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 byproduct of our calculations, an upper bound of the first eigenvalue of the Hodge Laplacian when the ambient manifold supports non-trivial parallel forms.
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Communicated by Peter B. Gilkey.
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Raulot, S., Savo, A. A Reilly Formula and Eigenvalue Estimates for Differential Forms. J Geom Anal 21, 620–640 (2011). https://doi.org/10.1007/s12220-010-9161-0
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DOI: https://doi.org/10.1007/s12220-010-9161-0