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Anti-de Sitter geometry and polyhedra inscribed in quadrics Joint work with Je Danciger and Sara Maloni Jean-Marc Schlenker University of Luxembourg Workshop on moduli spaces of geometric structures IMS, NUS, Aug 15-19, 2016 1/7 Jean-Marc Schlenker Anti-de Sitter geometry and polyhedra inscribed in quadrics Modern questions on polyhedra Jakob Steiner (17961863) started the study of the combinatorics of Euclidean/projective polyhedra. He introduced new questions (1832). • What are the possible of polyhedra ? the planar 3-connected graphs. combinatorics Ernst Steinitz (18711928), 1916 : 2/7 Jean-Marc Schlenker Anti-de Sitter geometry and polyhedra inscribed in quadrics Inscribed polyhedra • What are the possible combinatorics of polyhedra inscribed in the sphere, or in another quadric ? projectively invariant so there are only 3 quadrics to The question is consider : the sphere, one-sheeted hyperboloid, and cylinder. Steinitz (1927) : there are polyhedra not inscribable in a sphere. 3/7 Jean-Marc Schlenker Anti-de Sitter geometry and polyhedra inscribed in quadrics Inscribed polyhedra uninscribable.gif 304×313 pixels 24/02/2015 08:41 • What are the possible combinatorics of polyhedra inscribed in the sphere, or in another quadric ? projectively invariant so there are only 3 quadrics to The question is consider : the sphere, one-sheeted hyperboloid, and cylinder. Steinitz (1927) : there are polyhedra not inscribable in a sphere. http://www.ics.uci.edu/~eppstein/junkyard/uninscribable/uninscribable.gif Jean-Marc Schlenker Page 1 of 1 3/7 Anti-de Sitter geometry and polyhedra inscribed in quadrics Polyhedra inscribed in spheres In 1993, Craig Hodgson, Igor Rivin and Warren Smith gave a partial answer to Steiner's question. A polyhedron can be inscribed in a sphere i a certain system of linear equations and inequalities has a solution. Consequence : deciding whether a graph can be realized can be decided in polynomial time. Their result is based ideal hyperbolic polyhedra. 4/7 Jean-Marc Schlenker Anti-de Sitter geometry and polyhedra inscribed in quadrics An answer to Steiner's question Theorem A (Danciger-Maloni-S.) A planar 3-connected graph can be realized as a polyhedron inscribed in the hyperboloid (resp. cylinder) i it is inscribable in a sphere and it admits a Hamiltonian cycle. A Hamiltonian cycle is a path going through each vertex exactly once. Note : harder to decide whether a polyhedron is inscribable in a hyperboloid or in a cylinder than in a sphere. 5/7 Jean-Marc Schlenker Anti-de Sitter geometry and polyhedra inscribed in quadrics Ideal polyhedra in AdS 3 Ideal polyhedra have vertices on and edges in AdS3 . ∂∞ AdS3 So their faces are space-like (hyperbolic). Two types of faces : past and future. Equator is a Hamiltonian cycle, with angles w < 0. 6/7 Jean-Marc Schlenker Anti-de Sitter geometry and polyhedra inscribed in quadrics Main statements Theorem A (Danciger-Maloni-S.) A planar 3-connected graph can be realized as a polyhedron inscribed in the hyperboloid (resp. cylinder) i it is inscribable in a sphere and it admits a Hamiltonian cycle. Theorem B (Andreev, Rivin) Let Γ ⊂ S 2 be the 1-skeleton of a cell decomposition, and let w : Γ1 → (0, π)P. ∃P ⊂ H 3 with exterior dihedral angles w i i. ∀v ∈ Γ0 , e ∈v w (e ) =P2π , ii. ∀ other path c in Γ∗ , e ∈c w (e ) > 2π . Theorem C (Danciger, Maloni, S.) Let Γ ⊂ S 2 be the 1-skeleton of a cell decomposition, and let w : Γ1 → R6=0 . ∃P ⊂ AdS3 with exterior dihedral angles w i i. w < 0 onPa Hamiltonian path γ ⊂ Γ1 , ii. ∀v ∈ Γ0 , e ∈v w (e ) = 0, P iii. ∀ other path c in Γ∗ intersecting γ twice, e ∈c w (e ) > 0. Jean-Marc Schlenker 7/7 Anti-de Sitter geometry and polyhedra inscribed in quadrics