A note on the first nonzero eigenvalue of the Laplacian acting on p- forms (Q921608)

Let \(M^ m\) be a smooth compact Riemannian manifold without boundary. Let \(\Delta_ p\) be the Laplacian on p forms and let \(\lambda_ 1(\Delta_ p,M)\) be the first non-zero eigenvalue. If \(\lambda_ 1(\Delta_ 0,M)\) is small, then \(M^ m\) looks like a dumb-bell which means Cheeger's isoperimetric constant \[ h(M)=\inf \{vol(\partial \Omega)/vol(\Omega):\;vol(\Omega)\leq vol(M)\} \] is small. This arises from the inequalities: \[ \frac{1}{4}h^ 2(M)\leq \lambda_ 1(\Delta_ 0,M)\leq C(m)(\delta h(M)+h^ 2(M)) \] where \(\delta\) is the lower bound of the Ricci curvature of \(M^ m\). h(M) does not control \(\lambda_ 1(\Delta_{p,M})\) for \(p>0\). The authors construct \(M^ m_ i\) for \(m\geq 3\) with \(h(M_ i)\geq \epsilon >0\) and \(\lambda_ 1(\Delta_ p,M_ i)\to 0\) for any \(p>0\). There are examples with bounded diameter and with diameter going to infinity; all \(M^ m_ i\) have bounded sectional curvature. The authors also discuss relationships with the injectivity radius and topology of \(M^ m\).