For any sufficiently large $T\gg 1$, let $\mathcal{M}_{T}$ be the neck endowed with the $C^{2,\alpha}$-structure $(\omega_{T},\Omega_{T})$. Then there exists a constant $C_{N}>0$ independent of $T\gg 1$ such that for all

(6.32)

$$\varrho\in(0,\frac{1}{2})$$

and

(6.33)

$$\sqrt{-1}\partial\bar{\partial}\phi_{2}\in\overline{B_{\varrho}(\bm{0})}\subset\mathfrak{S}_{1},\quad\sqrt{-1}\partial\bar{\partial}\phi_{2}\in\overline{B_{\varrho}(\bm{0})}\subset\mathfrak{S}_{1},$$

we have the pointwise estimate

(6.34)

$\displaystyle\|\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_{1})-\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_{2})\|_{\mathfrak{S}_{2}}\leq C_{N}\cdot\varrho\cdot\|\sqrt{-1}\partial\bar{\partial}(\phi_{1}-\phi_{2})\|_{\mathfrak{S}_{1}}.$