Since $g_{\mathbb{C}^{3}}$ is a Ricci-flat metric, the Bochner formula implies

$$\Delta(\frac{1}{2}|\nabla u|^{2})=|\nabla^{2}u|^{2}+\langle\Delta\nabla u,\nabla u\rangle=|\nabla^{2}u|^{2}+\langle\nabla\Delta u,\nabla u\rangle=|\nabla^{2}u|^{2},$$

so $|du|^{2}$ is a non-negative subharmonic function. The decay condition implies it converges to zero at infinity, so maximum principle gives $du=0$. ∎