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Pré-Publication, Document De Travail Année : 2008

Fuchsian polyhedra in Lorentzian space-forms

François Fillastre
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Résumé

Let S be a compact surface of genus >1, and g be a metric on S of constant curvature K\in\{-1,0,1\} with conical singularities of negative singular curvature. When K=1 we add the condition that the lengths of the contractible geodesics are >2\pi. We prove that there exists a convex polyhedral surface P in the Lorentzian space-form of curvature K and a group G of isometries of this space such that the induced metric on the quotient P/G is isometric to (S,g). Moreover, the pair (P,G) is unique (up to global isometries) among a particular class of convex polyhedra, namely Fuchsian polyhedra. This extends theorems of A.D. Alexandrov and Rivin--Hodgson concerning the sphere to the higher genus cases, and it is also the polyhedral version of a theorem of Labourie--Schlenker.
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Dates et versions

hal-00131683 , version 1 (18-02-2007)
hal-00131683 , version 2 (13-05-2008)
hal-00131683 , version 3 (27-02-2009)

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François Fillastre. Fuchsian polyhedra in Lorentzian space-forms. 2008. ⟨hal-00131683v2⟩
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