By a result of W. P. Thurston, the moduli space of flat metrics on the sphere with prescribed n cone singularities of positive curvature is a complex hyperbolic orbifold of dimension $n3$. 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 $n3$ and $\leq 1/2(n1)$.By a result of W. Veech, the moduli space of flat metrics on a compact surface with prescribed cone singularities of negative curvature has a foliation whose leaves have a local structure of complex pseudospheres, coming again from the area of the metric. The form can be degenerate; its signature depends on the collection of angles. Using polyhedral surfaces in Minkowski space, we show that this moduli space has a natural decomposition into spherical convex polyhedra.
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A remark about the space of flat metrics with conical singularities on a compact surface
Research Group:
Francois Fillastre
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Université de CergyPontoise
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A137
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Wednesday, May 9, 2018  14:00
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