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July 2015 Geometric properties of rank one asymptotically harmonic manifolds
Gerhard Knieper, Norbert Peyerimhoff
J. Differential Geom. 100(3): 507-532 (July 2015). DOI: 10.4310/jdg/1432842363


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.


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Gerhard Knieper. Norbert Peyerimhoff. "Geometric properties of rank one asymptotically harmonic manifolds." J. Differential Geom. 100 (3) 507 - 532, July 2015.


Published: July 2015
First available in Project Euclid: 28 May 2015

zbMATH: 1327.53052
MathSciNet: MR3352797
Digital Object Identifier: 10.4310/jdg/1432842363

Rights: Copyright © 2015 Lehigh University


Vol.100 • No. 3 • July 2015
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