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.
Highlights
A complete Riemannian manifold X is called harmonic if about any point the mean curvature of a geodesic sphere of sufficiently small radius is constant
If X is a connected harmonic manifold without conjugate points, Szabo [Sza90] observed that X is “globally” harmonic: the mean curvature of any geodesic sphere is constant and this constant only depends on the radius of the sphere
By Szabo’s observation, every harmonic manifold without conjugate points is asymptotically harmonic and it is natural to ask if the class of asymptotically harmonic manifolds can be characterized
Summary
A complete Riemannian manifold X is called harmonic if about any point the mean curvature of a geodesic sphere of sufficiently small radius is constant. If X is a connected harmonic manifold without conjugate points, Szabo [Sza90] observed that X is “globally” harmonic: the mean curvature of any geodesic sphere is constant and this constant only depends on the radius of the sphere. One can consider the so-called asymptotically harmonic manifolds, these are the complete Riemannian manifolds without conjugate points such that the mean curvature of their horospheres is a universal constant. A complete Riemannian manifold M with universal Riemannian cover X is called asymptotically harmonic if there exists α ∈ R such that for all v ∈ SX the horofunction bv is C2 and ∆bv ≡ α. (4) X has purely exponential volume growth: let hvol be the volume growth entropy of X for each p ∈ X there exists a constant C > 0 such that for all R ≥ 1:
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