Geometric eigenvalue estimates of Kuttler-Sigillito type on differential forms

TL;DR

This work extends Kuttler–Sigillito-type spectral inequalities to differential forms on compact Riemannian manifolds with boundary. It introduces a novel biharmonic Steklov problem with Dirichlet-type boundary conditions (BSD2), proves ellipticity and discrete spectrum, and develops variational characterizations for the associated eigenvalues. A generalized Rellich identity for differential forms, together with curvature-based comparison theorems, yields sharp inequalities connecting the Steklov and Neumann spectra to higher-order BSD eigenvalues and geometric data. The results deepen the link between spectral theory and geometry for form-valued problems and extend curvature-free and curvature-dependent KS-type bounds to the realm of differential forms.

Abstract

We introduce a new biharmonic Steklov problem on differential forms with Dirichlet-type boundary conditions and show that it is elliptic. We prove the existence of a discrete spectrum for this problem and give variational characterizations for eigenvalues associated to it. We establish eigenvalue estimates known as Kuttler-Sigillito inequalities, that connect the eigenvalues of different problems on differential forms with curvature quantities on the manifold.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.7
• Theorem 1.8
• Conjecture 1.9
• Proposition 2.1
• Definition 2.2