Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces
Annales de l'Institut Fourier, Volume 57 (2007) no. 1, p. 163-195

A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric to a hyperbolic metric with conical singularities of positive singular curvature on a compact surface of genus greater than one. We prove that these metrics are actually realised by exactly one convex Fuchsian polyhedron (up to global isometries). This extends a famous theorem of A.D. Alexandrov.

Un polyèdre fuchsien de l’espace hyperbolique est une surface polyédrale invariante sous l’action d’un groupe fuchsien d’isométries (c.a.d. un groupe d’isométries qui laissent globalement invariante une surface totalement géodésique et sur laquelle il agit de manière cocompacte). La métrique induite sur un polyèdre fuchsien convexe est isométrique à une métrique hyperbolique avec des singularités coniques de courbure singulière positive sur une surface compacte de genre plus grand que un. On démontre que ces métriques sont en fait réalisées par un unique polyèdre fuchsien convexe (modulo les isométries globales). Ce résultat étend un théorème célèbre de A.D. Alexandrov.

DOI : https://doi.org/10.5802/aif.2255
Classification:  53C45,  52A55,  52B70,  53C24
Keywords: Fuchsian, convex, polyhedron, hyperbolic, conical singularities, infinitesimal rigidity, Pogorelov map, Alexandrov
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     author = {Fillastre, Fran\c cois},
     title = {Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {1},
     year = {2007},
     pages = {163-195},
     doi = {10.5802/aif.2255},
     mrnumber = {2313089},
     zbl = {1123.53033},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2007__57_1_163_0}
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Fillastre, François. Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces. Annales de l'Institut Fourier, Volume 57 (2007) no. 1, pp. 163-195. doi : 10.5802/aif.2255. http://www.numdam.org/item/AIF_2007__57_1_163_0/

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