Fuchsian polyhedra in Lorentzian space-forms
Résumé
A Fuchsian polyhedron in a Lorentzian space-form is a polyhedral surface invariant under the action of a group of isometries fixing a point $x_0$ and acting cocompactly on the time-like units vectors at $x_0$. The induced metric on a convex Fuchsian polyhedron is isometric to a constant curvature metric with conical singularities of negative singular curvature on a compact surface of genus greater than one. e prove that these metrics are actually realised by exactly one convex Fuchsian polyhedron (up to global isometries) --- in the spherical case, we must add the condition that the lengths of the contractible geodesics are $>2\pi$. This extends theorems of A.D. Alexandrov and Rivin--Hodgson \cite{alex42,RivHod} concerning the sphere to the higher genus cases, and it is also the polyhedral version of a theorem of Labourie--Schlenker \cite{SchLab}.
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