Download Canonical Wick rotations in 3-dimensional gravity by Riccardo Benedetti, Francesco Bonsante PDF

By Riccardo Benedetti, Francesco Bonsante

The authors advance a canonical Wick rotation-rescaling conception in three-dimensional gravity. This comprises: a simultaneous class: this indicates how maximal globally hyperbolic area occasions 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 diverse materializations of 'more basic' encoding buildings; Canonical geometric correlations: this exhibits how area occasions of other curvature, that percentage a related encoding constitution, are with regards to one another through canonical rescalings, and the way they are often remodeled through canonical Wick rotations in hyperbolic 3-manifolds, that hold the proper 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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9. Notice that if k is an arc transverse to a lamination of H there exists a transverse piece-wise geodesic arc homotopic to k through a family of transverse arcs. Indeed there exists a finite subdivision of k in sub-arcs ki for i = 1, . . , n such that ki intersects a leaf in a point and a 2-stratum in a sub-arc. If pi−1 , pi are the end-points of ki it is easy to see that each ki is homotopic to the geodesic segment [pi−1 , pi ] through a family of transverse arcs. It follows that a transverse measure on a lamination of H is determined by the family of measures on transverse geodesic arcs.

We refer, for instance, to [56, 15, 55], and we assume that the reader is familiar with the usual concrete models of the hyperbolic plane H2 and space H3 . We just note that the hyperboloid model of Hn , embedded as a spacelike hypersurface in the Minkowski space Mn+1 , establishes an immediate relationship with Lorentzian geometry. The restriction to Hn of the natural projection of Mn+1 \ {0} onto the projective space, gives the Klein projective model of Hn . On the other hand, the Poincar´e disk (or half-space) model of H3 2 concretely shows its natural boundary at infinity S∞ = CP1 , and the identification + 3 of the isometry group Isom (H ) with the group P SL(2, C) of projective transformations of the Riemann sphere (H3 is oriented in such a way that the boundary orientation coincides with the complex one).

N converges to λ on B. Proof : It is clear that any arc k ⊂ B transverse to λ is transverse to λn for any n. So (1) is verified. On the other hand any arc k ⊂ B transverse to λ is a finite composition of transverse arcs k1 , . .

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