We will focus on $\bar{\alpha}_{1}$. The periodic version of equation (2.7) on $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ is the measure equation

Integrating in the periodic $x$-variable from 0 to 1,

where $\Delta_{a}^{\prime}$ is the Laplacian of the metric $g_{a}^{\prime}=a_{ij}d\mu_{i}d\mu_{j}+Ady^{2}$ on $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{R}$, whose volume form is $d\text{Vol}_{a}^{\prime}=Ad\mu_{1}\wedge d\mu_{2}\wedge dy$.

Now the basic strategy is to build a function satisfying the same measure equation and then compare. For a large positive cutoff $\Lambda$, we calculate the Green representation

If we subtract $\frac{1}{2\sqrt{a_{22}}}\log(2\Lambda)$ and take the limit $\Lambda\to\infty$, we obtain the function

which by construction satisfies the same measure equation as (3.7).

We claim that this function differs from $\bar{\alpha}_{1}$ by a constant. By the Liouville theorem, it suffices to show that the function $\bar{\alpha}_{1}$ on $\mathbb{R}^{2}\times\mathbb{R}$ has the logarithmic growth estimate

which is easy to deduce from Lemma 3.2.

Now to pin down the constant, we can evaluate $\bar{\alpha}_{1}$ for $\mu_{1}=\mu_{2}=0,y\neq 0$. Then the $\arctan$ term drops out, and

Comparing the expressions give the formula for $\bar{\alpha}_{1}$. ∎