Paper detail

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 $Ω\subset M$ be a suitable domain, and let $λ(Ω)$ be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on $Ω$. We prove several bounds for the rate of decrease of $λ(Ω)$ and $Ω$ increases, and a result comparing the rate of decrease of $λ$ 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