Prescribing the behaviour of geodesics in negative curvature

Parkkonen, Jouni¹ ; Paulin, Frédéric²

¹ Department of Mathematics and Statistics, PO Box 35, 40014 University of Jyväskylä, Finland
² Département de Mathématique et Applications, UMR 8553 CNRS, Ecole Normale Supérieure, 45 rue d’Ulm, 75230 PARIS Cedex 05, France

Geometry & topology, Tome 14 (2010) no. 1, pp. 277-392.

Résumé

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.

Détail

DOI : 10.2140/gt.2010.14.277

Keywords: geodesics, negative curvature, horoballs, Lagrange spectrum, Hall ray

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