IUMJ

Title: Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of non-positive curvature

Authors: Tom Carroll and Jesse Ratzkin

Issue: Volume 65 (2016), Issue 1, 353-376

Abstract:

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.