A Reilly formula and eigenvalue estimates for differential forms

TL;DR

This paper develops a Reilly-type formula for differential forms on manifolds with boundary, providing sharp eigenvalue bounds for the Hodge Laplacian and establishing rigidity results under specific geometric conditions.

Contribution

It introduces a new Reilly formula for differential forms and applies it to derive eigenvalue estimates and rigidity results for manifolds with boundary.

Findings

Sharp lower bounds for the Hodge Laplacian spectrum on hypersurfaces

Rigidity results under non-negative curvature and boundary conditions

Upper bounds for the first eigenvalue in manifolds with parallel 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.