Energy functions on moduli spaces of flat surfaces with erasing forest

TL;DR

This paper studies energy functions on moduli spaces of flat surfaces with erasing forests, proving their integrals are finite and applying these results to classical theorems in the field.

Contribution

It introduces energy functions on these moduli spaces and proves their finiteness, extending classical results by Masur-Veech and Thurston.

Findings

Proved the finiteness of the integral of energy functions on the moduli space.

Reproduced classical results of Masur-Veech and Thurston using this finiteness.

Established a new framework for analyzing flat surfaces with erasing forests.

Abstract

Flat surfaces with erasing forest are obtained by deforming the flat metric structure of translation surfaces, the moduli space of such surfaces is a deformation of the moduli space of translation surfaces. On the moduli space of flat surfaces with erasing forest, one can define some energy function involving the area of the surface, and the total length of the erasing forest. Note that on this space, we have a volume form which is defined by using geodesic triangulations. The aim of this paper is to prove that the integral of the energy functions mentionned above with respect to this volume form is finite. As applications, we will use this result to recover some classical results due to Masur-Veech, and Thurston.