Structure of hyperbolic 3-manifolds

双曲3流形的结构

• 批准号: 0504019
• 负责人: Yair Minsky
• 金额: --
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504019&HistoricalAwards=false
• 关键词: Structure; hyperbolic; manifolds

The field of hyperbolic 3-manifolds and Kleinian groups has seen considerable progress in the last three years, with the resolution of most of the main motivational conjectures, such as Tameness, the Ending Lamination Conjecture, and the Density conjecture. These advances confirm much about our expected picture of hyperbolic 3-manifolds and their deformation spaces, and place the field in a moment of transition and opportunity. the techniques introduced in the proofs have much potential for further applications. Minsky will focus on deepening our understanding of the structure and deformation theory of hyperbolic 3-manifold, applying in particular the tools that have come out of his contribution to the solution of the Ending Lamination Conjecture. The models and estimates provided by these tools should provide an approach to a number of open questions, notably that of local connectivity of limit sets, geometric description of closed manifolds from the Heegaard decompositions, and uniformity theorems for deformation spaces (some of this work will be in collaboration with Brock, Bromberg and Canary). Another area of applications (jointly with Brock and Masur) involves the structure of geodesics in the Teichmuller space endowed with its Weil-Petersson metric. These have up till now resisted analysis but appear to be quite intimately connected to the geometry of 3-manifolds.The interactions between geometry, topology and dynamics have been a beautiful and powerful feature of mathematics and physics for more than a hundred years. Dynamics is the study of time-evolution of mathematical or physical systems, whereas geometry and topology involve "static" objects such as surfaces or higher-dimensional analogues, often the background for a dynamical process. Henri Poincare already knew that the standard round sphere, the setting of classical analysis and geometry, functioned also as a "horizon at infinity" for an exotic non-Euclidean geometry that we now call Hyperbolic space. Dynamical properties of transformations of the sphere translate to geometric properties of rigid motions of this space, and give rise to families of symmetric tilings whose structure we can study by geometric and topological methods. The complexity of these systems can constrain them so much that a combinatorial (or topological) description suffices to determine them uniquely, and this is what we call rigidity. This phenomenon occurs in many guises throughout geometry and dynamics, and is relevant to issues such as classification of systems, mapping out regions of stability and instability, deformation and bifurcation of families of systems, and probabilistic properties such as ergodicity, all of which havesignificance in both pure and applied mathematics. The particular aspects studied in this project are typical in some ways and special in others. They focus on the intricate relationships between geometry in two and three dimensions, and also on the ways in which topology, particularly of systems of curves within surfaces, determines geometry. There is also a strong emphasis on studying families of geometric structures on surfaces and three-dimensional manifolds, which are closely analogous to other families of dynamical systems.

双曲3流形和克莱因群领域在过去三年中取得了长足的进步，解决了大多数主要动机猜想，例如驯服、结束层合猜想和密度猜想。 这些进展在很大程度上证实了我们对双曲 3 流形及其变形空间的预期图像，并将该领域置于转型和机遇的时刻。 证明中介绍的技术具有进一步应用的巨大潜力。 明斯基将专注于加深我们对双曲 3 流形的结构和变形理论的理解，特别是应用他为解决终结叠层猜想所做的贡献所得出的工具。这些工具提供的模型和估计应该为许多开放性问题提供一种方法，特别是极限集的局部连通性、Heegaard 分解的闭流形的几何描述以及变形空间的均匀性定理（其中一些工作将与 Brock、Bromberg 和 Canary 合作）。另一个应用领域（与布洛克和马苏尔共同）涉及具有 Weil-Petersson 度量的 Teichmuller 空间中的测地线结构。到目前为止，这些都还无法进行分析，但似乎与三流形的几何关系非常密切。一百多年来，几何、拓扑和动力学之间的相互作用一直是数学和物理学的一个美丽而强大的特征。动力学是对数学或物理系统随时间演化的研究，而几何和拓扑涉及“静态”对象，例如表面或高维类似物，通常是动态过程的背景。 亨利·庞加莱已经知道，标准的球体、经典分析和几何的设置，也充当了我们现在称为双曲空间的奇异非欧几里得几何的“无限地平线”。球体变换的动力学特性转化为该空间刚性运动的几何特性，并产生对称平铺系列，我们可以通过几何和拓扑方法研究其结构。 这些系统的复杂性可以极大地限制它们，以至于组合（或拓扑）描述足以唯一地确定它们，这就是我们所说的刚性。这种现象以多种形式出现在几何学和动力学中，并且与诸如系统分类、绘制稳定和不稳定区域、系统族的变形和分岔以及遍历性等概率性质等问题相关，所有这些都在数学中具有重要意义。纯数学和应用数学。 该项目研究的特定方面在某些方面是典型的，在其他方面是特殊的。他们关注二维和三维几何之间复杂的关系，以及拓扑（特别是曲面内的曲线系统）决定几何的方式。还非常重视研究表面和三维流形上的几何结构族，它们与动力系统的其他族非常相似。