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We first prove that given a hyperbolic metric $h$ on a closed surface $S$,any flat metric on $S$ with negative singular curvatures isometrically embedsas a convex polyhedral Cauchy surface in a unique future-complete flat globallyhyperbolic maximal (2+1)-spacetime whose linear part of the holonomy is givenby $h$. The Gauss map allows to translate this statement to a purely2-dimensional problem of finding a balanced geodesic cellulation on thehyperbolic surface, from which the flat metric can be easily recovered.  We show next that given two such flat metrics on the surface, there exists aunique pair of future- and past-complete flat globally hyperbolic maximal(2+1)-spacetimes with the same holonomy, in which the flat metrics embedrespectively as convex polyhedral Cauchy surfaces. The proof follows fromconvexity properties of the total length of the associated balanced geodesiccellulations over Teichm\"uller space.

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