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

Abstract

Let $(M^n,g)$ be a compact Riemannian manifold with $Ric\ge -(n-1)$. It is well known that the bottom of spectrum ${\lambda }_0$ of its unverversal covering satisfies ${\lambda }_0\le (n-1{)}^2/4$. We prove that equality holds iff $M$ is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy.

to appear in J. Geom. Anal