Global perturbation potential function on complete special holonomy manifolds

Abstract

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 is a function $f$ on $X$ such that ${\omega }^{\prime }=\omega -{\mathcal{L}}_{\nabla f}\omega$ 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.

arXiv comment22 pages, to appear in The Asian J. Math