The Hodge Laplacian, which generalises the Laplace-Beltrami operator on smooth functions to the framework of differential forms on Riemannian manifolds, is a central analytic tool in spectral geometry revealing geometric and topological features of the underlying space through its interaction with homology. In this talk, we present 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. We also discuss extensions to perforated domains in general. The proofs employ local-to-global arguments via an explicit isomorphism between Čech cohomology and de Rham cohomology to obtain Poincaré-type inequalities with explicit geometric dependence, together with certain generalised versions of the Cheeger-McGowan glueing lemma. This is a joint work with Pierre Guerini.
