Paper detail

Global rigidity of 3-dimensional cone-manifolds

We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles $\leq π$ (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles $\leq π$, possibly with boundary consisting of totally geodesic hyperbolic turnovers. To that end we first generalize our local rigidity result to the setting of hyperbolic cone-3-manifolds of finite volume as above. We then use the geometric techniques developed by Boileau, Leeb and Porti to deform the cone-manifold structure to a complete non-singular or a geometric orbifold structure, where global rigidity holds due to Mostow-Prasad rigidity in the hyperbolic case, resp. a result of de Rham in the spherical case. This strategy has already been implemented successfully by Kojima in the compact hyperbolic case if the singular locus is a link using Hodgson-Kerckhoff local rigidity.