Recall that from (4.13) we see that on $B\times\mathbb{C}^{n-m}$

$$\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}=p^{*}(\omega_{0}+\omega_{SF})+\sqrt{-1}\partial\overline{\partial}u_{t},$$

where the functions $u_{t}=\tilde{\varphi}_{t}-t\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\xi$ have uniform $C^{\infty}$ bounds on compact sets. We need to show that as $t$ goes to zero we have $u_{t}\to(f\circ p)^{*}\varphi$ in $C^{\infty}_{loc}(B\times\mathbb{C}^{n-m},\delta)$, where $f^{*}\varphi$ is the $C^{1,\alpha}$ limit of $\varphi_{t}$ from [38]. To prove this we need another estimate from the second-named author’s work [38, (3.9)], which implies that there is a constant $C$ (that depends on the initial choice of $B$) so that for all $0<t\leqslant 1$ we have

(4.21)

$$\sup_{y\in B}\mathrm{osc}_{M_{y}}\varphi_{t}\leqslant Ct.$$

We now use this together with the fact that $\varphi_{t}\to f^{*}\varphi$ in $C^{0}$ to get that for any $(y,z)$ in $B\times\mathbb{C}^{n-m}$ we have

$$\begin{split}|\tilde{\varphi}_{t}(y,z)-(f\circ p)^{*}\varphi(y,z)|&=\left|\varphi_{t}\circ T_{-\sigma}\circ p\left(y,\frac{z}{\sqrt{t}}\right)-\varphi(y)\right|\\ &\leqslant\left|\varphi_{t}\circ p\left(y,\frac{z}{\sqrt{t}}-\tilde{\sigma}(y)\right)-\varphi_{t}\circ p(y,z)\right|\\ &\ \ \ \ +|\varphi_{t}\circ p(y,z)-((f^{*}\varphi)\circ p)(y)|\\ &\leqslant Ct+\sup_{U}|\varphi_{t}-f^{*}\varphi|,\end{split}$$

where in the last line we used (4.21) because the points $p(y,\frac{z}{\sqrt{t}}-\tilde{\sigma}(y))$ and $p(y,z)$ lie in the same fiber $M_{y}$. Letting $t$ go to zero we see that $\tilde{\varphi}_{t}\to(f\circ p)^{*}\varphi$ in $C^{0}(B\times\mathbb{C}^{n-m})$. On the other hand we have that $t\lambda_{t}^{*}p^{*}\xi\to 0$ in $C^{0}(B\times\mathbb{C}^{n-m})$, and so $u_{t}\to(f\circ p)^{*}\varphi$ in $C^{0}(B\times\mathbb{C}^{n-m})$. Thanks to the higher order estimates for $u_{t}$, we also have that $u_{t}\to(f\circ p)^{*}\varphi$ in $C^{\infty}_{loc}(B\times\mathbb{C}^{n-m},\delta)$, up to shrinking $B$ slightly. ∎