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 $M \setminus \Omega$ 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.