Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of non-positive curvature Tom CarrollJesse Ratzkin 58J50 (31C12 53C21)Dirichlet eigenvaluenon-positive curvatureSchwarz Lemma For a complete Riemannian manifold $(M,g)$ with nonpositive scalar curvature and a suitable domain $\Omega\subset M$, let $\lambda(\Omega)$ be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on $\Omega$. We obtain bounds for the rate of decrease of $\lambda(\Omega)$ as $\Omega$ increases, and a result comparing the rate of decrease of $\lambda$ before and after a conformal diffeomorphism. Along the way, we obtain a reverse-H"older inequality for the first eigenfunction, which generalizes results of Chiti to the manifold setting, and may be of independent interest. Indiana University Mathematics Journal 2016 text pdf 10.1512/iumj.2016.65.5757 10.1512/iumj.2016.65.5757 en Indiana Univ. Math. J. 65 (2016) 353 - 376 state-of-the-art mathematics http://iumj.org/access/