(Taub-NUT type Calabi-Yau metric on $\mathbb{C}^{3}$) There exists a complete metric $\omega_{\mathbb{C}^{3}}=\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}\phi^{\mathbb{C}^{3}}$ on $\mathbb{C}^{3}$ satisfying $\omega_{\mathbb{C}^{3}}^{3}=\frac{3}{4}\sqrt{-1}\Omega\wedge\overline{\Omega}$, with metric deviation estimate

$$\left\lVert d\phi^{\mathbb{C}^{3}}\right\rVert_{C^{k+1,\alpha}_{-\epsilon,-1+\epsilon}(\mathbb{C}^{3},\Lambda^{1})}\leq CA^{-1/4},\quad\left\lVert\sqrt{-1}\partial\bar{\partial}\phi^{\mathbb{C}^{3}}\right\rVert_{C^{k,\alpha}_{-1-\epsilon,-1+\epsilon}(\mathbb{C}^{3},\Lambda^{1,1})}\leq C.$$

Here $0<\epsilon\ll 1$ is an arbitrarily small given number, and the constants depend only on $k,\alpha,\epsilon$ and the scale invariant uniform ellipticity bound (2.11). This metric inherits all the symmetries of $\omega^{(2)}$.