Fuchsian polyhedra in Lorentzian space-forms
Abstract
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.
Keywords
Cite
@article{arxiv.math/0702532,
title = {Fuchsian polyhedra in Lorentzian space-forms},
author = {François Fillastre},
journal= {arXiv preprint arXiv:math/0702532},
year = {2009}
}