who proved discrete spectrum of Laplace-Beltrami on compact Riemannian manifolds?

Question

The fact (!) that the Laplace-Beltrami operator on compact (connected) Riemannian manifolds has compact resolvent, and therefore has purely discrete spectrum, is well known. So far as I can tell, the most common textbook or monograph discussions of things in this vein are treatments of Hodge theory, where most of the foundational analytical details and facts seem to be taken for granted.

It seems also to be well known that the idea of a proof is to smoothly cut up the given manifold into small-enough pieces that they can be continuously imbedded into Euclidean cubes, and, then, into products of circles. On the latter, with the usual flat Laplace-Beltrami operator, the theory of Fourier series easily gives a Rellich compactness lemma... which then gets transported back to the manifold.

Not that the proof details are necessarily interesting... but... who did such a thing first? I'm having great trouble following bibliographic links back to anything...

Answer 1

I believe that this was proved in full generality for the first time by Hilbert, in the series of papers Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, 1904-1910, later published as a book (Leipzig und Berlin: B. G. Teubner. XXVI u. 282 S.)

Many special cases were known earlier, in particular by Poincare.

Remark. However, it should be mentioned that the formal definition of a smooth manifold was only given by Veblen and Whitehead in 1931, see

Remmert, Reinhold, From Riemann surfaces to complex spaces. Matériaux pour l'histoire des mathématiques au XXe siècle (Nice, 1996), 203–241, Sémin. Congr., 3, Soc. Math. France, Paris, 1998.

So this is a case when the result was established long before the proper definitions were given, the thing which happens frequently.