Annals of Global Analysis and Geometry

, Volume 56, Issue 1, pp 17–36 | Cite as

\(L^{2}\) harmonic forms on complete special holonomy manifolds

  • Teng HuangEmail author


In this article, we consider \(L^{2}\) harmonic forms on a complete non-compact Riemannian manifold X with a nonzero parallel form \(\omega \). The main result is that if \((X,\omega )\) is a complete \(G_{2}\)- (or \(\textit{Spin}(7)\)-) manifold with a d(linear) \(G_{2}\)- (or \(\textit{Spin}(7)\)-) structure form \(\omega \), then the \(L^{2}\) harmonic 2-forms on X vanish. As an application, we prove that the instanton equation with square-integrable curvature on \((X,\omega )\) only has trivial solution. We would also consider the Hodge theory on the principal G-bundle E over \((X,\omega )\).


\(L^{2}\) harmonic form \(G_{2}\hbox {- } (\textit{Spin}(7)\hbox {-})\)manifold d(linear)-form Gauge theory 



I would like to thank the anonymous referee for careful reading of my manuscript and helpful comments. I would like to thank Professor Verbitsky for kind comments regarding his article [36]. Also I would like to thank Yuguo Qin for further discussions about this work. This work is supported by Nature Science Foundation of China No. 11801539 and Postdoctoral Science Foundation of China No. 2017M621998, No. 2018T110616.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesSoochow UniversitySuzhouPeople’s Republic of China
  2. 2.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China

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