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.

在过去的三年中，双曲线3个manifords和克莱恩群体的领域取得了很大的进步，分辨出大多数主要的动机猜想，例如驯服，结束层压构想和密度猜想。 这些进步证实了我们预期的双曲线3型物体及其变形空间的预期图片，并将该领域放在过渡和机会的时刻。 证明中引入的技术具有更多的进一步应用潜力。 Minsky将专注于加深我们对双曲线3个序列的结构和变形理论的理解，特别是应用了他对最终层压猜想的解决方案的贡献的工具。这些工具提供的模型和估计值应为许多开放问题提供一种方法，尤其是限制集的局部连接性，Heegaard分解的封闭歧管的几何描述以及变形空间的均匀定理（这项工作中的一些将与Brock，Brock，Bromberg和Canary合作）。应用的另一个领域（与Brock和Masur共同）涉及带有其Weil-Petersson指标的Teichmuller空间中的大地测量结构。到现在为止，这些都具有抗拒分析，但似乎与3个manifold的几何形状密切相关。几百多年来，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美丽而有力的特征。动力学是对数学或物理系统的时间进化的研究，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。 Henri Poincare已经知道，标准的圆形球体（经典分析和几何形状的设置）也作为异国情调的非欧盟几何形状的“无穷大的地平线”起作用，我们现在称之为双曲线空间。球体的转化的动力学特性转化为该空间刚性运动的几何特性，并引起了对称瓷砖的家族，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限制它们很大，以至于组合（或拓扑）描述足以确定它们，这就是我们所说的刚性。这种现象发生在整个几何和动态的许多伪造中，并且与系统分类，绘制稳定性和不稳定性区域，系统家族家族的变形和分叉以及概率特性（如诸如诸如纯粹的数学和应用数学的概率）。 该项目中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。他们专注于在两个维度和三个维度上的几何形状之间的复杂关系，也集中在拓扑（尤其是表面内曲线系统的拓扑结构）上决定几何形状的方式。在表面和三维歧管上研究几何结构家族也有很大的重视，这些家族与其他动态系统的家族非常相似。