Geodesics avoiding open subsets in surfaces of negative curvature
by
Viktor Schroeder
University of Zurich
Coauthors: S. Buyalo, M. Walz

The work is related to a question of F. Labourie: Does there exist a proper closed subset W of the unit tangent bundle of a compact surface M with negative curvature, such that W is invariant under the geodesic flow and \pi(W) = M, where \pi is the foot point projection.

Burns and Pollicott gave an affirmative in the case of constant curvature. The crucial point of their proof is a construction of a geodesic avoiding an a-separated subset of the universal covering space X.

Definition:   A subset \Omega of a metric space (X, d) is called a-separated (for some a > 0) if d(x, x') >= a for all distinct x, x' in \Omega.

One of our main results is an example which shows that the methods of Burns and Pollicott do not work in the variable curvature case.

Theorem A:   Given \delta > 0 and a > 0, the space R² can be equipped with a complete smooth Riemannian metric g of sectional curvature -1 - \delta <= K <= - 1 + \delta such that there exists an a-separated subset \Omega subset R² enjoying the properties

1. d (0, x) >= a for all x in \Omega.
2. If \gamma: R --> (R², g) is a complete geodesic with \gamma(0) = 0 then d(\gamma(R), \Omega) = 0.

We also discuss the case if a geodesic ray can avoid an a-separeated set in the context of an upper curvature bound. It is interesting that the result depends on the separating constant a.

Theorem B:   For every a > ln2 \thickapprox 0.693 ... there exists an \epsilon > 0 with the property:

• Let X be a complete simply connected surface with curvature K <= -1 and let \Omega subset X be an a-separated subset.
  Then for each 0 in X, d (0, \Omega) >= a/2 there exists a geodesic ray \gamma: [0, \infty) --> X with d(\gamma([0, \infty)), \Omega) >= \epsilon and \gamma(0) = 0.

Theorem C:   For every a with 0 < a <= 1/3 there exists a complete simply connected surface X with curvature K <= -1 which contains an a-separated subset \Omega subset X such that every geodesic ray \gamma: [0, \infty) --> X passes arbitrary close to points of \Omega.

Date received: May 13, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cabb-48.