Polyhedral surfaces in flat (2+1)-spacetimes and balanced cellulations on hyperbolic surfaces

We first prove that given a hyperbolic metric $h$ on a closed surface $S$, any flat metric on $S$ with negative singular curvatures isometrically embeds as a convex polyhedral Cauchy surface in a unique future-complete flat globally hyperbolic maximal (2+1)-spacetime whose linear part of the holonomy is given by $h$. The Gauss map allows to translate this statement to a purely 2-dimensional problem of finding a balanced geodesic cellulation on the hyperbolic surface, from which the flat metric can be easily recovered. We show next that given two such flat metrics on the surface, there exists a unique pair of future- and past-complete flat globally hyperbolic maximal (2+1)-spacetimes with the same holonomy, in which the flat metrics embed respectively as convex polyhedral Cauchy surfaces. The proof follows from convexity properties of the total length of the associated balanced geodesic cellulations over Teichm\"uller space.