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-L∇ fω is sufficiently small in L∞-norm. We establish some vanishing theorems on the L² harmonic forms under some conditions on the global perturbation potential function.