The central analytic difficulty comes from three sources:

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  The metric ansatz behaves very differently in various characteristic regions, and for different Fourier modes. In short, the geometry is multi-scaled.

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  The volume form error becomes larger at large distance, a problem closely related to the incompleteness of the metric.

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  We wish to treat the error estimates with relatively high precision, incorporating features such as exponential decay of higher Fourier modes.

These difficulties require us to introduce some weighted Hölder norms which are more complicated than the ones used in a standard gluing problem. The purpose of this Section is to give precise estimates on the volume form errors of the ansatz, in the complement of a small ball near the origin in $M^{+}$; the small ball itself will be later replaced in our gluing construction by a region in $\mathbb{C}^{3}$ equipped with the Taub-NUT type metric. The task of developing the requisite linear analysis will be deferred to later Sections.

There are 3 useful weight parameters or characteristic length scales:

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  The $g_{a}^{\prime}$-distance to the origin is $\varrho=|(\mu_{1},\mu_{2},y)|_{a}^{\prime}$.

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  The regularity scale is controlled by the parameter $\ell\sim A^{-1/4}+\text{dist}_{g_{a}}(\cdot,\mathfrak{D}).$

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  The parameter $\tilde{\ell}=2\pi A^{-1/2}\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D})$ is useful for measuring the rate of exponential decay of higher Fourier modes.

The key quantity to understand is the volume form error:

$$\tilde{E}^{(1)}=\frac{\tilde{W}_{(1)}}{\det(\tilde{V}^{ij}_{(1)})}-1=\frac{A+\tilde{w}}{A+Aa^{ij}\tilde{v}^{ij}+\det(\tilde{v}^{ij})}-1=-\frac{\det(\tilde{v}^{ij})}{A+\tilde{w}+\det(\tilde{v}^{ij})},$$

where $\det(\tilde{v}^{ij})=\tilde{\alpha}_{1}\tilde{\alpha}_{2}+\tilde{\alpha}_{1}\tilde{\alpha}_{3}+\tilde{\alpha}_{2}\tilde{\alpha}_{3}$ and $\tilde{w}=a_{22}\tilde{\alpha}_{1}+a_{11}\tilde{\alpha}_{2}+(a_{11}+2a_{12}+a_{22})\tilde{\alpha}_{3}$. The weighted Hölder norms will be taylor made for the volume form error. Familiarity with Section 2.2, 2.3 and 2.6 will be assumed.

Let $\delta\leq 0$. We shall define the weighted Hölder norms $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ for $T^{2}$-invariant tensor fields $T$ on $M^{+}\cap\{|\vec{\mu}|_{a}\gtrsim A^{-1/4}\}$, by prescribing the norm on a number of overlapping regions up to uniform equivalence.

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  In the region close to $\mathfrak{D}$ characterised by $\{\ell\lesssim A^{1/2}\}$, the ansatz metric is approximated by $g_{\text{Taub}}$ (cf. Section 2.3), and $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ is uniformly equivalent to the norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ for $g_{\text{Taub}}$ introduced in Section 2.3.

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  The region $\{\ell\gtrsim A^{1/2}\}$ can be covered by subregions of diameter $\sim\ell$, where the $T^{2}$-bundle is topologically trivial. Over each subregion the metric is approximated by the periodic version of the constant solution $g_{\text{flat}}$ (cf. Section 2.2). The $x$-variable defines an $S^{1}$ direction. We decompose $T$ into the part $\bar{T}$ independent of $x$ (the ‘zeroth Fourier mode’) and the oscillatory part $T-\bar{T}$ (the ‘higher Fourier mode’), and define the weighted Hölder norm separately on the two parts.

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  On the zeroth Fourier mode, the norm $\left\lVert\bar{T}\right\rVert_{C^{k,\alpha}_{\delta,0}}$ is equivalent to

  $$A^{-3\delta/4}(\sum_{j=0}^{k}\left\lVert\ell^{j}\nabla^{j}\bar{T}\right\rVert_{L^{\infty}}+[\ell^{k}\nabla^{k}\bar{T}]_{\alpha}),$$

  where $[]_{\alpha}$ denotes the appropriately normalised Hölder seminorm. Here the $\ell$-dependence is inserted to reflect the regularity scale.

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  On the higher Fourier modes we build in the exponential decay. Fix a parameter $0<\kappa<1$. The norm $\left\lVert T-\bar{T}\right\rVert_{C^{k,\alpha}_{\delta,0}}$ in this region is equivalent to

  $$A^{-3\delta/4}\sup_{\ell(p)\gtrsim A^{1/2}}e^{\kappa\tilde{\ell}}(\sum_{j=0}^{k}\left\lVert A^{j/2}\nabla^{j}(T-\bar{T})\right\rVert_{L^{\infty}}+A^{k/2}[\nabla^{k}(T-\bar{T})]_{\alpha}).$$

  An estimate in this norm is the higher order version of $|T-\bar{T}|\leq CA^{3\delta/4}e^{-\kappa\tilde{\ell}}.$

We will also need a variant weighted Hölder norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta}}$. The only difference from $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ is that in the region $\{\ell\gtrsim A^{1/2}\}$ on the zeroth Fourier mode, $\left\lVert\bar{T}\right\rVert_{C^{k,\alpha}_{\delta}}$ is equivalent to

$$A^{-\delta/4}(\sum_{j=0}^{k}\left\lVert\ell^{j-\delta}\nabla^{j}\bar{T}\right\rVert_{L^{\infty}}+[\ell^{k-\delta}\nabla^{k}\bar{T}]_{\alpha}),$$

so an estimate in this norm is the higher order version of $|\bar{T}|=O(A^{3\delta/4}(A^{-1/2}\ell)^{\delta})$. We have inserted an extra decay factor $(A^{-1/2}\ell)^{\delta}$.

The following Lemmas are simple consequences of asymptotes in Section 3.1 and 3.2. The higher order estimates are taken care by $\Delta_{a}$-harmonicity of $\tilde{\alpha}_{i}$.

Finally, multiplication property for the weighted Hölder norms implies