We develop a canonical Wick rotation-rescaling theory in 3-dimensional gravity. This includes (a) A simultaneous classification: this shows how maximal globally hyperbolic spacetimes of arbitrary constant curvature, which admit a complete Cauchy surface and canonical cosmological time, as well as complex projective structures on arbitrary surfaces, are all different materializations of "more fundamental" encoding structures.(b) Canonical geometric correlations: this shows how spacetimes of different curvature, that share a same encoding structure, are related to each other by canonical rescalings, and how they can be transformed by canonical Wick rotations in hyperbolic 3-manifolds, that carry the appropriate asymptotic projective structure. 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.This theory applies in particular to spacetimes with compact Cauchy surfaces. By the Mess/Scannell classification, for every fixed genus g ≥ 2 of a Cauchy surface S, and for any fixed value of the curvature, these spacetimes are parametrized by pairs (F, λ) ∈ T g × ML g , where T g is the Teichmüller space of hyperbolic structures on S, λ is a measured geodesic lamination on F . On the other hand, T g × ML g is also Thurston's parameter space of complex projective structures on S. The Wick rotation-rescaling theory provides, in particular, a transparent geometric explanation of this remarkable coincidence of parameter spaces, and contains a wide generalization of Mess/Scannell classification to the case of non-compact Cauchy surfaces. These general spacetimes of constant curvature are eventually encoded by a kind of measured geodesic laminations λ defined on some straight convex sets H in H 2 , possibly in invariant way for the proper action of some discrete subgroup of P SL(2, R). We specifically analyze the remarkable subsectors of the theory made by ML(H 2 )-spacetimes (H = H 2 ), and by QDspacetimes (associated to H consisting of one geodesic line) that are generated by quadratic differentials on Riemann surfaces. In particular, these incorporate the spacetimes with compact Cauchy surface of genus g ≥ 2, and of genus g = 1 respectively. We analyze broken T -symmetry of AdS ML(H 2 )-spacetimes and its relationship with earthquake theory, beyond the case of compact Cauchy surface.Wick rotation-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. This also provides further "classical amplitudes" of these interactions, beyond the volume of the hyperbolic convex cores. Vol(.) + iCS(.) Vol(.) and CS(.) being respectively the volume and the Chern-Simons invariant of both hyperbolic 3-manifolds of finite volume (compact and cusped ones) and of principal flat P SL(2, C)-bundles on compact closed manifolds (see [46], []).Volume ri...

Abstract. We show that any element of the universal Teichmüller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in AdS n+1 , any subset E of the boundary at infinity which is the boundary at infinity of a space-like hypersurface bounds a maximal space-like hypersurface. In AdS 3 , if E is the graph of a quasi-symmetric homeomorphism, then this maximal surface is unique, and it has negative sectional curvature. As a by-product, we find a simple characterization of quasi-symmetric homeomorphisms of the circle in terms of 3-dimensional projective geometry.

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Abstract. We prove two related results. The first is an "Earthquake Theorem" for closed hyperbolic surfaces with cone singularities where the total angle is less than π: any two such metrics in are connected by a unique left earthquake. The second result is that the space of "globally hyperbolic" AdS manifolds with "particles" -cone singularities (of given angle) along time-like lines -is parametrized by the product of two copies of the Teichmüller space with some marked points (corresponding to the cone singularities). The two statements are proved together.

Abstract. Let T be Teichmüller space of a closed surface of genus at least 2. For any point c ∈ T , we describe an action of the circle on T × T , which limits to the earthquake flow when one of the parameters goes to a measured lamination in the Thurston boundary of T . This circle action shares some of the main properties of the earthquake flow, for instance it satisfies an extension of Thurston's Earthquake Theorem and it has a complex extension which is analogous and limits to complex earthquakes. Moreover, a related circle action on T × T extends to the product of two copies of the universal Teichmüller space.