Eigenfunction basis of Laplacian on a manifold It is a well known result that for $\Omega$ bounded open set in $\mathbb{R}^n$, there exists a basis of $C^\infty$ eigenfunctions of the Laplacian for $L^2(\Omega)$. It is also known that there exists a basis of $C^\infty$ eigenfunctions of the Laplacian for $L^2(M)$, $M$ being a compact, connected Riemannian manifold. It can be proved with the fundamental solution of the heat equation (see http://www.math.harvard.edu/~canzani/math253/Lecture13.pdf). My question is : can the second result be proved with the first one ? In other words, can we use the eigenfunction basis of the spaces $L^2(U_i)$ ($U_i$ being the open sets in $\mathbb{R}^n$ associated to an atlas on $M$) to build the eigenfunction basis of $L^2(M)$ ? If anybody has an idea or a reference, that would be great.
Thanks !
 A: I think the answer is affirmative in some sense. By the spectral theorem for self-adjoint operators, the existence of a base of eigenfunctions follows from the compactness of the embeddings between Sobolev spaces (Rellich's lemma), wich is independent of the metric on compact manifolds. Rellich's lemma also holds on any open subset $U\subset\mathbb R^n$ considering only functions supported on a fixed compact subset $K=\overline{V}\subset U$, where $V$ is another open set. Usually, Rellich's lemma is checked on the torus using harmonic analysis, and then it is extended to compact manifolds using charts and partitions of unity (see e.g. Chapter 5 of Roe's book ``Elliptic operators, topology and asymptotic methods'', 2nd ed.). But, instead, you can assume Rellich's lemma is known for all $U$ and $K$ as above, apply this to a finite atlas $U_i$ of a compact manifold $M$ and an open covering of $M$ by sets $V_i$, where $K_i=\overline{V_i}\subset U_i$ is compact. Then, with the same arguments, you get Rellich's lemma for $M$, and therefore the base of eigenfunctions.
