$$L^{2}$$ L 2 harmonic forms on complete special holonomy manifolds

Abstract

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 $$(X,\omega )$$ .