Some sharp Hodge Laplacian and Steklov eigenvalue estimates for differential forms

TL;DR

This paper establishes sharp lower bounds for the first eigenvalue of the Hodge Laplacian and Steklov eigenvalues on differential forms, advancing spectral geometry understanding.

Contribution

It provides new sharp estimates for eigenvalues of differential forms, improving bounds for both Hodge Laplacian and Steklov problems on manifolds.

Findings

Sharp lower bounds for the first Hodge Laplacian eigenvalue

Sharp estimates for the first nonzero Steklov eigenvalue

Enhanced understanding of spectral properties of differential forms

Abstract

We give some sharp lower bounds of the first eigenvalue for the Hodge Laplacian acting on differential forms on the boundary of a Riemannian manifold. We also give some sharp estimates for the first nonzero Steklov eigenvalue for differential forms.