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

الأصل · EN

Let (M,g) be a complete manifold of nonpositive scalar curvature, let Ω⊂ 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