The first statement is a simple application of Lemma 4.7, since close to each puncture we have

$$\rho^{k}|\nabla^{k}(h_{\epsilon}-1)|\leq C\epsilon\rho^{-1},$$

where $\rho$ is the distance from the puncture. It follows that away from the punctures $g^{\textup{gh}}_{\epsilon}$ is $C^{k,\alpha}_{loc}$–close to the $\mathbb{Z}_{2}$ quotient of $g_{\mathbb{T}}+\epsilon^{2}\theta^{2}$ for any $k\geq 0$ and $\epsilon$ sufficiently small. ∎