Signature Calculation of the Area Hermitian Form on Some Spaces of Polygons

This chapter is motivated by the paper by Thurston on triangulations of the sphere and singular flat metrics on the sphere. Thurston gave a local parametrization of the moduli space of singular flat metrics on the sphere with prescribed positive curvature data at the singular points by a complex hyperbolic space of an appropriate dimension. This work can be considered as a generalization of the signature calculation of the Hermitian form that he made in his paper. The moduli space of singular flat metrics on the sphere having unit area and with prescribed curvature data at the singular points can be locally parametrized by certain spaces of polygons. This can be done by cutting singular flat spheres through length minimizing geodesics from a fixed singular point to the others. In that case the space of polygons is a complex vector space of dimension n − 1 when there are n + 1 singular points. There is natural area Hermitian form of signature (1, n − 2) on this vector space. In this chapter we calculate the signature of the area Hermitian form on some spaces of polygons which locally parametrize the moduli space of singular flat metrics having unit area on the sphere with one singular point of negative curvature. The formula we obtain depends only on the sum of the curvatures of the singular points having positive curvature.