Do we have compact Sobolev embeddings in complete noncompact Riemannian manifolds (under certain assumptions)?
In Hebey's book we see that for $n>kp$ the embedding $W^{k,p}(M)\hookrightarrow L^q(M)$ is compact for $q\in[1,p^*)$ where $p^*=\frac{np}{n-kp}$ and $M$ is a compact Riemannian $n$-manifold. Similarly, for Cartan-Hadamard manifolds with Ricci curvature bounded below we have continuous embeddings. Is there any equivalent compact embedding in case of complete noncompact Riemannian manifold under certain curvature assumptions? Any help is appreciated.