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

TL;DR

This paper investigates how the first Dirichlet eigenvalue of the Laplacian on manifolds with nonpositive curvature decreases as the domain expands, providing bounds and comparison results, including under conformal transformations.

Contribution

It introduces new bounds for the eigenvalue's rate of decrease, compares these rates before and after conformal maps, and generalizes a reverse-Holder inequality to the manifold setting.

Findings

Bounds for the decrease rate of eigenvalues as domains grow

Comparison of eigenvalue decrease rates under conformal diffeomorphisms

A generalized reverse-Holder inequality for eigenfunctions

Abstract

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