All three types of metric ansatzs are constructed in the generalised Gibbons-Hawking framework, by perturbing from the constant solution after incorporating topology. The constant solutions in Example 1.6 serve as zeroth order approximations to the metric ansatz, and can be thought as scaling limits (cf. Section 1.3.3). Geometrically they describe a flat torus fibration fibred over a Euclidean base with distinguished coordinates related to moment maps. The choice of this Euclidean metric is parametrised by a positive definite rank 2 real symmetric or Hermitian matrix, depending on the 3 cases. The principal difference between the Taub-NUT type metric on $\mathbb{C}^{3}$ and the Ooguri-Vafa type metrics is that the bases in the latter cases have periodic directions.

In order to build in the Gross-Ruan-Joyce topology (cf. Section 1.1) we need to make first order corrections to the constant solutions. Recall the generalised Gibbons-Hawking construction involves three sets of equations:

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  The integrability condition is responsible for the integrability of the complex structure and the Kähler condition.

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  The distributional equation captures the topology and the discriminant locus.

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  The Calabi-Yau condition is the only nonlinear equation.

It is natural to impose that the first order corrections satisfy the linearised version of these equations; in particular the linearisation of the Calabi-Yau condition gives rise to harmonic functions. These linearised equations combine into a coupled overdetermined system. Our method to solve this system is to first determine by educated guess the singularities of the harmonic functions along the discriminant locus, explicitly construct such harmonic functions using Green’s representation, and then verify the other equations in the overdetermined system by means of Liouville theorem type arguments. The first order corrections we obtain are canonical (up to constants) under mild growth constraints. For the Taub-NUT type $\mathbb{C}^{3}$ case the first order corrections admit elementary formulae. For the Ooguri-Vafa type metrics on the vertices the first order corrections involve infinite series and Green representation integrals, which are a priori divergent but become convergent after subtracting logarithmically divergent terms, much like what happens already for the Ooguri-Vafa metric.

We then extract various asymptotes of the first order ansatz. Transverse to the discriminant locus, the leading asymptotes can be interpreted geometrically as giving rise to Taub-NUT metrics; ultimately this is forced on us by the distributional equation coming from the topology. In the Ooguri-Vafa type situations, we can also perform Fourier analysis in the periodic variables. Suitably away from the discriminant locus, the zeroth Fourier mode is the dominant contribution, giving rise to a semiflat metric. The harmonicity condition implies that the higher Fourier modes satisfy Helmholtz equations, thereby decay exponentially. An additional problem in the Ooguri-Vafa type situations is that the metric ansatzs are only positive definite on a bounded region, whereby metrically incomplete.

The strategy to identify the holomorphic structure is to produce holomorphic differentials with integral periods, in a manner similar to the Taub-NUT metric and the Ooguri-Vafa metric (cf. Section 1.3.2). The functional equation satisfied by the holomorphic functions allows us to identify the holomorphic volume form. It should be emphasized that while topology is built a priori into the generalised Gibbons-Hawking construction, the holomorphic structure is a nontrivial a posteriori consequence.

The first order corrections are small perturbations suitably away from the discriminant locus, but near the discriminant locus they are large compared to the constant solution. This explains why the first order metric ansatz is approximately Calabi-Yau suitably away from the discriminant locus. In the suitable weighted Hölder norms this approximation continues to hold good near the discriminant locus, except on small balls near the origin in the Taub-NUT type $\mathbb{C}^{3}$ case and the positive vertex case. Geometrically this problem is caused by the 3 edges of $\mathfrak{D}$ interacting strongly at their intersection point. The same problem does not appear on the negative vertex because the discriminant locus $S$ has no singular point.

Our strategy trifurcates at this point. The small ball is a fully nonlinear region in which linear approximation methods fail completely. In the case of the Taub-NUT type metric on $\mathbb{C}^{3}$, we instead shift to the complex geometric perspective, and solve the complex Monge-Ampère equation with prescribed asymptotes at infinity. This is viable because the exterior of the small ball does admit an approximately Calabi-Yau ansatz. The output is a Calabi-Yau metric on $\mathbb{C}^{3}$ whose deviation from the first order ansatz satisfies an asymptotically good estimate.

The Ooguri-Vafa type metric on the positive vertex is best thought as the periodic version of the Taub-NUT type metric on $\mathbb{C}^{3}$, and is obtained by gluing the Taub-NUT type $\mathbb{C}^{3}$ to the first order ansatz on the positive vertex. The periodicity condition breaks down the scaling symmetry of the Taub-NUT type metrics, and instead results in the gluing picture, exactly analogous to the relation between the Taub-NUT metric and the usual Ooguri-Vafa metric. The nonlinear effect on the positive vertex is already fully present on the Taub-NUT type $\mathbb{C}^{3}$. It is worth comparing with the topological prediction of Gross-Ruan (cf. Section 1.1.3) where the neighbourhood of the origin is modelled on $\mathbb{C}^{3}$ with a $T^{2}$ fibration related to the Harvey-Lawson example 1.7. But for metric purposes we need to use an exotic Calabi-Yau metric on $\mathbb{C}^{3}$, rather than the Euclidean $\mathbb{C}^{3}$.

The Ooguri-Vafa type metric on the negative vertex, on the other hand, is constructed entirely perturbatively from the first order ansatz.