Eigenvalue and gap estimates for the Laplacian acting on 𝑝-forms

Abstract

We study the gap of the first eigenvalue of the Hodge Laplacian acting on p p -differential forms of a manifold with boundary, for consecutive values of the degree p p . We first show that the gap may assume any sign. Then we give sufficient conditions on the intrinsic and extrinsic geometry to control it. Finally, we estimate the first Hodge eigenvalue of manifolds whose boundaries have some degree of convexity.