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July 2012, 6(3): 377-403. doi: 10.3934/jmd.2012.6.377
Compact asymptotically harmonic manifolds
Andrew M. Zimmer 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.
Keywords: geodesic flow, Asymptotically harmonic manifolds, horospheres..
Mathematics Subject Classification: Primary: 53C24, 53C35; Secondary: 37D4.
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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Andrew M. Zimmer