The spectrum of the Hodge Laplacian on a Riemannian manifold

Question

The Hodge Laplacian operator on differential forms on a (compact?) Riemannian manifold carries useful information about the topology of the manifold. In particular, the multiplicity of the zero eigenvalue is equal to the betti number.

Does the non-zero spectrum carry similarly useful topological information (in particular, independent of the metric)? More generally, what is known about the spectrum?

What is the spectrum for instance on classical examples like Euclidean space, the flat torus, the spheres or the riemann surfaces?

From the wiki page, I know that on a compact manifold, the spectrum is non negative but this is about the limit of my knowledge.

Answer 1

This is an old question but I guess I can contribute with some information about the computation of the eigenvalues. Actually, to compute explicitly the spectrum of Hodge-Laplacian turns out to be a Representation-Theoretic task when you are dealing with sufficiently good spaces (i mean compact irreducible Symmetric Spacaes) like Spheres, Projective spaces, Lie groups, Grassmannians, some flag manifolds, etc. If you would like to take a look I refer this paper Ikeda-Taniguichi which gives how to translate the problem into Rep theory and explicitly computes the eigenvalues and eigenforms in when the base manifold is the Sphere and the Complex Projective Space.

The ideas of the paper of Ikeda and Taniguichi can be applied in other several settings, e.g. for computing the spectrum of Dirac operators.

To finish I would like to comment how Representation theory enter the game. A symmetric space can be written on the form $M = G/K$ where $G\supset K$ are Lie groups with some extra conditions. In this form, $M$ has a natural left $G$-action which lifts to an linear action on $\Omega^k(M)$, turning the latter an infinite dimensional $G$-representation (this can be identified with the induced $G$-representation defined by the $K$-representation $\Lambda^k (\mathfrak{g}/\mathfrak{k})^\ast$). What is shown in [Ikeda-Taniguchi] is that, under certain hypothesis, the Hodge-Laplacian on $\Omega^k(M)$ can be identified with the Casimir operator of $G$ acting on the induced representation mentioned above. Casimir operator is a classical object in representation theory, and its eigenvelues can be computed by the so called Freudenthal formula.

Answer 2

This "book" might interest you: http://math.bu.edu/people/sr/articles/book.pdf I haven't found a book title or an author inside the book, but on page 35 they claim, that even calculating the eigenvalues/eigenforms for the non-flat 2-torus of the laplacian on 1-forms is basically impossible. I don't know whether that is something one would expect.

Unfortunately all I have is the above link, as I only got it from another thread: https://math.stackexchange.com/questions/617557/relation-of-hodge-theorem-to-eigenfunction-basis-of-laplacian

But maybe you will find other information inside the book.

They have a "Hodge Theorem 1.30 for the Laplacian on k-forms" on page 34 that says:

For a closed connected orientable Riemannian manifold the eigenvalues of the Laplace operator on k-forms are all non-negative. There exists an orthonormal basis of $L^2(\Omega^k(M))$ consisting of eigenfunctions (or lets call them eigenforms I guess) of the Laplacian on k-forms. The eigenspaces are all finite-dimensional and the eigenvalues accumulate only at infinity.

So the eigenspaces of the Laplacian on k-forms are finite dimensional. Is that common knowledge? I haven't read much more of "the book".

Somewhere else I read that the spectrum of the Laplacian only consists of its eigenvalues (common knowledge?): https://arxiv.org/pdf/1710.09579.pdf on page 11.

This short text might also interest you: https://math.berkeley.edu/~alanw/240papers03/chen.pdf starting on page 8, he explains that one can build up Morse-homology via a deformed Laplacian, where critical points of index k of a morse function correspond to eigenforms of the Laplacian on k-forms (the same k) and the exterior differential d on the eigenforms corresponds to the dual of the connecting-flow-lines counting boundary operator in Morse Homology.