Eigenvalues of the Laplacian on Forms

Small eigenvalues of the Witten Laplacian acting on p-forms on a surface

In this article, we are interested in the exponentially small eigenvalues of the self adjoint realization of the semiclassical Witten Laplacian ∆ f,h, in the general framework of p-forms, on a connected compact Riemannian manifold without boundary. Our purpose is to notice that the knowledge of (the asymptotic formulas for) the smallest non zero eigenvalues of the self adjoint realization of ∆ ...

Eigenvalues of the Laplacian acting on p - forms and metric conformal deformations

Let (M,g) be a compact connected orientable Riemannian manifold of dimension n ≥ 4 and let λk,p(g) be the k-th positive eigenvalue of the Laplacian ∆g,p = dd +dd acting on differential forms of degree p on M . We prove that the metric g can be conformally deformed to a metric g, having the same volume as g, with arbitrarily large λ1,p(g ) for all p ∈ [2, n − 2]. Note that for the other values o...

Eigenvalues of the normalized Laplacian

A graph can be associated with a matrix in several ways. For instance, by associating the vertices of the graph to the rows/columns and then using 1 to indicate an edge and 0 otherwise we get the adjacency matrix A. The combinatorial Laplacian matrix is defined by L = D − A where D is a diagonal matrix with diagonal entries the degrees and A is again the adjacency matrix. Both of these matrices...

Eigenvalues of the Conformal Laplacian

We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimension at least three and we give lower and upper bounds in terms of the α-genus.

Bibliography on the Signless Laplacian Eigenvalues