A Reilly formula and eigenvalue estimates for differential forms

Abstract

We derive a Reilly-type formula for differential p-forms on a compact manifold with boundary and apply it to give a sharp lower bound of the spectrum of the Hodge Laplacian acting on differential forms of an embedded hypersurface of a Riemannian manifold. The equality case of our inequality gives rise to a number of rigidity results, when the geometry of the boundary has special properties and the domain is non-negatively curved. Finally we also obtain, as a by-product of our calculations, an upper bound of the first eigenvalue of the Hodge Laplacian when the ambient manifold supports non-trivial parallel forms.

22 pages

References in corpus (1)

• Rigidity of compact Riemannian spin Manifolds with Boundary

Cited by in corpus (8)

• On the first eigenvalue of the Dirichlet-to-Neumann operator on forms
• On the best constant in {G}affney inequality
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• Estimates for higher Steklov eigenvalues
• The first nonzero eigenvalue of the $p$-Laplacian on Differential forms
• Cohomology vanishing theorems for free boundary submanifolds
• Eigenvalue estimates for the magnetic Hodge Laplacian on differential forms
• A Poincaré formula for differential forms and applications