2024/09/21 by Kai Fu, Fu, Kai
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Geometric Topology (math.GT) #Mathematics and Applications #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2409.14188
openalex publication_date 2024/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.