Geometric properties of rank one asymptotically harmonic manifolds

Knieper, G.; Peyerimhoff, N.

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
• Public URL: https://durham-repository.worktribe.com/output/1441187
• Publisher URL: 10.4310/jdg/1432842363