(Ooguri-Vafa type metric on the negative vertex, cf. Chapter 4) There is a family of incomplete Calabi-Yau metrics with $S^{1}$-symmetry, which are parametrised by rank 2 Hermitian matrices $(a_{p\bar{q}})$, such that

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  The ambient space is topologically a singular $S^{1}$-bundle over a 5-dimensional base contained in $(\mathbb{C}^{*})^{2}_{z_{1},z_{2}}\times\mathbb{R}_{\mu}$, with discriminant locus along

  $$S=\{z_{1}+z_{2}=1,\mu=0\}\subset(\mathbb{C}^{*})^{2}_{z_{1},z_{2}}\times\mathbb{R}_{\mu}.$$

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  The holomorphic structure together with the holomorphic volume form agrees with the Zharkov prediction (cf. Section 1.1.6).

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  Suitably away from $S$ there is a $T^{3}$-fibration structure such that these Calabi-Yau metrics decay exponentially to semiflat metrics.

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  These metrics extend over an exponentially large region, under the unit homological volume normalisation on $T^{3}$.

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  Metric behaviour transverse to $S$ is modelled on a fibration by Taub-NUT metrics. Metric behaviour suitably away from $S$ is approximately a flat $S^{1}$-bundle over an open subset of $(\mathbb{C}^{*})^{2}_{z_{1},z_{2}}\times\mathbb{R}_{\mu}$ with a Euclidean metric.