  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 $W^{1, p} (Q^m; N^n)$.
