We may write

(7.108)

$$\phi_{t,\bf{II}_{-}}\circ(\pi_{\beta}^{\mathcal{N}})^{-1}(\bm{x})-\phi_{-}(\bm{x})=\sum_{\beta^{\prime}:q\in U_{1,\beta^{\prime}}}\chi_{\beta^{\prime}}^{1}(\bm{x})(\phi_{-}\circ\pi_{\beta^{\prime}}^{\mathcal{N}}\circ(\pi_{\beta}^{\mathcal{N}})^{-1}(\bm{x})-\phi_{-}(\bm{x})).$$

Write

(7.109)

$$\pi_{\beta^{\prime}}^{\mathcal{N}}\circ(\pi_{\beta}^{\mathcal{N}})^{-1}(\bm{x})=\bm{x}^{\prime\prime}=(\zeta_{3}^{\prime\prime},u_{i}^{\prime\prime}).$$

Then we write

(7.110)

$$\phi_{-}(\bm{x}^{\prime\prime})-\phi_{-}(\bm{x})=\int_{0}^{1}\langle\nabla\phi_{-}(t\bm{x}^{\prime\prime}+(1-t)\bm{x}),\bm{x}^{\prime\prime}-\bm{x}\rangle dt.$$

Claim: For any $k\geq 0$, there is a $C_{k}>0$ such at for all $\bm{x}\in\bf{II}_{-}$,

(7.111)

$$|\nabla^{k}_{\omega_{T}}\phi_{-}(\bm{x})|\leq e^{C_{k}T}.$$

To see this we notice by definition $\phi_{-}$ satisfies the equation

(7.112)

$$\Delta_{\omega_{T}}\phi_{-}=\Tr_{\omega_{T}}\omega_{T}=n.$$

Then we apply the local weighted Schauder estimate Proposition 4.22, (2). Notice by Corollary 4.11.1 Item (2), given $c\in(0,1/2)$ we have for all $\bm{x}\in\bf{II}_{-}$,

(7.113)

$$r(\bm{x})\geq cT^{-1}\log T.$$

Hence for all $\bm{x}$, every point $\bm{y}$ in the regularity ball $B_{\mathfrak{s}(x)}(\bm{x})$ satisfies

(7.114)

$$r(\bm{y})\geq\frac{c}{2}T^{-1}\log T.$$

So we can apply the Item (2) in Proposition 4.22, and it suffices to show a bound on the $C^{0}$ norm of $\phi_{-}$. By (7.66) it suffices to bound $\log|\zeta_{3}|$. By our definition for $\bm{x}\in\bf{II}_{-}$ we have

(7.115)

$$\log|\zeta_{3}(\bm{x})|\leq C-\log r_{-}\leq C.$$

Also since $z\geq-T$, by Proposition 4.11,

(7.116)

$$\log|\zeta_{3}(\bm{x})|\geq C-\log r_{-}\geq-CT^{2}.$$

So we get

(7.117)

$$|\phi_{-}(\bm{x})|\leq CT^{m},$$

for some $m>0$. This then proves the Claim.

Now it suffices to bound the norm of the vector field $\bm{x}^{\prime\prime}-\bm{x}$ and its convariant derivatives. To this end we divide into two cases.

Case 1: $z\leq-1$. Notice by Lemma 4.9 comparing with the cylindrical metric, we obtain the norm of the tangent vectors $|\bm{x}^{\prime\prime}-\bm{x}|\leq|v_{2}|T^{m}$ for some $m>0$. On the other hand we have $|v_{2}|\leq C|f_{2}|\leq\underline{\epsilon}_{T^{2}}$. So we obtain

(7.118)

$$|\phi_{-}(\bm{x}^{\prime\prime})-\phi_{-}(\bm{x})|=\underline{\epsilon}_{T^{2}}.$$

The higher order derivatives follows similarly by differentiating (7.110) and Lemma 4.9, using the fact that all derivatives of the vector field $\bm{x}^{\prime\prime}-\bm{x}$ in the cylindrical metric is bounded by $C|v_{2}|$.

Case 2. $z\geq-1$. Then we instead compare the metric $\omega_{T}$ with the standard metric

(7.119)

$$\omega_{std}\equiv\sum_{j=1}^{n-1}\sqrt{-1}dw_{j}\wedge d\bar{w}_{j}+\sqrt{-1}d\zeta_{3}\wedge d\bar{\zeta}_{3}.$$

As in the proof of Proposition 4.24 we first notice

(7.120)

$$\Delta_{\omega_{T}}w_{j}=\Delta_{\omega_{T}}\zeta_{3}=\Delta_{\omega_{T}}\zeta_{3}^{-1}=0.$$

By assumption we have $|\zeta_{3}|\leq C$ in this case, and also by Corollary 4.11.1, Item (3) we get $|\zeta_{3}^{-1}|\leq Ce^{CT}$. Then we again apply Schauder estimates Proposition 4.22, Item (2), to get

(7.121)

$$|\nabla^{k}w_{j}|\leq Ce^{C_{k}T},|\nabla^{k}\zeta_{3}|\leq Ce^{C_{k}T}.$$

Hence we get for all $k\geq 0$.

(7.122)

$$|\nabla^{k}_{\omega_{T}}\omega_{std}|_{\omega_{T}}\leq Ce^{C_{k}T}.$$

Now to get a lower bound we use the fact that

(7.123)

$$\omega_{T}^{n}\leq C\Omega_{T}\wedge\bar{\Omega}_{T}\leq C|\zeta_{3}|^{-2}\omega_{std}^{n}.$$

So we get that

(7.124)

$$\omega_{std}\geq Ce^{-CT}\omega_{T}.$$

Now we again can first estimate the norm of $\bm{x}^{\prime\prime}-\bm{x}$ and its derivatives using the standard metric, and use the above information to conclude. ∎