By a result of W. P. Thurston, the moduli space of flat metrics on the sphere with cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension . 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 and .
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.
@article{TSG_2016-2017__34__65_0, author = {Fillastre, Fran\c{c}ois and Izmestiev, Ivan}, title = {A remark on spaces of flat metrics with cone singularities of constant sign curvatures}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {65--92}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {34}, year = {2016-2017}, doi = {10.5802/tsg.355}, language = {en}, url = {http://www.numdam.org/articles/10.5802/tsg.355/} }
TY - JOUR AU - Fillastre, François AU - Izmestiev, Ivan TI - A remark on spaces of flat metrics with cone singularities of constant sign curvatures JO - Séminaire de théorie spectrale et géométrie PY - 2016-2017 SP - 65 EP - 92 VL - 34 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/tsg.355/ DO - 10.5802/tsg.355 LA - en ID - TSG_2016-2017__34__65_0 ER -
%0 Journal Article %A Fillastre, François %A Izmestiev, Ivan %T A remark on spaces of flat metrics with cone singularities of constant sign curvatures %J Séminaire de théorie spectrale et géométrie %D 2016-2017 %P 65-92 %V 34 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/tsg.355/ %R 10.5802/tsg.355 %G en %F TSG_2016-2017__34__65_0
Fillastre, François; Izmestiev, Ivan. A remark on spaces of flat metrics with cone singularities of constant sign curvatures. Séminaire de théorie spectrale et géométrie, Tome 34 (2016-2017), pp. 65-92. doi : 10.5802/tsg.355. http://www.numdam.org/articles/10.5802/tsg.355/
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