Unclouding the sky of negatively curved manifolds
- Cite this article as:
- Parkkonen, J. & Paulin, F. GAFA, Geom. funct. anal. (2005) 15: 491. doi:10.1007/s00039-005-0514-y
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.