Necessity of compactness of manifolds $M,N$ for smooth approximation of $W^{1,p}(M,N)$

I'm currently studying the Sobolev space $W^{1,p}(M,N)$ between manifolds $M,N$. One result by Schoen & Uhlenbeck is existence of approximation through $C^\infty(M,N)$-functions, if $M$ and $N$ are smooth and compact and $p \geq \dim M$.

A raw sketch of the result goes as follows. We use the embedding (by Whitney/Nash) of the target manifold $N$ into a Euclidean space. (Here we use smoothness.) Now we consider the smooth convolution approximation which is in $C^\infty(M, \mathbb{R}^k)$. A Poincaré inequality will show that the approximating sequence lies in some tubular neighbourhood of $N$ (here we use $p \geq \dim M$) and thus can be smoothly projected for the projected approximation to be in $C^\infty(M,N)$.

Where is compactness of the manifolds needed? Is it something in the proof? Or is something without compactness not welldefined at all?

I'd appreciate any help.

Note: Crossposted in Mathstackexchange.

• Poincare inequalities often require some form of compactness. I cannot be more precise than this since I don't know what kind Poincare inequality is used. – Liviu Nicolaescu Sep 22 '16 at 16:54
• @LiviuNicolaescu The Poincaré inequality is only used locally on some arbitrarily small ball. $\int_{B_\epsilon(x_0)} | u(x) - u(y) |^p dx \leq C r^{n+p-1} \int_{B_\epsilon(x_0)} |\nabla u|^p |x - y|^{1-n} dx$ is the inequality used here. – Nhat Sep 22 '16 at 17:00
• Are you talking about the construction in this paper? – Arctic Char Sep 24 '16 at 22:30