Canonical Wick rotations in 3-dimensional gravity

We develop a canonical Wick rotation-rescaling theory in 3-dimensional gravity''. This includes: (a) A simultaneous classification that shows how generic maximal globally hyperbolic spacetimes of constant curvature, which admit a complete Cauchy surface (in particular a compact one), as well as complex projective structures on arbitrary surfaces, are all encoded by pairs (H,L), H being a straight convex sets’’ in the hyperbolic plane, and L a measured geodesic laminations'' suitably defined on H. (b) Canonical geometric correlations: spacetimes of different curvature, that share a same encoding pair (H,L), are related to each other by canonical rescaling’‘; they can be transformed by canonical Wick rotations'' in hyperbolic 3-manifolds, that carry asymptotically the corresponding projective structures. Both Wick rotations and rescalings act along the "canonical cosmological time" and have universal rescaling functions’‘. These correlations are functorial with respect to isomorphisms of the respective geometric categories. We analyze the behaviour along a ray of measured laminations, (broken) T-symmetry by spacetimes of negative curvature, the relationship with earthquake theory'', beyond the case of compact Cauchy surface. WR-rescaling does apply on the ends’’ of geometrically finite hyperbolic 3-manifolds, that hence realize concrete interactions of their globally hyperbolic ``ending spacetimes’’ of constant curvature. It also provides further “classical amplitudes” of these interactions, beyond the volume of the hyperbolic convex cores.