First of all notice that after replacing $U$ with a slightly smaller open set, the semi-flat metric $\omega_{0}+\omega_{SF}$ is uniformly equivalent to $\omega_{M}$, which implies that

(4.4)

$$C^{-1}(\omega_{0}+t\omega_{SF})\leqslant\omega_{0}+t\omega_{M}\leqslant C(\omega_{0}+t\omega_{SF}),$$

for all small $t>0$. Thanks to Lemma 4.1 on $U$ we have that

$$C^{-1}(\omega_{0}+tT_{-\sigma}^{*}\omega_{M})\leqslant T_{-\sigma}^{*}\tilde{\omega}_{t}\leqslant C(\omega_{0}+tT_{-\sigma}^{*}\omega_{M}),$$

and since $T_{-\sigma}^{*}\omega_{M}$ is uniformly equivalent to $\omega_{M}$ we also have that

(4.5)

$$C^{-1}(\omega_{0}+t\omega_{M})\leqslant T_{-\sigma}^{*}\tilde{\omega}_{t}\leqslant C(\omega_{0}+t\omega_{M}),$$

and combining (4.4) and (4.5) we get

(4.6)

$$C^{-1}(\omega_{0}+t\omega_{SF})\leqslant T_{-\sigma}^{*}\tilde{\omega}_{t}\leqslant C(\omega_{0}+t\omega_{SF}),$$

on $U$. If we pull back (4.6) by $p\circ\lambda_{t}$ we get

(4.7)

$$C^{-1}(p^{*}\omega_{0}+t\lambda_{t}^{*}p^{*}\omega_{SF})\leqslant\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}\leqslant C(p^{*}\omega_{0}+t\lambda_{t}^{*}p^{*}\omega_{SF}),$$

on all of $B\times\mathbb{C}^{n-m}$. We claim that on the whole of $B\times\mathbb{C}^{n-m}$ we have that

(4.8)

$$t\lambda_{t}^{*}p^{*}\omega_{SF}=p^{*}\omega_{SF}.$$

In fact, the construction of $\omega_{SF}$ in section 3 gives that $p^{*}\omega_{SF}=\sqrt{-1}\partial\overline{\partial}\eta,$ for a function $\eta$ on $B\times\mathbb{C}^{n-m}$ that satisfies

(4.9)

$$\eta\circ\lambda_{t}(y,z)=\eta\left(y,\frac{z}{\sqrt{t}}\right)=\frac{1}{t}\eta(y,z),$$

for all $(y,z)$ in $B\times\mathbb{C}^{n-m}$ and any $t>0$. It follows then that

(4.10)

$$t\lambda_{t}^{*}p^{*}\omega_{SF}=t\lambda_{t}^{*}\sqrt{-1}\partial\overline{\partial}\eta=t\sqrt{-1}\partial\overline{\partial}(\eta\circ\lambda_{t})=\sqrt{-1}\partial\overline{\partial}\eta=p^{*}\omega_{SF},$$

as claimed. Combining (4.7) and (4.8) we get the bound (4.3). ∎