Given a convex domain $R$ in $\mathbb{R}^{k}\times\mathbb{R}^{l}$ and a $(\sigma,\tau)$-type solution $V,W$ of the split Monge-Ampère equation in $R$ there is a real one-parameter family of $(\sigma,\tau)$-solutions $V_{\lambda},W_{\lambda}$ to the Gibbons-Hawking ansatz in $\lambda R\times(S^{1})^{l}$ such that

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  The diameter of the circles both in the fiber $T^{n}$ and in the torus part $(S^{1})^{l}$ of the base away from the discriminant is roughly given by $\lambda^{-1}$.

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  The zero Fourier modes of the GH solutions $V_{\lambda}^{0}(u,x),W_{\lambda}^{0}(u,x)$ as functions of the rescaled variables $s,t$, where $u=\lambda s,x=\lambda t$, will converge (in some properly weighted norm on the function space) to $V(s,t),W(s,t)$ as $\lambda\to\infty$.

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  The higher Fourier modes decay exponentially away from the discriminant $\Pi(\tau)\times\Pi(\sigma)$ in $\lambda R$, uniformly in $\lambda$. That is, if $\beta(u,x)$ denotes the Euclidean distance from the point $(u,x)$ to the discriminant, then

  $$|V_{\lambda}^{m}(u,x)|\leq C_{1}e^{-\beta(u,x)|m|},\quad|W_{\lambda}^{m}(u,x)|\leq C_{2}e^{-\beta(u,x)|m|},$$

  for some constants $C_{1},C_{2}$, and large enough $\lambda$ and $\beta$.