Lemma 3.7 . [03H0]

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Lemma 3.7.

There exists a triple of $1$ -forms $\bm{a}$ with $\partial_{z}\,\lrcorner\,\bm{a}=0$ such that

(3.33)

\displaystyle\Phi^{*}\bm{\omega}_{TY}-\bm{\omega}_{\mathcal{C}}=d\bm{a}

and such that for all $k\geq 0$ and all $\epsilon>0$ ,

(3.34)

\displaystyle|\nabla^{k}_{g_{\mathcal{C}}}\bm{a}|_{g_{\mathcal{C}}}=O(e^{-(\underline{\delta}-\epsilon)z}).