For any simply connected open set $U\subseteq B_{0}$ with $\overline{U}\subseteq B_{0}$, and any $k\geq 2$, $0<\alpha<1$, there exists constants $C,C^{\prime}$, and $\epsilon_{0}$ such that for all choices of ${\bf B}$ in a fundamental domain for the B-field and any $\epsilon<\epsilon_{0}$,

$$\|u_{\epsilon}\|_{C^{k,\alpha}}\leq Ce^{-C^{\prime}/\epsilon}.$$

Here, the norm is as in Lemma 4.1 on the region $f^{-1}(U)$, and the constants $C,C^{\prime}$ are independent of $\epsilon$.