Rellich-Kondrakov embedding of the Laplacian resolvent on the torus

TL;DR

This paper demonstrates the compact embedding of the Laplacian's domain into L^2 on a closed Riemannian manifold, specifically proving the compactness of the Laplacian resolvent on the torus, extending classical embedding results.

Contribution

It establishes the compact embedding of the Laplacian domain into L^2 on the torus, providing new insights into spectral properties of the Laplacian on compact manifolds.

Findings

The Laplacian domain is compactly embedded in L^2 on closed Riemannian manifolds.

The resolvent of the Laplacian is compact on the torus.

Extension of Rellich-Kondrakov embedding to the Laplacian resolvent.

Abstract

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