Asymptotic behaviour of the Hodge Laplacian spectrum on graph-like manifolds (Q2361000)

Recall that a ``graph-like manifold'' consists of a family of compact connected \(m\)-dimensional Riemannian manifolds \(\{X_\varepsilon\}_{\varepsilon>0}\) made of building blocks for a metric graph which shrink to the given graph as \(\varepsilon\downarrow0\). Let \(\Delta_\varepsilon^p\) be the Laplacian on \(X_\varepsilon\) acting on \(p\)-forms. The decomposition of a \(p\)-form into an exact, co-exact, and harmonic form is preserved by the Laplacian; the exterior derivative intertwines the co-exact \(p\) forms and the exact \(p+1\) forms so it is only necessary to consider the exact \(p\)-forms. Assuming the \(X_\varepsilon\) are oriented, the Hodge Laplacian intertwines the Laplacian on \(p\) forms and on \(m-p\) forms. The authors examine the asymptotic behavior of the eigenvalues of the \(p\)-form value Laplacian on \(X_\varepsilon\) and produce manifolds with spectral gaps of arbitrarily large size in the spectrum of the Laplacian. Section 1 provides an introduction to the subject. The Laplacian of discrete and metric graphs is discussed in Section 2. Section 3 treats the eigenvalues of the Hodge Laplacian. Graph-like manifolds and their harmonic forms are discussed in Section 4. The main theorem is proved in Section 5 and examples are presented in section 6.