Prescribing the behaviour of geodesics in negative curvature

TL;DR

This paper develops methods to construct geodesic rays in negatively curved manifolds with prescribed penetration behaviors in convex subsets, leading to applications in geometric and dynamical problems.

Contribution

It introduces a new approach for prescribing geodesic behaviors in negatively curved spaces, improving previous results and extending the Hall ray phenomenon to general negative curvature.

Findings

Constructed geodesic rays with prescribed penetration properties.

Improved the unclouding problem in negatively curved manifolds.

Extended the Hall ray phenomenon to all negative curvature settings.

Abstract

Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods 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 in) them. Several applications are given, including a definite improvement of the unclouding problem of [PP1], 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.