Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of nonpositive curvature

Let $(M,g)$ be a complete manifold of nonpositive scalar curvature, let $\mathrm{\Omega }\subset M$ be a suitable domain, and let $\lambda (\mathrm{\Omega })$ be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on $\mathrm{\Omega }$. We prove several bounds for the rate of decrease of $\lambda (\mathrm{\Omega })$ and $\mathrm{\Omega }$ increases, and a result comparing the rate of decrease of $\lambda$ before and after a conformal diffeomorphism. Along the way, we prove a reverse-Holder inequality for the first eigenfunction, which generalizes results of Chiti to the monifold setting and may be of independent interest