A Generalization of Cheng's Theorem

Abstract

0. Introduction. In this paper, we prove a generalization of a theorem of S.Y. Cheng on the upper bound of the bottom of the L spectrum for a complete Riemannian manifold. In [C], Cheng proved a comparison theorem for the first Dirichlet eigenvalue of a geodesic ball. By taking the radius of the ball to infinity, he obtained an estimate for the bottom of the L spectrum. In particular, he showed that if M is an n-dimensional complete Riemannian manifold whose Ricci curvature is bounded from below by −(n− 1)K for some constant K > 0, then the bottom of the L spectrum, λ1(M), is bounded by λ1(M) ≤ (n− 1)K 4 .