Global perturbation potential function on complete special holonomy manifolds

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

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