Inequalities between Dirichlet and Neumann Eigenvalues on Surfaces

TL;DR

This paper establishes inequalities between Dirichlet and Neumann eigenvalues on curved surfaces, extending Euclidean results to various geometries using a novel variational approach involving the Hodge Laplacian.

Contribution

It introduces a new comparison method for eigenvalues on curved surfaces using the variational principle of the Hodge Laplacian on 1-forms, generalizing previous Euclidean results.

Findings

Proved inequalities relating Neumann and Dirichlet eigenvalues on surfaces.

Extended eigenvalue comparison results to hyperbolic and minimal surfaces.

Derived strict inequalities for simply connected smooth surfaces.

Abstract

For a bounded Lipschitz domain $\Sigma$ in a Riemannian surface $M$ satisfying certain curvature condition, we prove that $$\mu_{3-\beta_1} \leq \lambda_{1},$$ where $\mu_k$ ($\lambda_k$ resp.) is the $k$-th Neumann (Dirichlet resp.) Laplacian eigenvalue on $\Sigma$ and $\beta_1$ is the first Betti number of $\Sigma.$ If $\Sigma$ is smooth and simply connected, we can further derive the strict inequality $ \mu_{3}< \lambda_{1}. $ This extends previous results on the Euclidean space to various curved surfaces, including the flat cylinder, the hyperbolic plane, hyperbolic cusp, collar, funnel, and minimal surfaces such as catenoid and helicoid. The novelty of the paper lies in comparing Dirichlet and Neumann Laplacian eigenvalues via the variational principle of the Hodge Laplacian on $1$-forms on a surface, extending the variational principle on vector fields in the Euclidean plane as developed by Rohleder. The comparison is reduced to the existence of a distance function with appropriate curvature conditions on its level sets.