Monotonicity of the first dirichlet eigenvalue of the laplacian on manifolds of non-positive curvature

For a complete Riemannian manifold (M,g) with nonpositive scalar curvature and a suitable domain Ω ⊂ M, let λ(Ω) be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on Ω. We obtain bounds for the rate of decrease of λ(Ω) as Ω increases, and a result comparing the rate of decrease of λ before and after a conformal diffeomorphism. Along the way, we obtain a reverse-Hölder inequality for the first eigenfunction, which generalizes results of Chiti to the manifold setting, and may be of independent interest.