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Article Dans Une Revue Actes du Séminaire de Théorie Spectrale et Géométrie (Grenoble) Année : 2017

A remark on spaces of flat metrics with cone singularities of constant sign curvatures

A remark on spaces of flat metrics with cone singularities of constant sign curvatures

François Fillastre
Ivan Izmestiev
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Résumé

By a result of W. P. Thurston, the moduli space of flat metrics on the sphere with n cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension n − 3. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this space has a natural decomposition into real hyperbolic convex polyhedra of dimensions n-3 and \leq (n-1)/2. By a result of W. Veech, the moduli space of flat metrics on a compact surface with cone singularities of prescribed negative curvatures has a foliation whose leaves have a local structure of complex pseudo-spheres. The complex structure comes again from the area of the metric. The form can be degenerate; its signature depends on the curvatures prescribed. Using polyhedral surfaces in Minkowski space, we show that this moduli space has a natural decomposition into spherical convex polyhedra.

Dates et versions

hal-03793210 , version 1 (14-10-2022)

Identifiants

Citer

François Fillastre, Ivan Izmestiev. A remark on spaces of flat metrics with cone singularities of constant sign curvatures. Actes du Séminaire de Théorie Spectrale et Géométrie (Grenoble), 2017, 34, pp.65-92. ⟨10.5802/tsg.355⟩. ⟨hal-03793210⟩
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