Journal of Modern Dynamics (JMD)

Compact asymptotically harmonic manifolds

Pages: 377 - 403, Issue 3, July 2012      doi:10.3934/jmd.2012.6.377

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Andrew M. Zimmer - Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States (email)

Abstract: A complete Riemannian manifold without conjugate points is said to be asymptotically harmonic if the mean curvature of its horospheres is a universal constant. Examples of asymptotically harmonic manifolds include flat spaces and rank-one locally symmetric spaces of noncompact type. In this paper we show that this list exhausts the compact asymptotically harmonic manifolds under a variety of assumptions including nonpositive curvature or Gromov-hyperbolic fundamental group. We then present a new characterization of symmetric spaces amongst the set of all visibility manifolds.

Keywords:  Asymptotically harmonic manifolds, geodesic flow, horospheres.
Mathematics Subject Classification:  Primary: 53C24, 53C35; Secondary: 37D40.

Received: May 2012;      Available Online: October 2012.