On asymptotically harmonic manifolds of negative curvature

Details On asymptotically harmonic manifolds of negative curvature On asymptotically harmonic manifolds of negative curvature

2012-03-12

arxiv.org/abs/1203.2482v1

Mathematics

Differential Geometry

24 p

Scientific paper

We study asymptotically harmonic manifolds of negative curvature, without any cocompactness or homogeneity assumption. We show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces: in particular, we determine the volume entropy, the spectrum and the relative densities of visual and harmonic measures on the ideal boundary. Then, we prove an asymptotic analogue of the classical mean value property of harmonic manifolds, and we characterize asymptotically harmonic manifolds, among Cartan-Hadamard spaces of strictly negative curvature, by the existence of an asymptotic equivalent $\tau(u)\ex^{Er}$ for the volume-density of geodesic spheres (with $\tau$ constant in case $DR_M$ is bounded). Finally, we show the existence of a Margulis function, and explicitly compute it, for all asymptotically harmonic manifolds.

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