  Let $(M,g)$ be a complete manifold of nonpositive scalar curvature, let
$\Omega\subset M$ be a suitable domain, and let $\lambda(\Omega)$ be the first
Dirichlet eigenvalue of the Laplace-Beltrami operator on $\Omega$. We prove
several bounds for the rate of decrease of $\lambda(\Omega)$ and $\Omega$
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
