Hence $‾∇\hat n$ is in $L^p$. Since also $\abs{\hat n}=1$, $\hat n ∈ W^{1,p}(M, 𝕊^{N-1})$ and the norm is bounded by the norm of $\hat Q$.

Hence $‾∇\hat n$ is in $L^p$. Since also $\abs{\hat n}=1$, $\hat n ∈ W^{1,p}(M, 𝕊^{N-1})$, the norm of $\hat n$is bounded by the norm of $\hat Q$.

Since $(Q_{(k)})_k$ converges weakly, it is a bounded sequence by the Uniform Boundedness Principle.

Therefore the previous calculation \eqref{eq:∇ninQterms} shows that $(n_{(k)})_k$ is bounded as well.

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@@ -84,10 +84,10 @@ If we only used the intrinsic definition of the tensor fields it would be unclea

If we used it with $N = 𝒬M$, we would not have the guarantee that $u^{(k)}_p ∈ T_pM$. If we used it with $N = 𝒬_pM$ we have to somehow identify all tangent spaces which is -- in the general case -- not possible in a smooth way.

By embedding $M$ and considering non-tangent tensor fields we solve this problem but need \cref{thm:orientability_on_simply_connected_manifolds} and