On the defect of compactness in Sobolev embeddings on Riemannian manifolds

The defect of compactness for an embedding $ E\hookrightarrow F$ of two Banach spaces is the difference between a weakly convergent sequence in $ E$ and its weak limit, taken modulo terms vanishing in $ F$. We discuss the structure of the defect of compactness for (noncompact) Sobolev embeddings on manifolds, giving a brief outline of the theory based on isometry groups, followed by a summary of recent studies of the structure of bounded sequences without invariance assumptions.