Signature Calculation of Area Hermitian Form on Some Spaces of Polygons

TL;DR

This paper generalizes Thurston's signature calculation of the Hermitian form to spaces of polygons that parametrize moduli spaces of singular flat metrics on the sphere, focusing on cases with one negative curvature singularity.

Contribution

It computes the signature of the area Hermitian form on polygon spaces related to moduli spaces of singular flat metrics, extending Thurston's work.

Findings

Signature depends only on the sum of positive curvatures.

Provides explicit formula for the Hermitian form signature.

Connects polygon spaces with complex hyperbolic geometry.

Abstract

This chapter is motivated by the paper by Thurston on triangulations of the sphere and singular flat metrics on the sphere. Thurston locally parametrized the moduli space of singular flat metrics on the sphere with prescribed positive curvature data by the complex hyperbolic space of appropriate dimension. This work can be considered as a generalization of signature calculation of the Hermitian form that he made in his paper. The moduli space of singular flat metrics having unit area on the sphere with prescribed curvature data 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. Also 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. This paper will appear in the book "In the tradition of Thurston, Vol. II", Springer, 2022.