This is the periodic version of the original Gibbons-Hawking ansatz [GH78],[Haw77]. We consider the case when $k=l=1$ and both simplices $\tau$ and $\sigma$ are of length 1, although the construction works for a non-unimodular case as well.

The Gibbons-Hawking equation in this case is equivalent to the Laplace equation for $V(u,x,y)$ ($=W(u,x,y)$) on a domain in the cylinder $\mathbb{R}\times\mathbb{R}\times S^{1}$ with the Dirac $\delta$-function on the right hand side. We can write both the solution $V(u,x,y)$ and the $\delta$-function in the Fourier expansion:

$$V(u,x,y)=\sum_{m\in\mathbb{Z}}V_{m}e^{2\pi imy},\quad\gamma^{j}_{\tau}(u)\delta_{P_{\sigma}}(\eta)=-\delta(u,x,y)=-\sum_{m\in\mathbb{Z}}\delta(u,x)e^{2\pi imy}.$$

Here the minus sign takes into account the orientation of the circle action when passing from currents to generalized functions.

Being linear, the Gibbons-Hawking equation

$$\frac{\partial^{2}V_{\lambda}}{\partial u^{2}}+\frac{\partial^{2}V_{\lambda}}{\partial x^{2}}+\frac{\partial^{2}V_{\lambda}}{\partial y^{2}}=-2\pi\cdot\delta(u,x,y)$$

will decompose into the Helmholtz equations according to the Fourier modes:

$$\frac{\partial^{2}V_{\lambda}^{m}}{\partial u^{2}}+\frac{\partial^{2}V_{\lambda}^{m}}{\partial x^{2}}-(2\pi m)^{2}V_{\lambda}^{m}=-2\pi\cdot\delta(u,x),\quad m\in\mathbb{Z}.$$

On the other hand, the $(\sigma,\tau)$-type split Monge-Ampère equation in the rescaled coordinates $s=\lambda^{-1}u,t=\lambda^{-1}u$ is the two-dimensional Laplace equation:

$$\frac{\partial^{2}V}{\partial s^{2}}+\frac{\partial^{2}V}{\partial t^{2}}=-2\pi\cdot\delta(s,t),$$

whose fundamental solutions are in the form $V(s,t)=-\frac{1}{2}\log|s^{2}+t^{2}|+h(s,t)$, for a harmonic function $h$. Thus, one can take the zero mode of the corresponding Gibbons-Hawking solution to be $V_{\lambda}^{0}(u,x)=V(\lambda^{-1}u,\lambda^{-1}x)$, as long as $V(s,t)$ stays positive on $R$. As for the higher modes, it is known that a fundamental solution to the Helmholtz equation with $m\neq 0$ may be given by the Bessel function

$$V_{\lambda}^{m}=K_{0}(2\pi|m|r)\sim\frac{1}{\sqrt{4|m|r}}\,e^{-{2\pi|m|r}}\left(1+O(r^{-1})\right),\text{ where }r^{2}=u^{2}+x^{2},$$

which decays exponentially as required.