By Riccardo Benedetti, Francesco Bonsante

The authors boost a canonical Wick rotation-rescaling idea in three-d gravity. This comprises: a simultaneous class: this exhibits how maximal globally hyperbolic house instances of arbitrary consistent curvature, which admit an entire Cauchy floor and canonical cosmological time, in addition to complicated projective buildings on arbitrary surfaces, are all varied materializations of 'more basic' encoding buildings; Canonical geometric correlations: this exhibits how area instances of other curvature, that proportion a similar encoding constitution, are regarding one another by means of canonical rescalings, and the way they are often reworked via canonical Wick rotations in hyperbolic 3-manifolds, that hold the perfect asymptotic projective constitution. either Wick rotations and rescalings act alongside the canonical cosmological time and feature common rescaling capabilities. those correlations are functorial with appreciate to isomorphisms of the respective geometric different types

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Every T -level surface U(a), a ∈ (0, +∞), is a complete Cauchy surface of U. For every x ∈ U, there is a unique past-directed geodesic timelike segment γx that starts at x, is contained in U, has finite Lorentzian length equal to T (x). Proof : We only give a sketch of the proof of this theorem, referring to [7, 18] for a complete proof. A first very simple remark is that the cosmological time function T is finitevalued. In fact, since U is convex, the Lorentzian distance between two time-related points is realized by the geodesic segment between them.

The positive generator of ∂P (Id) (positive with respect to the orientation induced by P (Id)) is a positive combination of eL and eR . It easily follows that a positive basis of T ∂X−1 is given by (eR , eL ). 5. Complex projective structures on surfaces A complex projective structure on a oriented connected surface S is a 2 , P SL(2, C))-structure (respecting the orientation). (S∞ We will often refer to a parameterization of complex projective structures given in [41]. That work is a generalization (even in higher dimension) of a previous classification due to Thurston when the surface is assumed to be compact.

The group SL(2, R) = {A|q(A) = −1} is a Lorentzian sub-manifold of M2 (R), that is the restriction of η on it has signature (2, 1). 4. ANTI DE SITTER SPACE 27 preserves η. In particular, the restriction of η on SL(2, R) is a bi-invariant Lorentzian metric, that actually coincides with its Killing form (up to some multiplicative factor). For X, Y ∈ sl(2, R) we have the usual formula trXY = 2η(X, Y ) . ˆ −1 the pair (SL(2, R), η). Clearly X ˆ −1 is an orientable and timeWe denote by X orientable spacetime.

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