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. ***

小行星9626233 明斯基将调查集中在双曲3-流形和Kleinian群领域的基本分类的问题，以及它们与全纯动力学的联系。 这一领域的一个突出的开放问题是瑟斯顿的终结层压猜想，它指出双曲三维流形是由它的拓扑类型和描述其端点的渐近几何的不变量列表唯一确定的。 这一猜想的后果包括一个刚性定理的克莱因集团行动的领域，直接类似于突出的刚性austratures在全纯动力学，和拓扑描述参数空间的同构克莱因集团。 明斯基以前在特殊情况下建立了这些模型，开发的技术显示出扩展到一般情况的希望。 一些项目已经成长出这个计划，其成功完成应有助于解决的猜想，以及独立的利益。 这些项目研究了球面上Kleinian群作用的拓扑特征，与Kleinian群的几何极限相关的一些现象（R。Canary和J. Brock），以及曲面的Teichmueller空间的大尺度几何和组合学（H. Masur）。 明斯基也在考虑（与M。Lyubich）一种新的构造，它将一个类似于Kleinian群的商3-orbifold的三维双曲对象与一个有理映射联系起来。 这一目标使克莱因群和其他全纯动力系统之间的强有力的类比更加明确。 在低维几何、拓扑和动力学的研究中， 人们可以看到数学各领域之间相互联系的深度。 研究天体动力学和复分析的亨利·庞加莱（除其他事项外），观察到在19世纪，共形变换的Riema nn球-一些积木的复杂分析-延伸到行动的三维球有界的领域，他们的行动保持一个完整的，齐次度量，双曲或非欧空间的度量。 这打开了一扇大门，一个美丽的理论有关拓扑，几何和复杂的动力学，这是只有认真探讨在后半部分的这个世纪。 上面提到的刚性问题本质上是一个系统的拓扑或组合性质在多大程度上决定其几何性质的问题。 这个问题和该领域的其他基本问题可以用几何、拓扑或动力学术语来表述，并且仍然与庞加莱100年前就意识到的物理系统的一些激发性问题有关。 诸如系统分类、绘制稳定和不稳定区域、系统族的变形和分叉以及遍历性等概率性质等问题，在纯数学和应用数学中都具有重要意义。 拓扑学和几何学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取得成果。 ***