How close are cone singularities on a random flat surface?

TL;DR

This paper investigates the distribution of shortest geodesics on flat cone spheres with conical singularities, providing a recurrence relation for their length distribution within the moduli space.

Contribution

It introduces a recurrence relation for the distribution of shortest geodesic lengths on flat cone spheres, advancing understanding of their geometric properties.

Findings

Derived a recurrence relation for geodesic length distribution

Connected geodesic lengths to Thurston's volume form

Enhanced understanding of moduli space geometry

Abstract

We study the shortest geodesics on flat cone spheres, i.e. flat metrics on the sphere with conical singularities. The length of the shortest geodesic between two singular points can be treated as a function on the moduli space of flat cone spheres with prescribed angle defects. We prove a recurrent relation on the distribution of this function with respect to Thurston's volume form on the moduli space.