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June 2019 Fat Flats in Rank One Manifolds
D. Constantine, J.-F. Lafont, D. B. McReynolds, D. J. Thompson
Michigan Math. J. 68(2): 251-275 (June 2019). DOI: 10.1307/mmj/1549681300

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.

Citation

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D. Constantine. J.-F. Lafont. D. B. McReynolds. D. J. Thompson. "Fat Flats in Rank One Manifolds." Michigan Math. J. 68 (2) 251 - 275, June 2019. https://doi.org/10.1307/mmj/1549681300

Information

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

zbMATH: 07084762
MathSciNet: MR3961216
Digital Object Identifier: 10.1307/mmj/1549681300

Subjects:
Primary: 37D40
Secondary: 22E40, 37C40, 37D35

Rights: Copyright © 2019 The University of Michigan

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Vol.68 • No. 2 • June 2019
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