Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space

Let (Mn,g) be a compact Riemannian manifold with Ric ≥−(n−1). It is well known that the bottom of spectrum λ0 of its universal covering satisfies λ0≤(n−1)2/4. We prove that equality holds iff M is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy.