# Global perturbation potential function on complete special holonomy manifolds

## TL;DR

This paper introduces a new class of complete special holonomy manifolds characterized by a global perturbation potential function and proves vanishing theorems for $L^2$ harmonic forms under certain conditions.

## Contribution

It defines the concept of complete special holonomy manifolds via a global perturbation potential function and establishes vanishing theorems for harmonic forms in this context.

## Key findings

- Vanishing theorems for $L^2$ harmonic forms.
- Characterization of special holonomy manifolds with perturbation potentials.

## 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 $\omega'=\omega-\mathcal{L}_{\nabla f}\omega$ 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.