Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of non-positive curvature

Abstract

Let (M,g) be a complete Riemannian manifold with nonpositive scalar curvature, let ⊂ M be a suitable domain, and let λ() be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on . We prove several bounds for the rate of decrease of λ() as increases, and a result comparing the rate of decrease of λ before and after