Density of bounded maps in Sobolev spaces into complete manifolds

Given a complete noncompact Riemannian manifold , we investigate whether the set of bounded Sobolev maps on the cube is strongly dense in the Sobolev space for . The density always holds when p is not an integer. When p is an integer, the density can fail, and we prove that a quantitative trimming property is equivalent with the density. This new condition is ensured, for example, by a uniform Lipschitz geometry of . As a by-product, we give necessary and sufficient conditions for the strong density of the set of smooth maps in .