Deformations of 3-dimensional cone-manifold structures

三维锥形流形是一种具有奇异几何结构的三维流形。更准确地说，它带有一个长度度量，它是由一个常截面曲率的黎曼度量诱导的分段测地图的补充。在与奇异轨迹的边缘相交的圆盘上，度规具有孤立的圆锥奇点。每条奇边都有一个圆锥角，它是一个正实数。这一概念产生于三维奥氏空间的几何化，它可以看作是几何奥氏空间概念的自然推广。由W.瑟斯顿于1982年提出并由M.Boileau、B.Leeb和J.Porti在其一般形式下证明的orbillold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：在图的情况下，这些结果能推广到圆锥角pi以外吗？全球刚性定理是否可用？在欧几里得的案例中能说些什么呢？

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?