Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces

Fillastre, François

doi:10.5802/aif.2255

Cited by 20 publications

(38 citation statements)

References 17 publications

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“…A Fuchsian group of hyperbolic space is a discrete group of orientation-preserving isometries leaving globally invariant a totally geodesic surface, denoted by P H 2 , on which it acts cocompactly and without fixed points. In [Fil07] it is proved that: Theorem 1.3. A hyperbolic metric with conical singularities of positive singular curvature on a compact surface S of genus > 1 is realised by a unique (up to global isometries) convex Fuchsian polyhedron in hyperbolic space.…”

Section: Definitions and Statementsmentioning

confidence: 99%

“…Outline of the proof -organisation of the paper. Actually the general outline of the proof is very classical, starting from Alexandrov's work, and very close to the one used in [Fil07]. Roughly speaking, the idea is to endow with suitable topology both the space of convex Fuchsian polyhedra of M − K and the space of corresponding metrics, and to show that the map from one to the other given by the induced metric is a homeomorphism.…”

Section: Definitions and Statementsmentioning

confidence: 99%

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Fuchsian polyhedra in Lorentzian space-forms

Fillastre

2010

Math. Ann.

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Abstract. Let S be a compact surface of genus > 1, and g be a metric on S of constant curvature K ∈ {−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 > 2π. 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 [Ale42, RH93] concerning the sphere to the higher genus cases, and it is also the polyhedral version of a theorem of Labourie-Schlenker [LS00].Math. classification: 52B70(52A15,53C24,53C45)

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Section: Definitions and Statementsmentioning

confidence: 99%

Section: Definitions and Statementsmentioning

confidence: 99%

Fuchsian polyhedra in Lorentzian space-forms

Fillastre

2010

Math. Ann.

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“…This construction was communicated to us by Johannes Wallner. There exist numerous generalizations of Alexandrov's theorem: to surfaces of arbitrary genus, to hyperbolic and spherical polyhedral metrics, to singularities of negative curvature (see [12] and references therein). Perhaps our approach can be generalized to these cases.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes

2008

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We present a constructive proof of Alexandrov's theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation in the class of generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a relation with the weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of a positively curved generalized convex polytope. The latter is shown to be equal to the Hessian of the volume of the dual generalized polyhedron. We prove the non-degeneracy by generalizing the Alexandrov-Fenchel inequality. Our construction of a convex polytope from a given metric is implemented in a computer program.

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“…All of the curvatures of the cusp M must vanish due to (12). Thus (12) implies the existence part of Theorem A.…”

Section: Proof Of Theorem Amentioning

confidence: 82%

“…Corresponding realization theorems are proved by the first author [12] for hyperbolic space and by Schlenker [26] and the first author [11] for Lorentzian space-forms.…”

Section: Theorem 12 (Rivinmentioning

confidence: 99%

Hyperbolic cusps with convex polyhedral boundary

Fillastre

Izmestiev²

2009

Geom. Topol.

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We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthermore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex polyhedral cusp.The proof uses the discrete total curvature functional on the space of "cusps with particles", which are hyperbolic cone-manifolds with the singular locus a union of half-lines. We prove, in addition, that convex polyhedral cusps with particles are rigid with respect to the induced metric on the boundary and the curvatures of the singular locus.Our main theorem is equivalent to a part of a general statement about isometric immersions of compact surfaces.

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Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces

Cited by 20 publications

References 17 publications

Fuchsian polyhedra in Lorentzian space-forms

Fuchsian polyhedra in Lorentzian space-forms

Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes

Hyperbolic cusps with convex polyhedral boundary

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