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

François Fillastre, Ivan Izmestiev
2017 Séminaire de théorie spectrale et géométrie  
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
more » ... ne 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 fatcat:d6adx3ccljdcjmvglehoe5mwiy