(Ooguri-Vafa type metric on the positive vertex, cf. Chapter 3) There is a family of incomplete Calabi-Yau metrics with $T^{2}$-symmetry, which are parametrised by positive definite rank 2 real symmetric matrices $(a_{ij})$, such that

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  The ambient space has the same topology as the positive vertex predicted by Gross-Ruan-Joyce, namely it is a singular $T^{2}$-bundle over a 4-dimensional base contained in $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ with discriminant locus along

  $$\mathfrak{D}=\{\mu_{2}\geq 0,\mu_{1}=0,\eta=0\}\cup\{\mu_{1}\geq 0,\mu_{2}=0,\eta=0\}\cup\{\mu_{1}=\mu_{2}\leq 0,\eta=0\}.$$

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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 $\mathfrak{D}$ there is a $T^{3}$-fibration structure such that the 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 near the origin is modelled on the Taub-NUT type metrics on $\mathbb{C}^{3}$ mentioned above. Metric behaviour transverse to $\mathfrak{D}$ but suitably away from the origin is modelled on a fibration by Taub-NUT metrics. Metric behaviour suitably away from $\mathfrak{D}$ is approximately a flat $T^{2}$-bundle over an open subset of $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ with a Euclidean metric.

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  These Calabi-Yau metrics admit special Lagrangian $T^{3}$-fibrations.