Small eigenvalues of the rough and Hodge Laplacians under fixed volume (Q6607745)

This is a very well written paper of just over 25 pages and 25 references which discusses small eigenvalues of the rough and Hodge Laplacians. It studies the eigenvalue problems of two elliptic differential operators acting on differential p-forms on a connected oriented closed Riemannian manifold of dimension \(m \geq 2\). Two Laplacians are considered. One is the rough Laplacian, or connection Laplacian, acting on \(p\)-forms n \((M,g)\) given in terms of the covariant derivative induced from the Levi-Civita connection of the Riemannian metric \(g\). The spectrum consists of only non-negative eigenvalues with finite multiplicity. The other is the Hodge- Laplacian acting on \(p\)-forms on \((M,g)\), given in terms of \(d\) the exterior derivative and \(\delta\) its formal adjoint with respect to the \(L^2\) inner product. The paper is interested for one thing in the supremum and infimum of the \(k\)-th eigenvalues under all Riemannian metrics with fixed volume on \(M\). Results which seem to be generalizations of the results of Colbois and Maerten in 2010 to the case of higher degree forms are presented.