Energy functions on moduli spaces of flat surfaces with erasing forest (Q431231)

In a previous paper [Geom. Funct. Anal. 20, No. 1, 192--228 (2010; Zbl 1202.30074)], the author introduced the notion of an erasing forest on a flat surface and showed that the moduli spaces of flat surfaces with erasing forest are complex orbifolds with a natural volume form \(\mu\). The aim of the present paper is to show that the volume of those moduli spaces with respect to \(\mu\), normalized by some energy function involving the area and the total length of the erasing forest, is finite. As application, the author obtains new proofs of a result of Masur-Veech on the finiteness of the volume of the moduli space of translation surfaces, and of a result of Thurston on the finiteness of the volume of the moduli space of polyhedral flat surfaces.