(2+1)-Einstein spacetimes of finite type

The aim of this survey is to give an overview on the geometry of Einstein maximal globally hyperbolic 2+1 spacetimes of arbitrary curvature, conatining a complete Cauchy surface of finite type. In particular a specialization to the finite type case of the canonicla Wick rotation-rescaling theory, previously developed by the authors, is provided. This includes, for arbitrary curvatures, parameterizations in terms of suitable measured geodesic laminations on open hyperbolic surfaces of finite type. The same geometric objects also parameterize complex projective structures on the surfaces. The coincidence of such parameter space is explained by means of geometric correlations between spacetimes of different curvatures and projective surfaces realized via canonical WR-rescaling along the cosmological times. We also specialize on AdS case mostly referring to recent results achieved by other authors. In particular we describe maximal causal extensions of AdS globally hyperbolic spacetimes and an AdS approach to the theory of earthquakes for hyperbolic surfaces of finite type. A general earthquake theorem is proved for the so called enhanced Teichmuller space. The case of spacetimes with conical timelike singularities is also treated.