Flat spacetimes with compact hyperbolic Cauchy surfaces

Abstract

Given a closed hyperbolic n-manifold M, we study the flat Lorentzian structures on M×ℝ such that M×{0} is a Cauchy surface. We show there exist only two maximal structures sharing a fixed holonomy (one future complete and the other one past complete). We study the geometry of those maximal spacetimes in terms of cosmological time. In particular, we study the asymptotic behaviour of the level surfaces of the cosmological time. As a by-product, we get that no affine deformation of the hyperbolic holonomy ρ: π₁(M)→ SO(n,1) of M acts freely and properly on the whole Minkowski space. The present work generalizes the case n=2 treated by Mess, taking from a work of Benedetti and Guadagnini the emphasis on the fundamental rôle played by the cosmological time. In the last sections, we introduce measured geodesic stratifications on M, that in a sense furnish a good generalization of measured geodesic laminations in any dimension and we investigate relationships between measured stratifications on M and Lorentzian structures on M×ℝ.