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Geometric properties of rank one asymptotically harmonic manifolds.

Knieper, G. and Peyerimhoff, N. (2015) 'Geometric properties of rank one asymptotically harmonic manifolds.', Journal of differential geometry., 100 (3). pp. 507-532.

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.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
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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