The Kähler structure $(g^{(1)},\omega^{(1)},J,\Omega)$ extends smoothly over the region (2.14). The deviation from the model metric admits the estimates

$$\begin{cases}\left\lVert g^{(1)}-g_{\text{Taub}}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq C,\quad&\left\lVert\omega^{(1)}-\omega_{\text{Taub}}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq C,\\ \left\lVert J-J_{\text{Taub}}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq C,\quad&\left\lVert\Omega-\Omega_{\text{Taub}}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq C.\end{cases}$$

In particular, if $C_{1}$ is chosen large enough, then the magnitudes of the deviation

$$|g^{(1)}-g_{\text{Taub}}|\ll 1,\quad|\omega^{(1)}-\omega_{\text{Taub}}|\ll 1,\quad|\Omega-\Omega_{\text{Taub}}|\ll 1,\quad|J-J_{\text{Taub}}|\ll 1.$$

The volume form error function $E^{(1)}$ satisfies

$$\left\lVert E^{(1)}\right\rVert_{C^{k,\alpha}_{-1,-1}}\leq C.$$