We now study the metric near the discriminant locus but sufficiently far from the origin, which turns out to be locally modelled on a fibration by Taub-NUT metrics over a flat cylinder. A subtlety is that the smooth topology along $\mathfrak{D}_{i}$ is not a priori prescribed, and needs to be elucidated first.

We focus on the neighbourhood of $\mathfrak{D}_{1}$ far from the origin, where $\alpha_{1}\sim\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}$ and $\alpha_{2},\alpha_{3}$ are smooth. To leading order

$$V_{(1)}\sim V_{\text{Taub}}=\begin{bmatrix}\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix},W_{(1)}\sim W_{\text{Taub}}=A+\frac{a_{22}}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}.$$

The inverse matrix is

$$V_{(1)}^{-1}\sim V_{\text{Taub}}^{-1}=({A+\frac{a_{22}}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}})^{-1}\begin{bmatrix}a_{22}&-a_{21}\\ -a_{12}&\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+a_{11}\end{bmatrix}.$$

Now if we apply the generalised Gibbons-Hawking ansatz to $V_{\text{Taub}}$ and $W_{\text{Taub}}$, we obtain a model metric

$$g_{\text{Taub}}=V^{ij}_{\text{Taub}}d\mu_{i}d\mu_{j}+W_{\text{Taub}}|d\eta|^{2}+(V^{-1}_{\text{Taub}})^{ij}\vartheta_{i}^{\prime}\vartheta_{j}^{\prime}$$

where $\vartheta_{1}^{\prime},\vartheta_{2}^{\prime}$ are the connections. As $d\vartheta_{2}^{\prime}=0$ we may write $\vartheta_{2}^{\prime}=d\theta_{2}^{\prime}$. Rewriting the model metric,

(2.13)

$$\begin{split}g_{\text{Taub}}=(\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+\frac{A}{a_{22}})(d\mu_{1}^{2}+a_{22}|d\eta|^{2})+a_{22}(d(\mu_{2}+\frac{a_{12}}{a_{22}}\mu_{1}))^{2}\\ +(\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+\frac{A}{a_{22}})^{-1}(\vartheta_{1}^{\prime}-\frac{a_{12}}{a_{22}}d\theta_{2}^{\prime})^{2}+\frac{1}{a_{22}}(d\theta_{2}^{\prime})^{2}.\end{split}$$

Notice the dual basis for $\{\vartheta_{1}^{\prime}-\frac{a_{12}}{a_{22}}d\theta_{2}^{\prime},\vartheta_{2}^{\prime}\}$ is given by $\{\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}}+\frac{a_{12}}{a_{22}}\frac{\partial}{\partial\theta_{1}}\}$, which corresponds to the moment coordinates $\mu_{1}$ and $\mu_{2}+\frac{a_{12}}{a_{22}}\mu_{1}$.

The variables $\mu_{2}+\frac{a_{12}}{a_{22}}\mu_{1}$ and $\theta_{2}^{\prime}$ define a cylinder $\mathbb{R}\times S^{1}$. Translations in these variables are isometries of the model space. The model space fibres over this cylinder, and restricted to each fibre the metric is recognized as the Taub-NUT metric with parameter $\frac{A}{a_{22}}$. The fibration is not always a metric product, because for the generators $\frac{\partial}{\partial\theta_{1}}$ and $\frac{\partial}{\partial\theta_{2}}+\frac{a_{12}}{a_{22}}\frac{\partial}{\partial\theta_{1}}$ to give rise to an integral basis of $H_{1}(T^{2})$ we need $\frac{a_{12}}{a_{22}}$ to be an integer. On the universal cover the metric becomes the product of Taub-NUT metric with the flat $\mathbb{R}^{2}$, as $\theta_{2}$ becomes a real variable instead of a circle variable. In particular the universal cover is topologically $\mathbb{C}^{2}\times\mathbb{R}^{2}$, and the model space is a discrete $\mathbb{Z}$-quotient of $\mathbb{C}^{2}\times\mathbb{R}^{2}$, so inherits a smooth topology. The Riemannian curvature on the model metric is bounded but does not decay as we move to infinity along $\mathfrak{D}_{1}$.

Our next goal is to quantify the idea that the model $g_{\text{Taub}}$ is a good approximation to the metric ansatz $g^{(1)}$. We view both metrics as defined on the same smooth manifold, fibred over the region

(2.14)

$$|\vec{\mu}|_{a}\geq C_{1}A^{-1/4},\quad|\vec{\mu}|_{a}\geq C_{1}\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})$$

where $C_{1}$ is a large number as in Section 2.2. In this region the $g_{a}$-distance to $\mathfrak{D}_{2},\mathfrak{D}_{3}$ are both $O(|\vec{\mu}|_{a})$, and $\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})$ is comparable to $A^{1/4}\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}$, so

$$\ell=A^{-1/4}+\text{dist}_{g_{a}}(\cdot,\mathfrak{D})\sim A^{-1/4}+A^{1/4}\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}.$$

We introduce some weighted Hölder norms associated to the reference metric $g_{\text{Taub}}$. The regularity scale of $g_{\text{Taub}}$ is comparable to $\ell$. For any $T^{2}$-invariant tensor field $T$ over the region (2.14) , define the normalised Hölder seminorm

$$[T]_{\alpha}=\sup_{p}\ell(p)^{\alpha}\cdot\sup_{p^{\prime}\in B_{g_{\text{Taub}}(p,\ell/10)}}\frac{|T(p)-T(p^{\prime})|}{d_{\text{Taub}}(p,p^{\prime})^{\alpha}}$$

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

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

An estimate in this norm can be thought as the higher order version of $|T|=O(A^{\tau/4+\delta/4}\ell^{\delta}|\vec{\mu}|_{a}^{\tau})$. Similar weighted Hölder norms are defined in the neighbourhood of $\mathfrak{D}_{2}$ and $\mathfrak{D}_{3}$.

The deviation between $V^{ij}_{\text{Taub}}$ and $V^{ij}_{(1)}$ near $\mathfrak{D}_{1}$ is measured by the functions $\alpha_{1}-\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}$, $\alpha_{2}$ and $\alpha_{3}$.

Next we analyse the deviation between the connections $\vartheta_{i}$ and $\vartheta_{i}^{\prime}$ for $i=1,2$, corresponding to the ansatz $g^{(1)}$ and the model $g_{\text{Taub}}$ respectively. This involves the same gauge fixing issue as in Section 2.2. The defining condition on $\vartheta_{i}$ is

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

and similarly for $\vartheta_{i}^{\prime}$. Thus $\left\lVert d(\vartheta_{i}-\vartheta_{i}^{\prime})\right\rVert_{C^{0,-2}}\leq CA^{1/2}$ using the higher derivative estimates on $V^{ij}_{(1)}-V^{ij}_{\text{Taub}}$ etc. Using the d-Poincaré lemma, we can find a gauge fixed choice of the smooth 1-form $\vartheta_{i}-\vartheta_{i}^{\prime}$ such that $\left\lVert\vartheta_{i}-\vartheta_{i}^{\prime}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq CA^{1/4}.$ Combining the above, and recalling $|d\mu_{i}|\leq CA^{-1/4}$, $|d\eta|\leq CA^{-1/2}$, we obtain