Uniform Length Estimates for Trajectories on Flat Cone Surfaces

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.