Flat spacetimes with compact hyperbolic Cauchy surfaces

We study the flat (n+1)-spacetimes Y admitting a Cauchy surface diffeomorphic to a compact hyperbolic n-manifold M . Roughly speaking, we show how to construct a canonical future complete one, Yρ, among all such spacetimes sharing a same holonomy ρ. We study the geometry of Yρ in terms of its canonical cosmological time (CT). In particular we study the asymptotic behaviour of the level surfaces of the cosmological time. The present work generalizes the case n = 2 treated in [11] taking from [4] the emphasis on the fundamental role played by the canonical cosmological time. In particular Mess showed that if F is a closed surface of genus g ≥ 2 then the linear holonomies of the Lorentzian flat structures on R × F such that {0} × F is a spacelike surface are faithful and discrete representations of π1(F ) in SO (2, 1). Moreover he proved that every representation φ : π1(F ) → Iso(M3) whose linear part is faithful and discrete is a holonomy for some Lorentzian structure on R×F such that {0}×F is a spacelike surface. In particular he showed that there exists a unique maximal future complete convex domain of M3, called domain of dependence, which is φ(π1(F ))-invariant such that the quotient is a globally hyperbolic manifold homeomorphic to R × F with regular cosmological time. If we fix the linear holonomy f : π1(F ) → SO(2, 1) these domains (and so the affine deformations of the representation f) are parametrized by the measured geodesic laminations on H/f(π1(F )). The link between domains of dependence and measured geodesic laminations is the Gauss map of the CT-level surfaces. On the other hand Benedetti and Guadagnini noticed in [4] the singularity in the past of a domain of dependence is a real tree which is dual to the lamination. Moreover they argued that the action of π1(F ) on the CT-level surface Sa = T −1(a) converges in the Gromov sense to the action of π1(F ) on the singularity for a → 0 and to the action of π1(F ) on the hyperbolic plane