Compact asymptotically harmonic manifolds

A complete Riemannian manifold without conjugate points is said to beasymptotically harmonic if the mean curvature of its horospheres is auniversal constant. Examples of asymptotically harmonic manifoldsinclude flat spaces and rank-one locally symmetric spaces ofnoncompact type. In this paper we show that this list exhausts thecompact asymptotically harmonic manifolds under a variety ofassumptions including nonpositive curvature or Gromov-hyperbolicfundamental group. We then present a new characterization of symmetricspaces amongst the set of all visibility manifolds.