A remark on spaces of flat metrics with cone singularities of constant sign curvatures
François Fillastre1; Ivan Izmestiev2
1 Université de Cergy-Pontoise UMR CNRS 8088 95000 Cergy-Pontoise (France)
2 TU Wien Wiedner Hauptstraße 8-10/104 A-1040 Wien (Austria)
Séminaire de théorie spectrale et géométrie, Volume 34 (2016-2017), pp. 65-92.

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 ≤1 2(n-1).

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.

DOI: 10.5802/tsg.355
Keywords: Flat metrics, convex polyhedra, mixed volumes, Minkowski space, covolume
François Fillastre 1; Ivan Izmestiev 2

1 Université de Cergy-Pontoise UMR CNRS 8088 95000 Cergy-Pontoise (France)
2 TU Wien Wiedner Hauptstraße 8-10/104 A-1040 Wien (Austria) 
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François Fillastre; Ivan Izmestiev. A remark on spaces of flat metrics with cone singularities of constant sign curvatures. Séminaire de théorie spectrale et géométrie, Volume 34 (2016-2017), pp. 65-92. doi : 10.5802/tsg.355. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.355/

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