$L^{2}$ 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 $Spin(7)$-) manifold with a $d$(linear) $G_{2}$- (or $Spin(7)$-) structure form $\omega$, the $L^{2}$ harmonic $2$-forms on $X$ will be 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 $(X,\omega)$.