## The Michigan Mathematical Journal

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

### Fat Flats in Rank One Manifolds

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

#### 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 $k\ge 2$. 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**

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

**Zentralblatt MATH identifier**

07084762

**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