For each sufficiently large $T$, there exists an $S^{1}$-invariant Calabi-Yau metric $\omega_{T,CY}=\omega_{T}+\sqrt{-1}\partial\bar{\partial}\phi$ for $\phi\in\mathfrak{S}_{1}$, such that

(6.28)

$$\|\sqrt{-1}\partial\bar{\partial}\phi\|_{\mathfrak{S}_{1}}\leq C_{0}\cdot T^{\nu+\alpha},$$

where $C_{0}>0$ is a uniform constant independent of $T\gg 1$ and the weighted Hölder norm of $\mathfrak{S}_{1}$ is defined in (6.8) for parameters $\nu$, $\alpha$, $\delta$ and $\mu$ satisfying (6.10), (6.11), (6.12) and (6.13).