Since by construction

(4.338)

$$T^{\frac{n-2}{n}}\omega=T\pi^{*}\omega_{D}+dd^{c}\phi.$$

We have

(4.339)

$$\Delta_{T^{\frac{2n-2}{n}}\omega}\phi=n-T\cdot\Tr_{T^{\frac{2n-2}{n}}\omega}\pi^{*}\omega_{D}.$$

Since $\pi^{*}\omega_{D}$ is smooth on $\mathcal{V}$ and $\omega$ is parallel, again the above discussion gives that

(4.340)

$$|\Tr_{T^{\frac{2n-2}{n}}\omega}\pi^{*}\omega_{D}|_{C^{1,\alpha}_{\delta,\nu,\mu}(\mathcal{V})}\leq e^{CT}.$$

So by Proposition 4.22 we get that

(4.341)

$$|\phi|_{C^{3,\alpha}_{\delta,\nu,\mu}(\mathcal{V})}\leq e^{CT}+C|\phi|_{C^{0}_{\delta,\nu,\mu}(\pi^{-1}(U_{\beta})\cap\{|z|\leq 1\})}.$$

To bound the right hand side we use the formula

(4.342)

$$\phi=\int_{T_{+}}^{z}uh(u)du+\phi(T_{+})=\int_{T_{+}}^{0}uh(u)du+\phi(T_{+})+\int_{0}^{z}uh(u)du.$$

Hence

(4.343)

$$\phi=\frac{T}{2}z^{2}+\frac{1}{2}r+B_{T}+O(1),$$

which gives

(4.344)

$$|\phi|_{C^{0}_{\delta,\nu,\mu}(\pi^{-1}(U_{\beta})\cap\{|z|\leq 1\})}\leq O(T^{m})$$

for some $m>0$. The conclusion then follows. ∎