I'm in desperate need for help understanding a counterexample for the assertion that for (appropriate) manifolds $M,N$ the space $C^\infty (M,N)$ is dense in $L^p (M,N)$ if $\dim(M) > p$. (The statement holds true if $\dim(M) \leq p$.)
Here is the counterexample by Schoen & Uhlenbeck:
The map $u \in L^2_1(B^3_1, S^2)$ given by $u(x) = \frac{x}{\lvert x \rvert}$ is not an $L^2_1$ limit of a sequence $u_i \in C^\infty(B_1^3, S^2).$ To see this one can observe that if such a sequence $u_i$ did exist, then for almost every $r \in (\frac{1}{2}, 1)$ we would have $L^2(\partial B^3_r, S^2)$ convergence of $u_i$ to the map $\frac{x}{r}$. In particular, we would have a sequence $v_i \in C^\infty(S^2, S^2)$ (say $v_i(x) = u_i(rx))$, each $v_i$ having degree zero, converging to the identity map of $S^2$. By taking a subsequence we could assume $dv_i$ converges pointwise a.e. to the identity. Thus in particular the Jacobian $J(v_i) \rightarrow 1$ a.e. on $S^2$. Since $\lvert J(v_i) \rvert \leq \frac{1}{2} \lvert dv_i \rvert ^2$ and the latter converges in $L_1$ norm to a limit, the dominated convergence theorem implies that $$\lim_{i \to \infty} \int_{S^2}J(v_i) = 4 \pi.$$ The fact that each $v_i$ has degree zero implies that the integral of $J(v_i)$ is zero for each $i$, a contradiction.
I have two questions: I think I have a problem thinking in these spaces... Why is the degree of $v_i$ $0$? And furthermore how do I get the convergence of $v_i$'s derivative?
I would be really grateful for anyhelp.