Lemma 5.4 . [02HN]

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Lemma 5.4.

(i)

For all $i=1,\dots,n$ there exists a triple $\bm{\underline{a}}_{p_{i},\epsilon}$ of $1$ –forms on $H^{k_{i}}$ such that

$|\nabla^{k}\bm{\underline{a}}_{p_{i},\epsilon}|\leq C\epsilon^{3}\left(\frac{1}{\rho}\right)^{2+k}$

and $\epsilon^{2}\bm{\underline{\omega}}_{N_{i}}=\bm{\underline{\omega}}_{p_{i},\epsilon}+d\bm{\underline{a}}_{p_{i},\epsilon}$ .
(ii)

For all $j=1,\dots,8$ there exists a triple $\bm{\underline{a}}_{q_{j},\epsilon}$ of $\mathbb{Z}_{2}$ –invariant $1$ –forms on $H^{2m_{j}-4}$ such that

$|\nabla^{k}\bm{\underline{a}}_{q_{j},\epsilon}|\leq C\epsilon^{3}\left(\frac{1}{\rho}\right)^{2+k}$

and $\epsilon^{2}\bm{\underline{\omega}}_{M_{j}}=\bm{\underline{\omega}}_{q_{j},\epsilon}+d\bm{\underline{a}}_{q_{j},\epsilon}$ .

Proof.

The proof is identical in the two cases. Set $k=k_{i}$ in case (i) and $k=2m_{j}-4$ in case (ii). In case (ii) we work with $\mathbb{Z}_{2}$ –invariant forms on the double cover $H^{2m_{j}-4}$ .

By scaling we can assume that $\epsilon=1$ . It is enough to prove that every closed $2$ –form $\eta$ with $\eta=O(\rho^{-3})$ can be written as $\eta=da$ with $|\nabla^{k}a|=O(\rho^{-2-k})$ .

Since the restriction of $H^{k}$ to an exterior domain in $\mathbb{R}^{3}$ is diffeomorphic to $(R,\infty)\times\Sigma$ with $\Sigma$ an homology sphere, we can write $\eta=d\rho\wedge\alpha+\beta$ for some $\rho$ –dependent $1$ –form $\alpha$ and $2$ –form $\beta$ on $\Sigma$ with $|\alpha|+|\beta|=O(\rho^{-3})$ .

The condition $d\eta=0$ implies $\partial_{\rho}\beta-d_{\Sigma}\alpha=0$ . We then define $a=-\int_{\rho}^{\infty}{\alpha}$ . The Lemma follows. ∎