Hodge-Laplacian Eigenvalues on Surfaces with Boundary

TL;DR

This paper generalizes inequalities between Neumann and Dirichlet eigenvalues to the Hodge-Laplacian on surfaces with boundary using differential forms, broadening the scope of spectral analysis on manifolds.

Contribution

It introduces a differential forms framework to extend eigenvalue inequalities from planar domains to Riemannian manifolds with boundary.

Findings

Eigenvalues of the Hodge-Laplacian are expressed as unions of spectra on closed and co-exact forms.

An inequality relating Hodge-Laplacian eigenvalues to Laplace-Beltrami eigenvalues is established.

Rohleder's inequalities are derived as special cases of a more general theorem.

Abstract

Recently Rohleder proposed a new variational approach to an inequality between the Neumann and Dirichlet eigenvalues in the simply connected planar case using the language of classical vector analysis. Writing his approach in terms of differential forms permits to generalize these results to a much broader context. The spectrum of the absolute boundary problem for the Hodge-Laplacian on a Riemannian manifold with boundary is presented as a union of the spectra of the absolute boundary problem on the spaces of closed and co-exact forms. An inequality for the eigenvalues of the absolute boundary problem for the Hodge-Laplacian and the Dirichlet boundary problem for the Laplace-Beltrami operator in the Euclidean case is obtained using this presentation. The Rohleder's results are obtained as corollaries of a more general theorem.