The Alexandrov theorem for 2 + 1 flat radiant spacetimes

Brunswic, Léo Maxime

doi:10.2140/agt.2025.25.1321

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The Alexandrov theorem for $2+1$ flat radiant spacetimes

Léo Maxime Brunswic

Algebraic & Geometric Topology 25 (2025) 1321–1375

DOI: 10.2140/agt.2025.25.1321

Abstract

Fillastre showed that one can realize the universal covering of any locally Euclidean surface $Σ$ with conical singularities of angle bigger than $2 π$ as the boundary of a convex Fuchsian polyhedron in $3$ -dimensional Minkowski space in a unique manner, up to the action of $SO (1, 2) ⋉ ℝ^{3}$ , 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 $𝜃_{i}$ of $Σ$ and for any choice of nonnegative ${(κ_{i})}_{i \in [[1, s]]}$ such that $κ_{i} < 𝜃_{i}$ and $κ_{i} \leq 2 π$ , 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 $κ_{i}$ , and $P$ is a convex polyhedron in $M$ whose boundary $\partial P$ is a Cauchy surface isometric to $Σ$ , the $i^{th}$ conical singularity of $\partial P$ 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 $κ_{i} = 2 π$ and $κ_{i} = 0$ for all $i$ .

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

Mathematical Subject Classification

Primary: 51M05, 52B10, 52B70, 53C50, 57K35

Secondary: 53C42, 57M60

References

Publication

Received: 4 February 2022

Revised: 17 November 2023

Accepted: 3 January 2024

Published: 20 June 2025

Authors

Léo Maxime Brunswic
Centre de recherche astrophysique de Lyon Lyon France	Huawei Noah Ark Laboratories Montreal QC Canada
https://leo.brunswic.fr

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Volume 25, issue 3 (2025)