Thanks for your answer. At last, I has a puzzle about the "$\Gamma$-invariant metric". Is this metric on the base manifold $X$ or on the lifted manifold $\tilde{X}$ ?

Yes， the harmonic forms should be $\Gamma$-invariant.

I want to prove some vanising results by the method of Donnelly-Fefferman Ann.Math.118(1983). But $\bar{\partial}f$ and $\partial{f}$ in the process of calculation are always appearing. In my question, I only suppose $f$ has a bounded. I has no way to bounded $\bar{\partial}f$ and $\partial{f}$ by $f$.

In my question, I suppose that $f$ is bounded and $\omega$ is induced by a complete metric.

Through pull back the bundle $E$ over CY-3 fold to $\pi^{\ast}E$ over the $G_{2}$ manifold $CY^{3}\times S^{1}$. I had proved this assumed, but I can't belive my result.

Yes,your are right, $E$ is a principal $G$-bundle.But I can't understand why this has no natural Chern classes?