Global perturbation potential function on complete special holonomy manifolds

In this article, we introduce and study the notion of a complete special holonomy manifold $(X,\omega)$ which is given by a global perturbation potential function, i.e., there are a function $f$ and a smooth differential $\omega'$ on $X$ such that $\omega=\mathcal{L}_{\nabla f}\omega+\omega'$. We establish some vanishing theorems on the $L^{2}$ harmonic forms under some conditions on the global perturbation potential function.