Areas of two-dimensional moduli spaces (Q2731916)

By a hyperbolic surface \(R\) is meant a 2-dimensional cone manifold, that is, a surface equipped with a metric of constant curvature \(-1\) which admits conical singular points and all of whose boundary components are totally geodesic closed curves. If the Teichmüller space of a hyperbolic orbifold \(R\) is two-dimensional, the signature is either of type \((1;\theta)\) or the type is parameterized by a Fenchel-Nielsen coordinate system \((l,s)\), \(l>0\), \(-\infty <s<\infty ,\) then the 2-form NEWLINE\[NEWLINE\omega _{WP} = dl\wedge ds \tag{1}NEWLINE\]NEWLINE is invariant under the mapping class group, and hence it can be considered in the moduli space. The form (1) is Wolpert's formula for the Weil-Petersson 2-form. NEWLINENEWLINENEWLINEIn this paper, the authors compute the areas of 2-dimensional moduli spaces of hyperbolic cone-surfaces with respect to the form (1) when the corresponding Teichmüller spaces admit global Fenchel-Nielsen coordinates. In order to calculate the areas, they use the classical result that the Teichmüller space for the signature \((1;2\theta)\) is represented by the sublocus of the equation \(x^2 + y^2 + z^2 - 2xyz - \sin ^2\frac \theta 2 = 0\) satisfying \(x, y, z > 1\). NEWLINENEWLINENEWLINETheorem 1.1. If \(\theta _1, \theta _2, \theta _3\) and \(\theta _4\) are numbers in \([0, \frac \pi 2]\cup iR_{+},\) then the area of the moduli space for the signature \((0; 2\theta _1, 2\theta _2, 2\theta _3, 2\theta _4)\) is \(2(\pi ^2 - \theta _1^2 - \theta _2^2 - \theta _3^2 - \theta _4^2).\) NEWLINENEWLINENEWLINETheorem 2.2. The 2-form \(\omega _{WP}\) is invariant under the action of the mapping class group.