On the first eigenvalue of the Dirichlet-to-Neumann operator on forms

TL;DR

This paper investigates the first eigenvalue of the Dirichlet-to-Neumann operator on differential forms, providing bounds, characterizations of equality, and relations to other spectral operators on manifolds.

Contribution

It generalizes the classical Steklov problem to differential forms and derives bounds and relations for the first eigenvalue in this new context.

Findings

Derived upper and lower bounds for the first eigenvalue.

Characterized cases of equality in the bounds.

Connected the eigenvalues to those of the Hodge Laplacian and biharmonic Steklov operator.

Abstract

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of these estimates will be sharp, and for some of them we characterize equality. We also relate these new eigenvalues with those of other operators, like the Hodge Laplacian or the biharmonic Steklov operator.