Proposition 4.18 . [045K]

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Proposition 4.18.

Via the local diffeomorphism $\Psi$ , on the chart $\{r\lesssim A^{1/4}\}$ ,

\begin{cases}|\nabla_{g_{a}}\{\Psi^{*}(w^{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q})-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}d\xi_{1}\otimes d\bar{\xi}_{1}\}|_{g_{a}}\leq CA^{-1},\\ |\nabla_{g_{a}}\{\Psi^{*}v-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}\}|_{g_{a}}\leq C.\end{cases}

Furthermore

\begin{cases}|\nabla^{2}_{g_{a}}\{\Psi^{*}(w^{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q})-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}d\xi_{1}\otimes d\bar{\xi}_{1}\}|_{g_{\text{NUT}}}\leq CA^{-3/4}\ell^{-2},\\ |\nabla^{2}_{g_{a}}\{\Psi^{*}v-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}\}|_{g_{\text{NUT}}}\leq CA^{1/4}\ell^{-2}.\end{cases}