The Michigan Mathematical Journal

Fat Flats in Rank One Manifolds

D. Constantine, J.-F. Lafont, D. B. McReynolds, and D. J. Thompson

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Abstract

We study closed nonpositively curved Riemannian manifolds M that admit “fat k-flats”; that is, the universal cover M˜ contains a positive-radius neighborhood of a k-flat on which the sectional curvatures are identically zero. We investigate how the fat k-flats affect the cardinality of the collection of closed geodesics. Our first main result is to construct rank 1 nonpositively curved manifolds with a fat 1-flat that corresponds to a twisted cylindrical neighborhood of a geodesic on M. As a result, M contains an embedded closed geodesic with a flat neighborhood, but M 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 M with a fat k-flat contains an immersed, totally geodesic k-dimensional flat closed submanifold. This guarantees the existence of uncountably many closed geodesics when k2. Finally, we collect results on thermodynamic formalism for the class of manifolds considered in this paper.

Article information

Source
Michigan Math. J., Volume 68, Issue 2 (2019), 251-275.

Dates
Received: 17 April 2017
Revised: 9 March 2018
First available in Project Euclid: 9 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1549681300

Digital Object Identifier
doi:10.1307/mmj/1549681300

Mathematical Reviews number (MathSciNet)
MR3961216

Subjects
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

Citation

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


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