Deformations of 3-dimensional cone-manifold structures

• Applicant: Professor Dr. Hartmut Weiß
• Subject Area: Mathematics
• Term: from 2003 to 2006
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5407518

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?

• DFG Programme: Priority Programmes
• Subproject of: SPP 1154: Global Differential Geometry