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Geometric properties of rank one asymptotically harmonic manifolds

Knieper, G.; Peyerimhoff, N.

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Authors

G. Knieper



Abstract

In this article we consider asymptotically harmonic manifolds which are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature h. We prove the following equivalences for asymptotically harmonic manifolds X under the additional assumption that their curvature tensor together with its covariant derivative are uniformly bounded: (a) X has rank one; (b) X has Anosov geodesic flow; (c) X is Gromov hyperbolic; (d) X has purely exponential volume growth with volume entropy equals h. This generalizes earlier results by G. Knieper for noncompact harmonic manifolds and by A. Zimmer for asymptotically harmonic manifolds admitting compact quotients.

Citation

Knieper, G., & Peyerimhoff, N. (2015). Geometric properties of rank one asymptotically harmonic manifolds. Journal of Differential Geometry, 100(3), 507-532. https://doi.org/10.4310/jdg/1432842363

Journal Article Type Article
Acceptance Date Mar 18, 2014
Online Publication Date May 28, 2015
Publication Date Jul 1, 2015
Deposit Date Nov 25, 2014
Publicly Available Date Dec 1, 2014
Journal Journal of Differential Geometry
Print ISSN 0022-040X
Electronic ISSN 1945-743X
Publisher International Press
Peer Reviewed Peer Reviewed
Volume 100
Issue 3
Pages 507-532
DOI https://doi.org/10.4310/jdg/1432842363
Publisher URL 10.4310/jdg/1432842363

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Copyright Statement
Copyright © International Press. First published in Journal of Differential Geometry in 100(3), 2015, published by International Press.




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