## The Michigan Mathematical Journal

### Fat Flats in Rank One Manifolds

#### Abstract

We study closed nonpositively curved Riemannian manifolds $M$ that admit “fat $k$-flats”; that is, the universal cover $\tilde{M}$ 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 $k\geq 2$. 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
Revised: 9 March 2018
First available in Project Euclid: 9 February 2019

https://projecteuclid.org/euclid.mmj/1549681300

Digital Object Identifier
doi:10.1307/mmj/1549681300

Mathematical Reviews number (MathSciNet)
MR3961216

Zentralblatt MATH identifier
07084762

#### 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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