By Riccardo Benedetti, Francesco Bonsante

The authors strengthen a canonical Wick rotation-rescaling concept in three-dimensional gravity. This contains: a simultaneous type: this indicates how maximal globally hyperbolic house occasions of arbitrary consistent curvature, which admit an entire Cauchy floor and canonical cosmological time, in addition to advanced projective buildings on arbitrary surfaces, are all various materializations of 'more primary' encoding constructions; Canonical geometric correlations: this exhibits how house occasions of other curvature, that proportion a similar encoding constitution, are relating to one another by way of canonical rescalings, and the way they are often remodeled by means of canonical Wick rotations in hyperbolic 3-manifolds, that hold definitely the right asymptotic projective constitution. either Wick rotations and rescalings act alongside the canonical cosmological time and feature common rescaling features. those correlations are functorial with recognize to isomorphisms of the respective geometric different types

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Then dev(p, ·) parameterizes in arc-length the geodesic ray of H3 with end-points ρ(D(p)) ∈ P and D(p) (see Fig. 4). 5. COMPLEX PROJECTIVE STRUCTURES ON SURFACES 33 ρ(D(p)) D(p) Figure 4. The construction of the H-hull. 3. [41] dev is a C1,1 developing map for a hyperbolic structure on S × (0, +∞). Moreover it extends to a map 3 dev : S˜ × (0, +∞] → H such that dev|S×{+∞} is a developing map for the complex projective structure on ˜ S. We call such a hyperbolic structure the H-hull of S and denote it by H(S).

In fact, ﬁx a point p0 ∈ L and consider a geodesic arc c transverse to the leaf l0 through p0 . There exists a neighbourhood K of p such that if a geodesic li meets K then it cuts c. Orient c arbitrarily and orient any geodesic li cutting c in such a way that respective positive tangent vectors at the intersection point form a positive base. Now for x ∈ L ∩ K deﬁne v(x) as the unitary positive tangent vector of the leaf through x at x. The following lemma ensures that v is a 1-Lipschitz vector ﬁeld on L ∩ K (see [29] for a proof).

Conversely, by a classical result of Choquet-Bruhat and Geroch [27], given a scalar product g and a symmetric bilinear form b on S satisfying the GaussCodazzi equation, there exists a unique (up to isometries) maximal globally hyperbolic Lorentzian structure of constant curvature κ on S × R, such that - S × {0} is a Cauchy surface; - The ﬁrst and the second fundamental form of S × {0} are respectively g and b. 2. Let us make precise what maximal means in this context. A constant curvature globally hyperbolic spacetime M is said maximal if every isometric embedding of M into a constant curvature spacetime M sending a Cauchy surface of M onto a Cauchy surface of M is an isometry.