Noncompact asymptotically harmonic manifolds

TL;DR

This paper establishes equivalences among geometric and dynamical properties of asymptotically harmonic manifolds with bounded curvature tensor and its derivative, extending previous results to a broader class of noncompact manifolds.

Contribution

It proves that for such manifolds, rank one, Anosov geodesic flow, Gromov hyperbolicity, and exponential volume growth are equivalent, generalizing earlier theorems.

Findings

Rank one implies Anosov geodesic flow

Gromov hyperbolicity is equivalent to volume growth

Volume entropy equals mean curvature h

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.