Asian Journal of Mathematics

Volume 25 (2021)

Number 3

Global perturbation potential function on complete special holonomy manifolds

Pages: 393 – 412



Teng Huang (School of Mathematical Sciences and the CAS Key Laboratory of Wu Wen-Tsun Mathematics, University of Science and Technology of China, Hefei, Anhui China)


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.


special holonomy manifolds, $L^2$-harmonic forms, global perturbation potential function

2010 Mathematics Subject Classification

53C29, 53C38, 57R57

This work was supported by Nature Science Foundation of China No. 11801539, and by Postdoctoral Science Foundation of China No. 2017M621998, No. 2018T110616.

Received 19 May 2020

Accepted 9 September 2020

Published 14 March 2022