Mathematical Sciences: Hyperbolic Geometry and Rigidity in Three Dimensions

9626233 Minsky将研究以双曲3-流形和Kleinian群的基本分类猜想为中心的一系列问题，以及它们与全纯动力学的联系。该领域中一个突出的开放问题是Thurston的端点分层猜想，该猜想指出双曲3流形是唯一由其拓扑类型和描述其端点的渐近几何的不变量列表决定的。该猜想的结果包括球上Kleinian群作用的刚性定理，直接类似于全纯动力学中突出的刚性猜想，以及同构Kleinian群参数空间的拓扑描述。明斯基以前在特殊情况下建立了这些猜想，而所开发的技术显示出扩展到一般情况的希望。许多项目已经从这个计划中发展出来，它们的成功完成应该有助于猜想的解决，以及具有独立的兴趣。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（与R. Canary和J. Brock合作），以及表面的Teichmueller空间的大尺度几何和组合学（与H. Masur合作）。Minsky也在考虑（与M. Lyubich）一种新的结构，它将一个类似于Kleinian群的商3-轨道的三维双曲对象与有理映射联系起来。这一对象使克莱因群和其他全纯动力系统之间的强大类比更加明确。在低维几何、拓扑学和动力学的研究中，人们可以看到数学领域之间的深度联系。亨利·庞加莱（Henri Poincare）研究了天体动力学和复杂分析（以及许多其他事物），他在19世纪观察到，黎玛球的共形变换——复杂分析的一些组成部分——扩展到以球体为界的三维球上，它们的作用保持了一个完整的、齐次的度规，即双曲或非欧几里德空间的度规。这打开了一扇门，通向一个与拓扑学、几何学和复杂动力学有关的美丽理论，直到本世纪后半叶才得到认真的探索。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，并且仍然与庞加莱一百年前就意识到的关于物理系统的一些激励问题有关。诸如系统的分类，绘制稳定和不稳定区域，系统族的变形和分叉，以及概率性质（如遍历性）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。＊＊＊

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. ***