Hyperbolic Geometry

9803445 Bonahon这个项目的主要主题是研究三维流形上双曲结构的刚度问题。特别地，它的目的是证明三维双曲凸壳的几何完全由其边界上的部分数据决定。该项目还涉及双曲和欧几里得锥流形上的几个相关问题，以及称为测地线分层的曲面上曲线的推广。这些问题本身很有趣，但它们也是一个实验室，用于开发和测试新的技术工具，以应用于更广泛的低维几何问题。预计这些问题的解决将加深我们对低维双曲几何的理解，并将其应用于数学的其他分支。该项目将研究三维流形上的几个几何问题，即三维空间上的几何问题。二十年前，低维拓扑学领域的革命性发展表明，双曲(非欧几里德)几何是分析三维流形拓扑的一个非常强大的工具。这一突破还导致低维拓扑学与其他数学分支之间建立了意想不到的联系，如微分几何、复杂分析和动力系统。***

9803445 Bonahon The leading theme of this project is the study of rigidity problems for hyperbolic structures on 3-manifolds. In particular, it aims at proving that the geometry of a 3-dimensional hyperbolic convex hull is completely determined by partial data on the geometry of its boundary. The project also involves several connected problems on hyperbolic and euclidean cone manifolds, and on a generalization of curves on a surface called geodesic laminations. These problems are of interest by themselves, but they are also a laboratory for developing and testing new technical tools for application to a wider range of problems in low-dimensional geometry. It is anticipated that a solution to these problems will deepen our understanding of hyperbolic geometry in low dimensions and have applications to other branches of mathematics as well. The project will study several geometric problems on 3-manifolds, namely on spaces of dimension 3. Twenty years ago, revolutionary developments in the field of low-dimensional topology showed that hyperbolic (non-euclidean) geometry was a very powerful tool to analyze the topology of 3-manifolds. This breakthrough also led to the establishment of unexpected connections between low-dimensional topology and other branches of mathematics, such as differential geometry, complex analysis and dynamical systems. ***