Knieper, G. and Peyerimhoff, N. (2015) 'Geometric properties of rank one asymptotically harmonic manifolds.', Journal of differential geometry., 100 (3). pp. 507-532.
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.
|Full text:||(AM) Accepted Manuscript|
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|Publisher Web site:||10.4310/jdg/1432842363|
|Publisher statement:||Copyright © International Press. First published in Journal of Differential Geometry in 100(3), 2015, published by International Press.|
|Date accepted:||18 March 2014|
|Date deposited:||01 December 2014|
|Date of first online publication:||28 May 2015|
|Date first made open access:||No date available|
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