The Michigan Mathematical Journal
- Michigan Math. J.
- Volume 68, Issue 2 (2019), 251-275.
Fat Flats in Rank One Manifolds
We study closed nonpositively curved Riemannian manifolds that admit “fat -flats”; that is, the universal cover contains a positive-radius neighborhood of a -flat on which the sectional curvatures are identically zero. We investigate how the fat -flats affect the cardinality of the collection of closed geodesics. Our first main result is to construct rank nonpositively curved manifolds with a fat -flat that corresponds to a twisted cylindrical neighborhood of a geodesic on . As a result, contains an embedded closed geodesic with a flat neighborhood, but nevertheless has only countably many closed geodesics. Such metrics can be constructed on finite covers of arbitrary odd-dimensional finite volume hyperbolic manifolds. Our second main result is a proof of a closing theorem for fat flats, which implies that a manifold with a fat -flat contains an immersed, totally geodesic -dimensional flat closed submanifold. This guarantees the existence of uncountably many closed geodesics when . Finally, we collect results on thermodynamic formalism for the class of manifolds considered in this paper.
Michigan Math. J., Volume 68, Issue 2 (2019), 251-275.
Received: 17 April 2017
Revised: 9 March 2018
First available in Project Euclid: 9 February 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 37C40: Smooth ergodic theory, invariant measures [See also 37Dxx] 37D35: Thermodynamic formalism, variational principles, equilibrium states
Constantine, D.; Lafont, J.-F.; McReynolds, D. B.; Thompson, D. J. Fat Flats in Rank One Manifolds. Michigan Math. J. 68 (2019), no. 2, 251--275. doi:10.1307/mmj/1549681300. https://projecteuclid.org/euclid.mmj/1549681300