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

在佩雷尔曼的工作之后，已知大多数3-流形都承认一个双曲度规，即一个常负曲率的度规。不幸的是，除了少数例外，这个度量的其他属性都不为人所知。在这个项目中，P.I.将研究3流形的拓扑结构和双曲几何的性质是如何相互关联的。第一个目标是根据拓扑和组合信息获得几何数据的显式估计，例如流形的Heegaard属的基本群的秩。第二个目标是在适当的假设下，使用这些几何信息来重建双曲度规本身。3流形是一个基本的数学对象。例如，我们生活的空间是一个三流形。最近研究3流形的一个趋势是在有限组合模型中尽可能多地编码它们精细的、无限复杂的几何结构。在这个过程中，一定数量的信息丢失了。该项目的目标是量化实际丢失的信息数量。获得具体的先验估计是至关重要的。例如，它们打开了用有限模型预测3流形现象的大门。令人惊讶的是，在许多情况下，似乎很可能足够精确的先验估计将允许恢复所有的几何信息。具体的估计将使精确的计算机模拟成为可能。

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.