Unclouding the sky of negatively curved manifolds

Let M be a complete simply connected Riemannian manifold, with sectional curvature K ≤ −1. Under certain assumptions on the geometry of ∂M, which are satisfied for instance if M is a symmetric space, or has dimension 2, we prove that given any family of horoballs in M, and any point x₀ outside these horoballs, it is possible to shrink uniformly, by a finite amount depending only on M, these horoballs so that some geodesic ray starting from x₀ avoids the shrunk horoballs. As an application, we give a uniform upper bound on the infimum of the heights of the closed geodesics in the finite volume quotients of M.Received: January 2004 Accepted: August 2004