On compact subsets of Sobolev spaces on manifolds

Abstract

The paper considers compactness of Sobolev embeddings of non-compact manifolds, restricted to subsets (typically subspaces) defined either by conditions of symmetry (or quasisymmetry) relative to actions of compact groups, or by restriction in the number of variables, i.e. consisting of functions of the form f ∘ φ f\circ \varphi with a fixed φ \varphi . The manifolds are assumed to satisfy general common conditions under which Sobolev embeddings exist. We provide sufficient conditions for compactness of the embeddings, which in many situations are also necessary.