Geometric Cone Surfaces and (2+1) - Gravity coupled to Particles

We introduce the (2+1)-spacetimes with compact space of genus g ≥ 0 and r gravitating particles which arise by Minkowskian suspensions of flat or hyperbolic cone surfaces, by distinguished deformations of hyperbolic suspensions and by patchworking of suspensions. Similarly to the matter-free case, these spacetimes have nice properties with respect to the canonical Cosmological Time Function. When the values of the masses are sufficiently large and the cone points are suitably spaced, the distinguished deformations of hyperbolic suspensions determine a relevant open subset of the full parameter space; this open subset is homeomorphic to U × R, where U is a non empty open set of the Teichmüller Space T r g . By patchworking of suspensions one can produce examples of spacetimes which are not distinguished deformations of any hyperbolic suspensions, although they have the same masses; in fact, we will guess that they belong to different connected components of the parameter space.