Hcfyk

Let $(M,g)$ be a compact Riemannian manifold, and let $Delta_g$ be its Laplace-Beltrami operator. A "well-known fact" is that the eigenvalues of $Delta_g$ have finite multiplicity and tend to infinity. What is the easiest/simplest way to estblish this fact? By simplest, I mean an abstract, formal, approach to the problem, not too heavily based on technical calculation. In the same line of thought, is there a philosophical/intuitive reason to see why the spectrum should behave like this?

A modern "simple philosophical" explanation is that this problem can be restated as an eigenvalue problem for a compact operator in an appropriate Hilbert space, whose eigenvalues are reciprocal to those of the Laplacian. Then two properties that you stated follow from the general properties of compact operators.

This approach is due to Hilbert. He wrote a long series of papers on the subject in 1904-1912.
(see also his book with Courant, Methods of mathematical physics). Modern expositions are usually based on Hilbert's ideas. The notions of Hilbert space and compact operator were essentially distilled from these works.

An earlier philosophy of Poincaré interprets these eigenvalues as poles of certain
meromorphic function in the plane (in modern language it is essentially the resolvent), and the poles of a meromorphic function are isolated and tend to infinity. Poincaré was the first to prove under general conditions the existence of an infinite sequence of eigenvalues tending to infinity.
(Sur les équations de la physique mathématique, Rend. Circ. mat. Palermo, 1894 8, 57-155.)

Three remarks should be made:

a) At the time of Poincaré and Hilbert, the modern formal notion of compact Riemannian manifold did not exist. (The notion of compact was introduced by Aleksandrov and Urysohn in 1924, and the notion of manifold by Weyl 1913, and only for dimension 2). Even in the classical book on the subject by Titchmarsh, Eigenfunction expansions..., 1958, the words "manifold" and "compact" are not mentioned!

b) There was a very large number of problems about vibrations which were solved "explicitly" in 18th and 19th century. So Hilbert and Poincaré had a lot of "empirical material" to generalize. Fourier should be mentioned: his work inspired Hilbert and Poincare. He solved many concrete eigenvalue problems but had no tools to attack the general problem.

c) As a physical fact, existence of infinite discrete spectrum was first discovered (for the case of a string) by a music theorist Marin Mersenne in
1637. This started a long story of research about these eigenvalues.

What follows is only an answer to the philosophical/intuitive question and follows only one specific point of view (there are probably many others).

Eigenvalue problem for the Laplace operator on a Riemannian manifold $(M,g)$ is a quantization of the problem of the classical motion of a particle "freely moving", i.e. following geodesics, on $(M,g)$. More precisely, the phase space of this classical mechanics problem is the symplectic manifold $T^{*}M$ (cotangent bundle, with its standard symplectic form) and the Hamiltonian is the function on $T^{*}M$ given by $H=|p|_g^2/2$, where $p$ is linear form on cotangent fibers.

Eigenspaces of the Laplace operator with eigenvalues less than E are quantization of the part of the phase space with $H <E$. If M is compact, then $H<E$ is a subset of $T^{*}M$ of finite volume and so its quantization will produce a finite dimensional vector space (of dimension roughly the volume in units of Planck constant $hbar$): in particular, there will be finitely many eigenvalues below $E$, all with finitely many multiplicities. When $E$ goes to infinity, the volume of $H<E$ goes to infinity and so eigenvalues go to infinity.

In fact, this picture tells you that the number of eigenvalues less than $E$ should be of the order of the volume of the set $H<E$, which is of order $vol(M,g) E^{dim(M)/2}$, where $vol(M,g)$ is the volume of $(M,g)$, and $dim(M)$ is the dimension of $n$. This is indeed true (Weyl law).

(Maybe a trivial comment, but one never knows: it is probably helpful to think about the case of the circle, and more generally flat tori, where everything is trivial Fourier analysis, before thinking about more general situations).