Geometric & Functional Analysis GAFA

, Volume 15, Issue 2, pp 491–533

Unclouding the sky of negatively curved manifolds

Original Paper

DOI: 10.1007/s00039-005-0514-y

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.

Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläFinland
  2. 2.Département de Mathématique et Applications, UMR 8553CNRS, Ecole Normale SupérieureParis Cedex 05France