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 0 of its unverversal covering satis…es 0 (n 1) =4. We prove that equality holds i¤ M is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy. 1. Introduction Complete Riemannian manifolds with nonnegative Ricci curvature have been intensively studied by many people and there are various methods and many beautiful results (see e.g., the book [P]). One of the most important theorems on such manifolds is the following Cheeger-Gromoll splitting theorem: Theorem 1. (Cheeger-Gromoll) If (N; g) contains a line and has Ric 0, then (N; g) is isometric to a product R ; dt + h . This theorem has the following important corollaries on the structure of manifolds with nonnegative Ricci curvature: A complete Riemannian (N; g) with Ric 0 either has only one end or is isometric to a product R ; dt + g , with compact. If (M; g) is compact with Ric 0 then its universal covering f M splits isometrically as a product R n , where is a simply connected compact manifold with Ric 0. If furthermore f M has Euclidean volume growth, then f M is isometric to R. Riemannian manifolds with a negative lower bound for Ricci curvature are considerably more complicated and less understood. It is naive to expect such splitting results in general. Nevertheless there have been very interesting results due to Li and J. Wang recently. It has been discovered that the bottom of the L spectrum plays an important role (see also the earlier work [W] in the conformally compact case). Let us assume that (N; g) is a complete Riemannian manifold with Ric (n 1). The bottom of the L spectrum of the Laplace operator on functions is denoted by 0 (N) and can be characterized as 0 (N) = inf R N jruj R N u2 ; 1991 Mathematics Subject Classi…cation. 53C24, 31C05, 58J50.