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 $(W^{1,p}\cap L^{\mathrm{\infty }})(Q^m;N^n)$ on the cube $Q^m$ is strongly dense in the Sobolev space $W^{1,p}(Q^m;N^n)$ for $1 \le p \le m$\mathrm{.}Thedensityalwaysholdswhen$p$isnotaninteger\mathrm{.}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 $C^{\mathrm{\infty }}(\overline{Q^m};N^n)$ in $W^{1,p}(Q^m;N^n)$.