UPPER BOUNDS FOR THE FIRST EIGENVALUE OF THE LAPLACE OPERATOR ON COMPLETE RIEMANNIAN MANIFOLDS

Abstract

Let M be a complete Riemannian manifold with infinite volume and Omega be a compact subdomain in M. In this paper we obtain two upper bound estimates for the first eigenvalue of the Laplacian on the punctured manifold MOmega subject to volume growth and lower bound of Ricci curvature, respectively. The proof hinges on asymptotic behavior of solutions of second order differential equations, the max-min principle and Bishop volume comparison theorem.