Volume 14, issue 1 (2010)

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Prescribing the behaviour of geodesics in negative curvature

Jouni Parkkonen and Frédéric Paulin

Geometry & Topology 14 (2010) 277–392

Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighbourhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much into) them. Several applications are given, including a definite improvement of the unclouding problem of our paper [Geom. Func. Anal. 15 (2005) 491–533], the prescription of heights of geodesic lines in a finite volume such M, or of spiraling times around a closed geodesic in a closed such M. We also prove that the Hall ray phenomenon described by Hall in special arithmetic situations and by Schmidt–Sheingorn for hyperbolic surfaces is in fact only a negative curvature property.

geodesics, negative curvature, horoballs, Lagrange spectrum, Hall ray
Mathematical Subject Classification 2000
Primary: 53C22, 11J06, 52A55
Secondary: 53D25
Received: 1 June 2007
Revised: 21 July 2009
Accepted: 15 April 2009
Preview posted: 27 October 2009
Published: 2 January 2010
Proposed: Martin Bridson
Seconded: Walter Neumann, Jean-Pierre Otal
Jouni Parkkonen
Department of Mathematics and Statistics
PO Box 35
40014 University of Jyväskylä
Frédéric Paulin
Département de Mathématique et Applications
Ecole Normale Supérieure
45 rue d’Ulm
75230 PARIS Cedex 05