| We explore the geometric rigidity of negatively curved homogeneous spaces. We characterize negatively curved symmetric spaces by a necessary condition involving their hyperbolic rank, and we present an example of a higher hyperbolic rank manifold which is not symmetric. We also characterize asymptotic harmonicity in terms of various natural measures. We show that these spaces are distinguished from compact manifolds in that their Bowen-Margulis, harmonic, and Liouville measures on the unit tangent bundle along with their corresponding measures on the boundary are always in the same measure class. We then show that Cheeger's constant, the Kaimanovich entropy, and the bottom of the spectrum of the Laplacian are all maximal for these spaces. Along the way we present sharp asymptotic estimates for Jacobi fields and the Poisson and Green's kernels. Finally, we present examples showing that in general these manifolds are not asymptotically harmonic. |
