Geometric cone surfaces and (2+1)-gravity coupled to particles

We introduce the (2 + 1)-spacetimes with compact space of genus g greater than or equal to 0 and r gravitating particles which arise by three kinds of construction called: (a) the Minkowskian suspension of flat or hyperbolic cone surfaces; (b) the distinguished deformation of hyperbolic suspensions; (c) the 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 UxR(6g-6+2r), where U is a non empty open set of the Teichmuller space T-g(r). By patchworking of suspensions one can produce examples of spacetimes which are not distinguished deformations of any hyperbolic suspensions, although they have the same topology and same masses; in fact, we will guess that they belong to different connected components of the parameter space.