Spectrum, harmonic functions, and hyperbolic metric spaces

The main result of the paper says, in particular, that if M is a complete simply connected Riemannian manifold with Ricci curvature bounded from below and without focal points, which is also a hyperbolic metric space in the sense of Gromov, then the top λ of the L ²-spectrum of the Laplace-Beltrami operator Δ is negative, the Martin boundary of M corresponding to Δ is homeomorphic to the sphere at infinity S(∞), and the harmonic measures on S(∞) have positive Hausdorff dimensions. These generalize the results of [AS], [An1], [Ki], [KL] and [BK]. Moreover, if dim M=2, then in the presence of the other conditions the hyperbolicity is also necessary for λ<0. The machinery consists of a combination of geometrical and probabilistic means.