Constant Mean Curvature Foliations of Simplicial Flat Spacetimes

Benedetti and Guadagnini [5] have conjectured that the constant mean curvature foliation Mτ in a 2 + 1 dimensional flat spacetime V with compact hyperbolic Cauchy surfaces satisfies limτ→−∞ lMτ = sT , where lMτ and sT denote the marked length spectrum ofMτ and the marked measure spectrum of the R-tree T , dual to the measured foliation corresponding to the translational part of the holonomy of V , respectively. We prove that this is the case for n+1 dimensional, n ≥ 2, simplicial flat spacetimes with compact hyperbolic Cauchy surface. A simplicial spacetime is obtained from the Lorentz cone over a hyperbolic manifold by deformations corresponding to a simple measured foliation.