There exists $0<\rho_{0}<\tfrac{1}{4}\textrm{inj}\,g_{\mathbb{T}}$ such that the balls $B_{2\rho_{0}}(q_{j})$, $j=1,\dots,8$, and $B_{2\rho_{0}}(\pm p_{i})$, $i=1,\dots,n$, in $\mathbb{T}$ are all disjoint and such that the following holds.

1. (i)

   For $j=1,\dots,8$ there exists $\lambda_{j}\in\mathbb{R}$ such that in $B_{2\rho_{0}}(q_{j})$

   $$h_{\epsilon}=(1+\epsilon\lambda_{j})+\frac{\epsilon(m_{j}-2)}{\rho_{j}}+O(\epsilon\,\rho_{j}^{2}).$$

2. (ii)

   For each $i=1,\dots,n$ there exists $\lambda_{i}\in\mathbb{R}$ and a linear function $\ell_{i}$ on $\mathbb{R}^{3}$ with $|\ell_{i}|\leq C\rho$ such that in $B_{2\rho_{0}}(\pm p_{i})$

   $$h_{\epsilon}=(1+\epsilon\lambda_{i})+\frac{\epsilon k_{i}}{2\rho_{i}}+\epsilon\,\ell_{i}+O(\epsilon\,\rho_{i}^{2}).$$

Moreover, $\rho_{0},\lambda_{j},\lambda_{i},\ell_{i}$ depend continuously on the position of the punctures $p_{1},\dots,p_{n}$ and on the flat metric $g_{\mathbb{T}}$ and $f=O(\epsilon\,\rho^{3})$ means that there exists a constant $C$ depending continuously on these data such that $|\nabla^{k}f|\leq C\epsilon\rho^{3-k}$ for $k=0,1,2,3$.