是使用这些几何信息在适当的假设下重建双曲线指标本身。3-manifold是基本兴趣的数学对象。例如，我们居住的空间是一个3个manifold。 3个manifolds研究的最新趋势是在有限组合模型中编码它们的精细，无限复杂的几何结构。在此过程中丢失了一定数量的信息。该项目的目的是量化实际丢失的信息。然后，获得具体的先验估计值至关重要。例如，他们从3个策略中发生的有限现象模型中为预测打开了大门。令人惊讶的是，在许多情况下，先验估计值允许恢复所有几何信息似乎很有可能。具体的估计将使进行准确的计算机模拟。