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 3dimensional 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 2sphere 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 3dimensional 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 3manifold $M$ 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.01275
 Bibcode:
 2020arXiv201201275B
 Keywords:

 Mathematics  Geometric Topology;
 Mathematical Physics;
 Mathematics  Differential Geometry;
 51H20;
 52B70;
 52B10;
 57Z05;
 57K20;
 57K35;
 83A99
 EPrint:
 57 pages, 10 figures