This Section uses weighted Hölder norms to quantify the idea that sufficiently away from the discriminant locus the metric $g^{(1)}$ is approximated by the constant solution.

Given a large number $C_{1}\gg 1$, we consider $M$ over the base region

(2.12)

$$\begin{cases}A^{1/4}|\vec{\mu}|_{a}\geq C_{1},\\ |\vec{\mu}|_{a}\leq 2C_{1}\ell,\end{cases}$$

meaning that the region is far from the origin, and the $g_{a}$-distance to $\mathfrak{D}$ is comparable to the $g_{a}$-distance to the origin. Topologically the base region is obtained by removing the apex from a cone over a thrice-punctured 3-sphere. The $A$-dependence is inserted for convenience when we analyse the scaling behaviours.

The flat model metric is simply constructed by applying the generalised Gibbons-Hawking ansatz to $V^{ij}_{\text{flat}}=a_{ij}$ and $W_{\text{flat}}=A$:

$$g_{\text{flat}}=a_{ij}d\mu_{i}d\mu_{j}+A|d\eta|^{2}+a^{ij}\vartheta_{i}^{\text{flat}}\vartheta_{j}^{\text{flat}},$$

where $\vartheta_{i}^{\text{flat}}$ for $i=1,2$ are flat connections. Likewise we define $\omega_{\text{flat}}$ and $\Omega_{\text{flat}}$. A subtlety is that $g_{\text{flat}}$ cannot model $g^{(1)}$ globally over the region defined by (2.12), because the Chern class of the $T^{2}$-bundle for $g^{(1)}$ evaluates nontrivially on the $S^{2}$ cycles wrapping the 3 puncture points in $S^{3}$, which obstructs the flat connection $\vartheta_{i}^{\text{flat}}$. It is thence understood that we are comparing the model metric with $g^{(1)}$ over a finite number of contractible conical subregions which cover (2.12).

The deviation of $V^{ij}$ from $a_{ij}$ is measured by $\alpha_{1},\alpha_{2},\alpha_{3}$. To estimate these quantities over these regions we introduce some weighted Hölder norms associated to the reference metrics $g_{\text{flat}}$. For any $T^{2}$-invariant tensor field $T$ defined over the region, we define the normalised Hölder seminorm

$$[T]_{\alpha}=\sup_{p}|\vec{\mu}|_{a}^{\alpha}\cdot\sup_{|p-p^{\prime}|_{a}<\frac{1}{10}\ell}\frac{|T(p)-T(p^{\prime})|}{d_{\text{flat}}(p,p^{\prime})^{\alpha}}$$

where we compare $T(p)$ and $T(p^{\prime})$ using parallel transport along minimal geodesics. The weighted norm of $T$ is then defined by

$$\left\lVert T\right\rVert_{C^{k,\alpha}_{\tau^{\prime}}}=A^{-\tau^{\prime}/4}\sum_{j=0}^{k}\left\lVert|\vec{\mu}|_{a}^{-\tau^{\prime}+j}\nabla^{j}T\right\rVert_{L^{\infty}}+A^{-\tau^{\prime}/4}[|\vec{\mu}|_{a}^{-\tau^{\prime}+k}\nabla^{k}T]_{\alpha}.$$

An estimate in this norm is thought as the higher order version of $|T|=O(A^{\tau^{\prime}/4}|\vec{\mu}|_{a}^{\tau^{\prime}})$.

Next we estimate the deviation of $\vartheta_{i}$ from $\vartheta_{i}^{\text{flat}}$. Since gauge equivalent choices of $\vartheta_{i}$ give rise to the same Kähler structure $(g^{(1)},\omega^{(1)},J,\Omega)$ up to holomorphic isometry, we may make any convenient gauge choice. The defining condition on $\vartheta_{i}$ is

$$d\vartheta_{i}=\sqrt{-1}\left(\frac{1}{2}\frac{\partial W_{(1)}}{\partial\mu_{i}}d\eta\wedge d\bar{\eta}+\frac{\partial V^{ij}_{(1)}}{\partial\eta}d\mu_{j}\wedge d\eta-\frac{\partial V^{ij}_{(1)}}{\partial\bar{\eta}}d\mu_{j}\wedge d\bar{\eta}\right),$$

and $d\vartheta_{i}^{\text{flat}}=0$. Thus $\left\lVert d(\vartheta_{i}-\vartheta_{i}^{\text{flat}})\right\rVert_{C^{k,\alpha}_{-2}}\leq CA^{1/2}$ using the higher derivative estimates on $\alpha_{i}$. Using the d-Poincaré lemma, we can find a gauge fixed choice of the 1-form $\vartheta_{i}-\vartheta_{i}^{\text{flat}}$ such that $\left\lVert\vartheta_{i}-\vartheta_{i}^{\text{flat}}\right\rVert_{C^{k,\alpha}_{-1}}\leq CA^{1/4}$. Combining these discussions, and noticing $|d\mu_{i}|\leq CA^{-1/4},|d\eta|\leq CA^{-1/2}$, we obtain