On asymptotically harmonic manifolds of negative curvature

TL;DR

This paper investigates the geometric and spectral properties of asymptotically harmonic manifolds with negative curvature, revealing their asymptotic structure, measure relations, and the existence of a Margulis function.

Contribution

It provides new characterizations of asymptotically harmonic manifolds among negatively curved spaces and computes the Margulis function explicitly.

Findings

Determines volume entropy and spectrum of these manifolds.

Establishes an asymptotic mean value property for harmonic functions.

Proves the existence and explicit form of the Margulis function.

Abstract

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.