# American Institute of Mathematical Sciences

July  2012, 6(3): 377-403. doi: 10.3934/jmd.2012.6.377

## Compact asymptotically harmonic manifolds

 1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States

Received  May 2012 Published  October 2012

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.
Citation: Andrew M. Zimmer. Compact asymptotically harmonic manifolds. Journal of Modern Dynamics, 2012, 6 (3) : 377-403. doi: 10.3934/jmd.2012.6.377
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