TITLE: Eigenvalues of the Laplace-Beltrami operator on a compact Riemannnian manifold
QUESTION [16 upvotes]: 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?

REPLY [11 votes]: 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.