  We give a proof of the Donnelly-Fefferman growth bound of Laplace-Beltrami
eigenfunctions which is probably the easiest and the most elementary one. Our
proof also gives new quantitative geometric estimates in terms of curvature
bounds which improve and simplify previous work by Garofalo and Lin. The proof
is based on a convexity property of harmonic functions on curved manifolds,
generalizing Agmon's Theorem on a convexity property of harmonic functions in
R^n.
