Fuchsian polyhedra in Lorentzian space-forms
A. D. Aleksandrov proved that a hyperbolic cone metric on the sphere \(\mathbb{S}^2\) with positively curved singularities can be realized (uniquely, up to congruence) by the induced metric of a convex polyhedron in \(\mathbb{H}^3 .\) Much later, the characterization of compact convex polyhedral surfaces in the de Sitter space \(\mathbb{S}^2_1 \) was obtained by \textit{C. D. Hodgson} and \textit{I. Rivin} in [Invent. Math. 111, No.~1, 77--111 (1993; Zbl 0784.52013)]. The paper under review greatly generalizes the above results to Lorentzian space forms of dimension 3 with constant curvature \(K, K\in \{-1,0,1\}\): Let \(S\) be a compact surface of genus \(>1\) and \(g\) a metric of constant curvature \(K\) with conical singularities of negative singular curvature (in the spherical case the lengths of the contractible geodesics are assumed to be \(>2\pi\)). It is shown that there exists a convex polyhedral surface \(P\) in the Minkowski space (\(K=0\)), in the de Sitter space (\(K=1\)) and in the anti-de Sitter space (\(K=-1\)) and a group \(G\) of isometries of this space such that the induced metric on \(P/G\) is isometric to \((S,g).\) Moreover, uniqueness (up to global isometries) is shown within a particular class of convex polyhedra.
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