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.
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Received: January 2004 Accepted: August 2004
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Parkkonen, J., Paulin, F. Unclouding the sky of negatively curved manifolds. GAFA, Geom. funct. anal. 15, 491–533 (2005). https://doi.org/10.1007/s00039-005-0514-y
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DOI: https://doi.org/10.1007/s00039-005-0514-y