Mathematical Sciences: Hyperbolic Geometry and Rigidity in Three Dimensions

数学科学：双曲几何和三维刚性

• 批准号: 9626233
• 负责人: Yair Minsky
• 金额: $ 6.81万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626233&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Hyperbolic; Geometry; Rigidity

9626233 Minsky Minsky will investigate a collection of problems centered on the basic classification conjectures in the field of hyperbolic 3-manifolds and Kleinian groups, and their connections to holomorphic dynamics. A prominent open question in this field is Thurston's Ending Lamination Conjecture, which states that a hyperbolic 3-manifold is uniquely determined by its topological type and a list of invariants that describe the asymptotic geometry of its ends. Consequences of this conjecture include a rigidity theorem for Kleinian group actions on the sphere, directly analogous to outstanding rigidity conjectures in holomorphic dynamics, and a topological description of parameter spaces of isomorphic Kleinian groups. Minsky previously established these conjectures in special cases, and the techniques developed show promise for extension to the general case. A number of projects have grown out of this program, whose successful completion should contribute to the solution of the conjecture, as well as being of independent interest. These projects investigate topological characterizations of Kleinian group actions on the sphere, some phenomena associated with geometric limits of Kleinian groups (with R. Canary and J. Brock), and the large scale geometry and combinatorics of the Teichmueller space of a surface (with H. Masur). Minsky is also considering (with M. Lyubich) a new construction that associates to a rational map a 3-dimensional hyperbolic object analogous to the quotient 3-orbifold of a Kleinian group. This object renders more explicit the powerful analogies between Kleinian groups and other holomorphic dynamical systems. In the study of low-dimensional geometry, topology and dynamics, one witnesses the depth of interconnection between fields of mathematics. Henri Poincare, who studied both celestial dynamics and complex analysis (among many other things), observed in the 19th century that the conformal transformations of the Riema nn sphere -- some of the building blocks of complex analysis -- extend to act on the three-dimensional ball bounded by the sphere, and their action preserves a complete, homogeneous metric, the metric of Hyperbolic or Non-Euclidean Space. This opened the door to a beautiful theory relating topology, geometry, and complex dynamics, which was only seriously explored in the latter part of this century. The rigidity problem mentioned above is essentially the question of to what extent the topological, or combinatorial, properties of a system determine its geometric properties. This and other basic questions in the field can be phrased in geometric, topological or dynamical terms, and are still linked to some of the motivating questions about physical systems of which Poincare was aware a hundred years ago. 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 have significance in both pure and applied mathematics. Topology and geometry have already provided deep insights into such issues, and one hopes that further study of these interactions will continue to bear fruit. ***

9626233 Minsky Minsky将研究以双曲线3个manifolds和Kleinian群体领域的基本分类猜想的集合，及其与Holomorphic Dynamics的联系。 在该领域的一个突出的开放问题是瑟斯顿的终结层压猜想，该猜想指出，双曲线3个manifold由其拓扑类型和描述其末端渐近几何形状的不变性列表唯一决定。 该猜想的后果包括针对球体上克莱恩组作用的刚性定理，直接类似于霍明态动力学中出色的刚性猜想，以及同构克莱尼群体的参数空间的拓扑描述。 明斯基先前在特殊情况下建立了这些猜想，这些技术发展了向一般案例扩展的希望。 该计划的许多项目已经发展出来，其成功完成应有助于猜想的解决方案，并具有独立的兴趣。 这些项目调查了克莱琳小组在球体上的作用的拓扑特征，某些现象与克莱琳群体的几何极限相关（带有R. Canary和J. Brock），以及表面的Teichmueller空间的大规模几何形状和组合物（与H. Masur）。 Minsky还在考虑（与M. Lyubich）一起使用新结构，该结构将其与合理地图相关联的3维双曲对象类似于Kleinian组的商3-孔。 该对象更加明确地说明了克莱恩组与其他全态动力学系统之间的强大类比。 在对低维几何形状，拓扑和动力学的研究中，有人见证了数学领域之间互连的深度。 研究了天体动力学和复杂分析（除其他方面），在19世纪观察到，riema nn领域的保形转换（复杂分析的某些基础）扩展到面向球体的三维球，及其动作保持了一个完整的，同质性的空间，而不是一个同质的空间。 这为与拓扑，几何学和复杂动力学有关的美丽理论打开了大门，该理论只在本世纪后期才进行了认真的探索。 上面提到的僵化问题本质上是系统的拓扑或组合特性在多大程度上决定其几何特性。 该领域中的这个和其他基本问题可以用几何，拓扑或动态术语来表达，并且仍然与一百年前Poincare意识到的物理系统的一些激励性问题有关。 诸如系统分类，绘制稳定性和不稳定性区域，系统家族家族的变形和分叉等问题，以及诸如Ergodicity之类的概率特性，都在纯数学和应用数学中都具有重要意义。 拓扑和几何形状已经为这些问题提供了深刻的见解，人们希望对这些相互作用的进一步研究将继续结出果实。 ***