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This paper proves that the domain of the Laplacian, $\DEL,$ on a closedRiemannian manifold, $(M,g),$ is compactly embedded in $L^{2} (M) .$Particularly, the resolvent of the Laplacian, $(\DEL + 1)^{-1},$ is shown to becompactly embedded on the torus.

Rellich-Kondrakov embedding of the Laplacian resolvent on the torus