Deformations of 3-dimensional cone-manifold structures

3维锥流管结构的变形

• 批准号: 5407518
• 负责人: Professor Dr. Hartmut Weiß
• 金额: --
• 依托单位: Mathematisches Seminar
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2003
• 资助国家: 德国
• 起止时间: 2002-12-31 至 2005-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5407518?language=en
• 关键词: Deformations; dimensional; cone; manifold; structures

A 3-dimensional cone-manifold is a 3-manifold equipped with a singular geometric structure. More precisely, it carries a length metric, which is in the complement of a piecewise geodesic graph induced by a Riemannian metric of constant sectional curvature. On a disk transverse to an edge of the singular locus, the metric has an isolated conical singularity. One associates with each singular edge the cone-angle, which is a positive real number. This concept arises in the geometrization of 3-dimensional orbifolds, it can be considered as a natural generalization of the concept of geometric orbifold. The orbifold theorem, which was announced by W. Thurston in 1982 and recently proved by M. Boileau, B. Leeb and J. Porti in its general form, states that 3-manifolds with a certain kind of symmetry are geometrizable. ... I intend to study the following questions: Can these results be extended beyond cone-angle pi in the graph-case? Is a global rigidity theorem available? What can be said in the Euclidean case?

3维锥形流形是具有奇异几何结构的3维流形。更准确地说，它带有一个长度度量，它是由恒定截面曲率的黎曼度量导出的分段测地线图的补充。在横向于奇异轨迹边缘的圆盘上，度量具有孤立的圆锥形奇点。人们将每个奇异边缘与锥角相关联，锥角是正实数。这个概念出现在3维轨道折叠的几何化中，它可以被认为是几何轨道折叠概念的自然推广。轨道定理由 W. Thurston 于 1982 年提出，最近由 M. Boileau、B. Leeb 和 J. Porti 以其一般形式证明，该定理指出，具有某种对称性的 3 流形是可几何化的。 ...我打算研究以下问题：这些结果能否扩展到图形情况下的锥角 pi 之外？全局刚性定理可用吗？在欧几里得的例子中可以说什么呢？