Lemma 3.28 . [043U]

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Lemma 3.28.

Over the region $\{\frac{1}{6}A^{1/2}\leq|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\}$ ,

\left\lVert\tilde{g}^{(3)}-g^{(1)}\right\rVert_{C^{k,\alpha}_{0,0}}\leq C\nu A^{-3/4},\quad\left\lVert\Omega-\Omega_{\mathbb{C}^{3}}\right\rVert_{C^{k,\alpha}_{0,0}}\leq C\nu A^{-3/4},

and the volume form error $\tilde{E}^{(3)}$ of $\tilde{g}^{(3)}$ satisfies

\left\lVert\tilde{E}^{(3)}-\Delta_{a}\varphi_{3}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq C\nu^{2}A^{-3/4}.