Uniform Length Estimates for Trajectories on Flat Cone Surfaces

TL;DR

This paper establishes uniform length estimates for trajectories on flat cone surfaces, especially convex flat cone spheres, using self-intersection numbers and providing explicit constants.

Contribution

It introduces uniform lower bounds for trajectory lengths on convex flat cone spheres with positive curvature gap and fixed singularities, including explicit constants.

Findings

Derived lower bounds for trajectory lengths based on self-intersection numbers.

Provided explicit constants for length estimates on convex flat cone spheres.

Achieved uniform two-sided estimates combining previous upper bounds.

Abstract

This paper studies length estimates for trajectories on flat cone surfaces in terms of their self-intersection numbers. For an area-one flat cone surface, we obtain a lower bound for the length of a trajectory, with constants depending only on the flat metric. Our main focus is the case of convex flat cone spheres. We show that these constants can be chosen uniformly for such spheres with a positive curvature gap and a fixed number of singularities. Explicit values for these constants are also provided. Combined with a previously established upper bound, this yields uniform two-sided estimates for trajectory lengths on such flat cone spheres. As an application, we obtain uniform bounds for counting functions of trajectories on convex flat cone spheres and on convex polygonal billiards.