(The $C^{2,\alpha}$ estimate.) If $U\subseteq B$ is a simply connected open set with $\overline{U}\subseteq B_{0}=B\setminus\Delta$, then there exists constants $\alpha$ and $\epsilon_{0}$ and a polynomial $P$, depending on $J$ and $U$, such that

$$\|u_{\epsilon}\|_{C^{2,\alpha}}\leq P(\epsilon^{-1})$$

in $f^{-1}(U)$ for all $\epsilon<\epsilon_{0}$ and ${\bf B}$ in a fundamental domain for the B-field (see Remark 4.5). Here the $C^{2,\alpha}$ norm is on $f^{-1}(U)$ as defined in Lemma 4.1, and so $\alpha$, $\epsilon_{0}$ and $P$ also depend on the choice of holomorphic coordinate $y$ and fixed bounded domain $T^{\prime}$, as specified in the proof below.