Deformations of 3-dimensional cone-manifold structures

三维锥流形是具有奇异几何结构的三维流形。更准确地说，它带有一个长度度规，它是由恒定截面曲率的黎曼度规诱导的分段测地线图的补充。在与奇异轨迹边缘横向的圆盘上，度规有一个孤立的锥形奇异点。每条奇异边都有一个圆锥角，它是一个正实数。这个概念产生于三维轨道的几何化，它可以被认为是几何轨道概念的自然推广。由W. Thurston于1982年宣布，最近由M. Boileau， B. Leeb和J. Porti以其一般形式证明的轨道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：这些结果是否可以推广到图形情况下的锥角pi以外？全局刚性定理可用吗？在欧几里得的情况下我们能说些什么呢？

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?