Original Paper

Geometric & Functional Analysis GAFA

, Volume 15, Issue 2, pp 491-533

First online:

Unclouding the sky of negatively curved manifolds

  • J. ParkkonenAffiliated withDepartment of Mathematics and Statistics, University of Jyväskylä Email author 
  • , F. PaulinAffiliated withDépartement de Mathématique et Applications, UMR 8553, CNRS, Ecole Normale Supérieure

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Abstract.

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.