On the lowest eigenvalue of the Hodge Laplacian
The article gives a lower bound for the least positive eigenvalue of the Hodge-Laplacian acting on \(p\)-forms in a compact \(C^\infty\) manifold \(\mathcal M\) without boundary. The bound is related to the parameters of a certain kind of coverings \(\{{\mathcal U}_i\}\) of \(\mathcal M\) by finitely many geodesically convex sets. The parameters are the number of sets \({\mathcal U}_i\), their geodesic radii and bounds on the differential of the exponential map in each \({\mathcal U}_i\). The estimate is combined with the Łojasiewicz inequality to show that the positive eigenvalues of the Hodge-Laplacian on the regular level sets of a \(C^\omega\) function \(\Phi:S^n\to\mathbb{R}\) are larger than a constant times a power of the distance between the value of \(\Phi\) and the set of the critical values of \(\Phi\).
scientific article; zbMATH DE number 1052961