Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential

Abstract

Suppose X is the torus the complex plane C divided by the group of translations generated by z → z + 1 and 2 → z + i and q = dz ² is the unique up to scalar multiple quadratic differential on X. The trajectories of q are straight lines and a trajectory is closed if and only if its slope is a rational p/q. Its length is (p ₂ + q ₂)_1/2. Parallel closed trajectories fill up X. The number N(T) of parallel families of length ≤ T is then the number of lattice points (p, q) with p, q relatively prime inside a circle of radius T. It is classical that

$$ \mathop{{T \to \infty }}\limits^{{\lim }} \frac{{N\left( T \right)}}{{{T^{2}}}} = \frac{6}{{{\pi ^{2}}}}. $$

This work was supported in part by NSF Grant DMS-8601977.