Density of bounded maps in Sobolev spaces into complete manifolds

Given a complete noncompact Riemannian manifold $N^n$, we investigate whether the set of bounded Sobolev maps $(W^{1, p} \cap L^\infty) (Q^m; N^n)$ on the cube $Q^m$ is strongly dense in the Sobolev space $W^{1, p} (Q^m; N^n)$ for $1 \le p \le m$. 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 $N^n$. As a byproduct, we give necessary and sufficient conditions for the strong density of the set of smooth maps $C^\infty (\overline{Q^m}; N^n)$ in $W^{1, p} (Q^m; N^n)$.