Alexandrov Theorem for 2+1 flat radiant spacetimes

Abstract

A classical Theorem of Alexandrov states that the map associating its boundary to a convex polyhdedron of the 3-dimensional Euclidean space is a bijection from the set of convex polyhdedron up to congruence to the set of isometry classes of locally Euclidean metric on the 2-sphere with conical singularities smaller that $2\pi$. Fillastre proved a similar statement for locally Euclidean metric on higher genus surfaces with conical singularities bigger than $2\pi$ by embedding their universal covering in 3-dimensional Minkowski space as the boundary of Fuchsian polyhedra. The original proofs of Alexandrov and Fillastre both rely on invariance of domain Theorem hence are not effective. Volkov, in his thesis, provided a variational, hence effective, proof of Alexandrov Theorem which has then been generalised by Bobenko, Izmestiev and Fillastre. The present work goes further by adapting Volkov's variational method to provide an effective version of Fillastre Theorem and extend Fillastre's result: we show that for any closed locally Euclidean surface $\Sigma$ with conical singularities of arbitrary angles $(\theta_i)_{1 \leq i \leq s }$ and any choice of Lorentzian angles $(\kappa_i)_{1\leq i\leq s}$ such that $\kappa_i<\theta_i$ and $\kappa_i\leq 2\pi$, there exists a locally Minkoswki 3-manifold $M$ of linear holonomy with conical singularities $(\kappa_i)_{1\leq i\leq s}$ and a convex polyedron $P$ in $M$ whose boundary is isometric to $\Sigma$; furthermore such a couple $(M,P)$ is unique.