There exist smooth triples of $1$-forms $\bm{a}^{\pm}\in\Omega^{1}(DZ_{\pm})\otimes\mathbb{R}^{3}$ satisfying

(6.52)

$\displaystyle\bm{\omega}^{\pm}+d\bm{a}^{\pm}=\Psi_{\pm}^{*}\bm{\omega}^{N}\mbox{ in }DZ_{\pm},$

such that for any $k\in\mathbb{N}$,

(6.53)

$\displaystyle|\nabla^{k}\bm{a}^{\pm}|\leq C_{k}e^{-\delta z_{\pm}}\mbox{ in }DZ_{\pm},$

where $\delta>0$ and $C_{k}>0$ are uniform constants independent of $\beta$.