Bounds on saddle connections for flat spheres (2308.08940v1)

Abstract: We consider a flat metric with conical singularities on the sphere. Assuming no partial sum of angle defects is equal to $2\pi$, we draw on the geometry of immersed disks to obtain an explicit upper bound on the number of saddle connections with at most $k$ self-intersections. We also deduce a bound on their lengths for a surface with a normalized area.