Lower bounds for the eigenvalues of the Hodge Laplacian on certain non-convex domains

Abstract

We establish geometric lower bounds for the smallest positive eigenvalue of the Hodge Laplacian in the class of non-convex domains given by Euclidean annular regions with a convex outer boundary and a spherical inner boundary. These bounds are then extended to convex domains with multiple holes, where we derive lower bounds for certain higher order exact eigenvalues, and under additional geometric assumptions, also for the smallest positive eigenvalue. For $1$-forms on compact manifolds with boundary, we provide a general lower bound on the smallest exact eigenvalue - corresponding to the first positive Neumann eigenvalue - which, in certain respects, is better than the classical Cheeger inequality. Furthermore, we emphasise the necessity of the "contact radius" in the lower bounds of the main results. Our proofs employ local-to-global arguments via an explicit isomorphism between \v{C}ech cohomology and de Rham cohomology to obtain Poincar\'e-type inequalities with explicit geometric dependence, and utilise certain generalised versions of the Cheeger-McGowan gluing lemma.