The Alexandrov theorem for 2 + 1 flat radiant spacetimes

Brunswic, Léo Maxime¹

¹Centre de recherche astrophysique de Lyon, Lyon, France, Huawei, Noah Ark Laboratories, Montreal QC, Canada

Algebraic and Geometric Topology, Tome 25 (2025) no. 3, pp. 1321-1375

Résumé

Fillastre showed that one can realize the universal covering of any locally Euclidean surface Σ with conical singularities of angle bigger than 2π as the boundary of a convex Fuchsian polyhedron in 3-dimensional Minkowski space in a unique manner, up to the action of SO ⁡ (1,2) ⋉ ℝ3, the affine isometry group of Minkowski space. The proof used a so-called deformation method, which is nonconstructive. We adapt a variational method previously used by Volkov, Bobenko, Izmestiev, and Fillastre on similar problems to provide an effective proof of Fillastre’s theorem. In passing, we extend Fillastre’s theorem as follows. Without assumptions on the conical angles 𝜃i of Σ and for any choice of nonnegative (κi)i∈[[1,s]] such that κi < 𝜃i and κi ≤ 2π, there exists a unique couple (M,P) where M belongs to a class of singular locally Minkowski manifolds we define with s singular lines of respective conical angle κi, and P is a convex polyhedron in M whose boundary ∂P is a Cauchy surface isometric to Σ, the i th conical singularity of ∂P lying on the i th singular line of M. Our result unifies Fillastre’s theorem and instances of Penner–Epstein convex hull constructions, corresponding respectively to κi = 2π and κi = 0 for all i.

Détail

DOI : 10.2140/agt.2025.25.1321

Keywords: Euclidean surface, spacetime, conical singularities, polyhedral embedding, linear holonomy, massive particle, BTZ black hole, Minkowski space, Lorentzian geometry, $(G,X)$-manifold, singular $(G,X)$-structure, affine geometry, $3$-manifold, geometrical structure, Penner–Epstein convex hull, weighted Delaunay triangulation, flipping algorithm, hyperbolic surface, hyperbolic geometry, Einstein–Hilbert functional, total curvature functional, Weyl's embedding problem, Schläfli formula

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